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Iterative Observer-based Estimation Algorithms for
Steady-State Elliptic Partial Differential Equation Systems
Dissertation by
Muhammad Usman Majeed
In Partial Fulfillment of the Requirements
For the Degree of
Doctor of Philosophy
King Abdullah University of Science and Technology
Thuwal, Kingdom of Saudi Arabia
May, 2017
2
EXAMINATION COMMITTEE PAGE
The dissertation of Muhammad Usman Majeed is approved by the examination com-
mittee
Committee Chairperson: Taous Meriem Laleg-Kirati - Associate Professor
Committee Members: Ralph C. Smith - Professor, Jeff S. Shamma - Professor, David
E. Keyes - Professor, Ying Wu - Associate Professor
3
©May, 2017
Muhammad Usman Majeed
All Rights Reserved
4
ABSTRACT
Iterative Observer-based Estimation Algorithms for Steady-State
Elliptic Partial Differential Equation Systems
Muhammad Usman Majeed
Steady-state elliptic partial differential equations (PDEs) are frequently used to
model a diverse range of physical phenomena. The source and boundary data estima-
tion problems for such PDE systems are of prime interest in various engineering dis-
ciplines including biomedical engineering, mechanics of materials and earth sciences.
Almost all existing solution strategies for such problems can be broadly classified
as optimization-based techniques, which are computationally heavy especially when
the problems are formulated on higher dimensional space domains. However, in this
dissertation, feedback based state estimation algorithms, known as state observers,
are developed to solve such steady-state problems using one of the space variables
as time-like. In this regard, first, an iterative observer algorithm is developed that
sweeps over regular-shaped domains and solves boundary estimation problems for
steady-state Laplace equation. It is well-known that source and boundary estimation
problems for the elliptic PDEs are highly sensitive to noise in the data. For this, an
optimal iterative observer algorithm, which is a robust counterpart of the iterative
observer, is presented to tackle the ill-posedness due to noise. The iterative observer
algorithm and the optimal iterative algorithm are then used to solve source localiza-
tion and estimation problems for Poisson equation for noise-free and noisy data cases
respectively. Next, a divide and conquer approach is developed for three-dimensional
domains with two congruent parallel surfaces to solve the boundary and the source
data estimation problems for the steady-state Laplace and Poisson kind of systems
5
respectively. Theoretical results are shown using a functional analysis framework,
and consistent numerical simulation results are presented for several test cases using
finite difference discretization schemes.
6
ACKNOWLEDGEMENTS
This thesis document not only represents my research work but is a milestone
that is achieved after several years of dedicated work at Estimation Modeling and
Analysis Group (EMANG) at King Abdullah University of Science and Technology
(KAUST). Ever since I arrived at KAUST, I have felt it like my home. I have come
across several amazing people and I have had countless memorable experiences, some
of those I would love to recall in the following.
First and foremost, I would like to thank my research supervisor Prof. Dr. Taous
Meriem Laleg-Kirati for her consistent support and encouragement during my Ph.D.
work. Her guidance helped me in all the time of research and writing of this thesis.
The joy and enthusiasm that she has for research were contagious and motivational.
I could not have imagined having a better advisor and mentor for my Ph.D. studies.
Members of EMANG have contributed a lot to my personal and professional
growth. I am highly indebted to all of my colleagues here who bore me with patience
and helped me learn and grow over the years. I shall always cherish the fruitful time
spent with Fadi Eleiwi and Ayman Karam in our office desk space. I also like to thank
Abeer Aldoghaither, Sharefa Asiri, Zehor Belkhatir and Shahrazed Elmetennani for
very useful academic discussions and for their consistent feedback on my research. I
would like to acknowledge EMANG postdocs, Dayan Liu, Chadia Zayane, Ibrahima
N’Doye and Sarah Mechhoud for their benevolent help and encouragement during
rough times in my Ph.D. I am also grateful to a number of renowned researchers who
visited EMANG and provided valuable feedback on my research. I would certainly
like to thank my friends at KAUST particularly Tamour Javed, Bilal Janjua, Furrukh
Sana and Awad Alquaity, who always helped and motivated me along the ebbs and
flows of the journey and without whom my KAUST story would be incomplete.
7
Many thanks to Prof. Rabia Djellouli at Department of Mathematics, California
State University Northridge (CSUN) for giving me an opportunity to work in his group
for about two months during the summer of 2012. His rigorous supervision helped
me develop a strong interest in my area of research. I am thankful to Prof. Miroslav
Krstic from University of California San Diego (UCSD) for fruitful discussions during
his couple of visits at KAUST and for hosting me for a talk in his research group at
UCSD. I am also indebted to Prof. Alexandre Bayen from University of California
Berkeley (UC Berkeley) to host me for a couple of months as a visiting scholar to
work on some of the most interesting and challenging problems in control and partial
differential equations. I am also thankful to Prof. Ralph Smith from North Carolina
State University (NCSU) for taking out time to review my thesis work and to provide
constructive feedback.
A very heartily thanks to my whole family. Words cannot express how grateful I
am to my parents and my late uncles. They have raised me with love and discipline
and supported me in all my pursuits. I am also thankful to my brothers and only sister
who guided me through years and left no stone unturned to help me succeed. And
my loving wife Saira, whose help and support during my Ph.D. is highly appreciated.
Thank you all.
8
TABLE OF CONTENTS
Examination Committee Page 2
Copyright 3
Abstract 4
Acknowledgements 6
List of Abbreviations 12
List of Symbols 13
List of Figures 15
1 Introduction 20
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3 Proposed Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 State Observer Theory and Inverse Problems: Two Worlds Apart 26
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Observation Theory for Dynamical Systems . . . . . . . . . . . . . . 26
2.2.1 Finite-Dimensional Linear Time-Invariant Systems . . . . . . 27
2.2.2 Infinite-Dimensional Linear Partial Differential Equation (PDE)
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1 Forward and Inverse Problems . . . . . . . . . . . . . . . . . . 37
2.3.2 Ill-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.3 Some Solution Strategies for the Inverse Problems . . . . . . . 39
2.3.4 Ill-posedness of Boundary Data Estimation Problems for Laplace
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
9
3 Iterative Observers for Boundary Estimation Problems for Laplace
Equation 44
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Iterative Observer Design for Boundary Estimation . . . . . . . . . . 45
3.2.1 Problem Statment on a Rectangular Domain . . . . . . . . . . 45
3.2.2 Notations and Definitions . . . . . . . . . . . . . . . . . . . . 46
3.2.3 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.4 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.5 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.6 Numerical Implementation . . . . . . . . . . . . . . . . . . . . 62
3.2.7 Results and Simulations . . . . . . . . . . . . . . . . . . . . . 66
3.3 Robust Iterative Algorithm for Boundary Estimation . . . . . . . . . 71
3.3.1 Problem Statement on an Annulus Domain . . . . . . . . . . . 71
3.3.2 Problem Reformulation . . . . . . . . . . . . . . . . . . . . . . 71
3.3.3 Derivation of Optimal MSE Minimizer Algorithm . . . . . . . 73
3.3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4 Iterative Observer-based Approach for Source Localization and Es-
timation for Poisson Equation 88
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2 Iterative Observer-based Strategy for Point Source Localization . . . 89
4.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.2 Preliminary Analysis and Results . . . . . . . . . . . . . . . . 91
4.2.3 Point Source Localization Strategy . . . . . . . . . . . . . . . 94
4.2.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 96
4.2.5 Further Simulation Results . . . . . . . . . . . . . . . . . . . . 98
4.3 Robust Iterative Algorithm-based Strategy for Inverse Source Local-
ization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 103
4.3.2 Robust Iterative Observer Design . . . . . . . . . . . . . . . . 104
4.3.3 Two-step Process for Source Localization . . . . . . . . . . . . 112
4.3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 113
4.4 Distributed Potential Field Estimation for Poisson Equation . . . . . 117
4.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 117
4.4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 118
10
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 Dimension Decomposition Approach for 3D Domains 121
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Boundary Estimation Problem for Laplace Equation in 3D . . . . . . 121
5.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 122
5.2.2 Theoretical Analysis and Results . . . . . . . . . . . . . . . . 125
5.2.3 Dimension Decomposition . . . . . . . . . . . . . . . . . . . . 131
5.2.4 Observer Design for the Subproblem . . . . . . . . . . . . . . 133
5.2.5 Numerical Implementation and Simulation Results . . . . . . 134
5.3 Point Source Localization Problem for Poisson Equation in 3D . . . . 135
5.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 136
5.3.2 Point Source Localization as Boundary Estimation Problem . 137
5.3.3 Two-step Process for Source Localization: . . . . . . . . . . . 140
5.3.4 Boundary Estimation Problem for Laplace Equation . . . . . 140
5.3.5 Preliminary Theoretical Results . . . . . . . . . . . . . . . . . 140
5.3.6 Observability Result . . . . . . . . . . . . . . . . . . . . . . . 143
5.3.7 Dimension Decomposition . . . . . . . . . . . . . . . . . . . . 143
5.3.8 Observer Design for the Subproblem . . . . . . . . . . . . . . 145
5.3.9 Numerical Implementation and Results . . . . . . . . . . . . . 146
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6 Concluding Remarks 151
6.1 Summary of the Thesis Work . . . . . . . . . . . . . . . . . . . . . . 151
6.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2.1 Extensions to Arbitrary Shaped Domains . . . . . . . . . . . . 152
6.2.2 Iterative Observer Applications to Other Steady-State PDE
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2.3 Steady-State Energy Field Imaging Technique . . . . . . . . . 153
References 159
Appendices 164
A.1 An Example of Exactly Observable System based on String Equation [11]165
A.1.1 Semigroup Generated by A . . . . . . . . . . . . . . . . . . . 166
A.1.2 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
B.1 Journal Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
11
B.2 Reviewed Conference Papers & Proceedings . . . . . . . . . . . . . . 169
B.3 Talks & Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . 170
12
LIST OF ABBREVIATIONS
DPS Distributed Parameter Systems
ECG Electrocardiography
EEG Electroencephalography
EMANG Estimation Modeling and Analysis Group
KAUST King Abdullah University of Science and
Technology
MEG Magnetoencephalography
NCSU North Carolina State University
ODE Ordinary Differential Equation
PDE Partial Differential Equation
13
LIST OF SYMBOLS
A′ State operator matrix after full system dis-
cretization
C Discrete observation operator matrix
K Gain operator matrix
λmn Infinite set of eigenvalues corresponding to
Φmn
λn Infinite set of eigenvalues corresponding to Φn
C Observation operator matrix
Csub Observation operator matrix for the sub prob-
lem
Ω1 Annulus domain without boundaries
Ω2 Rectangular domain without boundaries
Ω3 3D domain with two congruent parallel sur-
faces without boundaries
Ω4 3D rectangular prism without boundaries
A Differential operator matrix in proper Hilbert
space
Asub Differential operator matrix in proper Hilbert
space defined over the rectangular cross-
section of Ω3 or Ω4
M Linear or nonlinear mapping operator between
two normed spaces
Ψ Data to output map operator
P Discrete error covariance matrix
Φmn Infinite set of orthonormal basis functions for
all m, n ∈ Z?
Φn Infinite set of orthonormal basis functions for
all n ∈ Z?
Q Discrete process noise covariance matrix
R Discrete measurement noise covariance matrix
14
T Exponential of A under certain conditions
S Exponential of A − KC under certain condi-
tions
W Exponential of an unbounded operator under
certain conditions
T Discrete observability matrix
X Complex Hilbert space with proper inner
product
Y Complex Hilbert space with proper inner
product
X1 Part of complex Hilbert space X that satisfies
the special conditions given in Theorem 4
Z? Non zero set of integers
15
LIST OF FIGURES
2.1 Concept of a general Luenberger observer, dashed lines represent the
auxiliary observer structure for the linear finite-dimensional time vary-
ing systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1 Rectangular domain Ω2. . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Idea of iterations over rectangular domain Ω2. . . . . . . . . . . . . . 49
3.3 Domain Ω2 after discretization and fictitious points outside ΓB, index
i = 0 represents fictitious points. . . . . . . . . . . . . . . . . . . . . . 63
3.4 Two dimensional rectangle domain with homogeneous Neumann side
boundaries, Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5 Comparison of exact and observer constructed solution on the bottom
boundary ΓB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6 Two dimensional rectangle domain with homogeneous Dirichlet side
boundaries, Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.7 Comparison of exact and observer constructed solution on the bottom
boundary ΓB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.8 Comparison of exact and observer constructed solution on the bottom
boundary ΓB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.9 Annulus domain Ω1 with inner boundary Γin and outer boundary Γout. 72
3.10 Fictitious points close to inner boundary Γin, with index i = 0. . . . . 76
3.11 True solution or solution obtained by solving the problem (3.70) with
h = sin(θ) + sin(3θ) and g = 0 over Ω1 . . . . . . . . . . . . . . . . . 83
3.12 Solution obtained from optimal iterative algorithm over Ω1, after a
number of iterations in the direction of time-like variable θ . . . . . . 83
3.13 Difference between the true and optimal observer algorithm solutions
over Ω1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.14 On Γin, comparison of true boundary h = sin(θ) + sin(3θ) to the one
recovered by optimal iterative algorithm using Cauchy data from Γout 83
3.15 Numerical solution over Ω1 obtained by solving the problem (3.70) with
pulse shaped h on Γin and g = 0 on Γout . . . . . . . . . . . . . . . . 84
16
3.16 Solution obtained from optimal iterative algorithm over Ω1, after a
number of iterations in the direction of time-like variable θ . . . . . . 84
3.17 Difference between the true and the recovered solutions over Ω1 . . . 84
3.18 On Γin, comparison of the true pulse-shaped boundary signal to the
one recovered by optimal iterative algorithm using only the Cauchy
data from Γout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.19 Noisy data on Γout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.20 On Γin: Comparison of true h = sin(θ) to the one recovered by robust
iterative algorithm using noisy Cauchy data with measurement noise
variance σ2 = 10−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.21 Percentage relative error in the Dirichlet measurement data on Γout vs.
percentage relative error in the recovered solution on Γin . . . . . . . 87
4.1 Left: Rectangular domain Ω2 with ∂Ω2 = Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4, Right:
Co-ordinate axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Numerical simulation result for the Poisson equation over domain Ω2
with a point source in the middle using homogeneous Neumann bound-
ary data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Dirichlet data g1 on bottom boundary Γ1. Because of symmetry g2|Γ2, g3|Γ3
and g4|Γ4 would be similar. . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4 Dirichlet data g1|Γ1 and estimated S1 = ξ1 on opposite boundary Γ2
using iterative observer. Because of symmetry, qualitatively similar
profiles for other three cases. . . . . . . . . . . . . . . . . . . . . . . . 98
4.5 Top plot: Solution profile S1 over Ω2 obtained by solving problem (4.7)
with g1 on Γ1 and corresponding S1 = ξ1 on opposite boundary Γ2 and
insulated (homogeneous Neumann) side boundaries.
From 2nd to 4th: Plots for S2, S3 and S4 obtained using similar procedure. 99
4.6 From top to bottom: Weight profiles w1, w2, w3 and w4 over Ω2 cor-
responding to solution profiles S1, S2, S3, S4 respectively, as shown in
Figure 4.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.7 Weighted sum as given in equation (4.16) over Ω2. Marker in the middle
represents the minima, where the negative point source is located. . . 100
4.8 Top: Non-centered point source inside Ω2. Bottom: Weighted sum
with marker representing minimum point and the location of point
source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
17
4.9 Top: Two opposite polarity well seperated point sources in Ω2. Bot-
tom: Weighted sum with markers representing minimum and maxi-
mum points and locations of two point sources. . . . . . . . . . . . . 102
4.10 Top: Three closely located point sources in Ω2. Bottom: Weighted
sum with minima locating approximate position of point sources. . . 102
4.11 Left: Domain Ω2 after discretization and fictitious points outside Γ2,
index i = 0 represents fictitious points (in blue). . . . . . . . . . . . . 107
4.12 Square domain Ω2 with a point source in the middle using homogeneous
Neumann boundary data. . . . . . . . . . . . . . . . . . . . . . . . . 114
4.13 Noisy measurement data g with σ1 = 5 × 10−4, because of symmetry
of the special case under consideration (one point source in the middle
of the domain), qualitatively similar profiles on all parts of ∂Ω2. . . . 114
4.14 Comparison of noisy measurement data g on Γi and robust iterative
observer solution on the opposite boundary. σ1 = 5 × 10−4 and σ2 =
1 × 10−1. Because of symmetry, qualitatively similar profiles on all
parts of ∂Ω2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.15 Top figure: Solution profile S1 with measurement data g1 on Γ1, in-
sulated side boundaries and boundary estimate obtained using robust
iterative observer on Γ2. From 2nd to 4th: Plots for S2, S3 and S4
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.16 From top to bottom: Weight profiles w1, w2, w3 and w4 respectively,
obtained by solving boundary value problem (4.15) for i ∈ 1, 2, 3, 4. 116
4.17 Weighted sum obtained using equation (4.16). Minimum point repre-
sented with white marker in the middle provides the location of point
source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.18 Top: Solution of problem (4.27),(4.28) with two point sources of op-
posite polarity and homogeneous Neumann boundary on ∂Ω2. Below:
Weighted sum obtained using equation (4.16). Minimum and maxi-
mum points represented with white markers provide locations of point
sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.19 Steady-state potential field u over Ω, obtained by numerically solving
Poisson equation (4.67) for various f with homogeneous Neumann side
boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.20 Weighted sum obtained from equation (4.16), which provides estimate
to the steady-state potential field u over Ω2 for various test cases shown
in Figure 4.19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
18
5.1 Left: Domain Ω3 with two congruent parallel surfaces ΓB and ΓT (ΓB
and ΓT Lipschitz continuous); Right: Plane containing time-like co-
ordinate x1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 Left: Cross-sectional plane ω of Ω3; Right: x2x3 plane orientation (in
gray); . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.3 Left: Cross-sectional plane ω of Ω3 parallel to x2x3 plane; Right: x2x3
plane orientation (in gray); . . . . . . . . . . . . . . . . . . . . . . . . 127
5.4 Analytical solution u on Γ1 . . . . . . . . . . . . . . . . . . . . . . . . 135
5.5 Analytical solution u on Γ2 . . . . . . . . . . . . . . . . . . . . . . . . 135
5.6 Recovered solution on Γ2 using observer algorithm . . . . . . . . . . . 135
5.7 Difference of the analytical solution and the one recovered by observer
algorithm on Γ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.8 Rectangular prism Ω4 with six boundary surfaces, ∂Ω4 = ∪6i=1Γi. Sur-
faces Γ1, . . . ,Γ6 represent bottom, top, and side surfaces respectively. 137
5.9 Rectangular cross-section ω at a particular value of x in yz-plane inside
Ω4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.10 Idea of iterations over rectangular cross-section ω. . . . . . . . . . . . 146
5.11 Domain Ω4 with three orthogonal cross-sectional planes. . . . . . . . 147
5.12 A single point source in the middle of a 1 × 1 × 1 cube on a uniform
20× 20× 20 grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.13 Cross-sectional plane ω from Figure 5.12 on a 200× 200 uniform grid. 148
5.14 Iterative observer solution from equation (5.71) with h1,sub = 0 and
g1,sub extracted from Figure 5.13 on Γb and recovered boundary data
on opposite boundary. Similarly iterative observer solution can be
computed on cross-sectional planes parallel to ω. . . . . . . . . . . . . 149
5.15 Solution profile S2 on a 20× 20× 20 uniform grid, obtained by solving
boundary estimation problem (5.43) for i = 2 using iterative observer. 149
5.16 Weight profile v2 obtained by solving boundary estimation problem
(5.44) for i = 2 on a uniform 20× 20× 20 grid. (only two orthogonal
cross-sectional planes displayed) . . . . . . . . . . . . . . . . . . . . . 149
5.17 Weighted sum computed using equation (5.45). Local maxima in the
center locate the position of the point source. . . . . . . . . . . . . . 150
6.1 Domains under consideration for the boundary estimation problem for
Laplace equation with Γ∗, the unknown data boundary. . . . . . . . . 153
19
6.2 Domains under consideration for the source localization and estimation
problems for Poisson equation. . . . . . . . . . . . . . . . . . . . . . . 153
6.3 Numerical solution of the Poisson equation 4u = f over Ω4 with f =
exp(−(2.5(x−0.5))2−(2.5(y−0.5))2−(2.5(z−0.5))2) and homogeneous
Neumann boundary data at ∂Ω4. The solution is computed over a 50×50× 50 uniform grid using 2nd order accurate centered finite difference
discretization schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4 A particular cross-sectional view of the numerical solution presented
in Fig. 6.3, at x = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.5 (In blue): Dirichlet data, extracted from Fig. 6.4 at Γb = ω ∩ Γ1,
corrupted with added white Gaussian noise with η1 = 0 and σ1 =
5×10−3; (In black) Numerical solution of boundary estimation problem
(5.66), obtained by using optimal iterative algorithm (4.49), (4.50) at
cross-section ω|x=0.5 at Γt = ω ∩ Γ2. . . . . . . . . . . . . . . . . . . . 156
6.6 Full solution profile over cross-section ω|x=0.5 obtained by using fi-
nite difference discretization schemes and the estimated boundary data
from Fig. 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.7 Tomographic image S1 over Ω4 obtained by solving boundary estima-
tion problem for Laplace equation (5.43) using dimension decomposi-
tion approach and optimal iterative algorithm. . . . . . . . . . . . . . 157
6.8 Weight profile v1 obtained by numerically solving boundary value prob-
lem (5.44) for index i = 1. . . . . . . . . . . . . . . . . . . . . . . . . 157
6.9 Weighted sum obtained from equation (5.45) by combining tomographic
image profiles S1, . . . , S6 and corresponding weights v1, . . . , v6. The
simulation result shows recovery of the distributed potential field pre-
sented in Fig. 6.3, using only the noise corrupted boundary data. . . . 158
6.10 A cross-sectional view of weighted sum over ω|x=0.5. The result above
to be compared with the cross-sectional view given in Fig. 6.4. . . . . 158
20
Chapter 1
Introduction
1.1 Motivation
Partial Differential Equations (PDEs) are widely used to model physical phenomena
including sound, heat, wave, fluid dynamics and quantum mechanics. Just as Or-
dinary Differential Equations (ODEs) are used to model one-dimensional space or
time evolving physical phenomena, PDEs are used to model multidimensional sys-
tems [1,2]. Over the last decades, there have been a lot of efforts to develop feedback
based strategies to control and observe dynamical PDE\ODE systems. The objec-
tives of these techniques are to drive the system in a particular way to achieve some
desired goals and to observe some system states and parameters respectively [3,4,5].
However, there is a subclass of PDEs, known as elliptic PDEs, that represents the
steady-state physical phenomena that do not evolve with time. Such phenomena
arise in mechanics of materials, bio-medical engineering and earth science applica-
tions [1, 6, 7, 8].
Customarily, the control and observation design techniques are only developed
for time varying PDE systems, broadly classified as hyperbolic and parabolic PDE
systems, with inherent time component [3,4,5,9,10,11,12]. Recently, there has been a
lot of work done on the feedback based system state estimation algorithms, commonly
know as state observer algorithms, for dynamical PDE systems. In dynamical systems
theory, the state observer is an algorithm that provides estimates of internal states
of a given real system from the measurements of inputs and outputs [13]. Various
21
kinds of state observer algorithms have been proposed for dynamical PDE systems in
literature. A limited introduction to the observation design for PDE systems can be
found in [11,14,15,16,17,18,19].
On the one hand state observers are only developed for time varying PDE sys-
tems, and on the other hand the unknown source and boundary data estimation
problems for steady-state elliptic PDEs belong to the class of ill-posed problems.
The ill-posedness is in the sense of stability, that is, a small amount of discrepancy
in the observed data destroys the estimated solution [20, 21]. In other words, the
mathematical problem is well-posed only for smooth data. Whereas, the real experi-
mental observations are always prone to measurement noise like sensor related issues
and other unavoidable artifacts. Nearly all existing solution techniques for estima-
tion problems for elliptic PDEs can be broadly classified as optimization based algo-
rithms [8,20,22,23,24]. Such techniques generally solve a series of similar well-posed
problems to achieve the convergence based on some preset criteria. These optimiza-
tion approaches are numerically costly, particularly, when the problem is posed on
a higher dimensional domain [25, 26]. In this scenario, the efforts to develop robust
state observer like algorithms, possibly using one of the space variables as time-like,
to achieve lower numerical complexity and robustness, becomes highly interesting.
This strongly motivates the study of observer design techniques for the steady-state
elliptic PDE systems.
The source and boundary data estimation problems for elliptic PDE systems have
been fundamental problems of interest in several areas of science and engineering
including corrosion problem and non-destructive testing in mechanics of materials,
steady-state reservoir modeling in earth science, mineral exploration in geo-science
and Electrocardiography (ECG) and Electroencephalography (EEG) applications in
biomedical discipline [27, 28,29,30,31,32].
22
1.2 Background
A vast literature is available for the observation design problems for dynamical sys-
tems including [4,11,13,14,15,33]. The pioneering works of Luenberger on the design
of feedback based system state estimation algorithms paved the way for the field of
state observer design for dynamical systems [13]. Early observer designs were pro-
posed for state estimation for finite dimensional lumped parameter systems governed
by ODEs. However, over the years, the concepts of observer design have been ex-
tended to infinite dimensional Distributed Parameter Systemss (DPSs) modeled by
time varying PDEs [11,14,16,34].
Traditionally for DPSs early or late lumping techniques are considered [35]. Early
lumping techniques transform DPS to a finite dimensional system of ODEs using
some approximation and discretization techniques [9, 10]. The resultant system of
ODEs is an approximation to the DPS and unknown states recovered by the state
observers may not be the estimate of true states [17]. On the other hand, late lump-
ing techniques exploit mathematical properties of underlying PDEs to develop ob-
server design. The potential challenges include the mathematical justification of
system observability and the design of observer gain. The system observability
ensures the existence of the solution. Various design techniques based on semi-
group theory, spectral theory, Lyapunov based design, backstepping approaches are
available [3, 11, 15, 16, 18, 19, 34, 36, 37, 38]. Further, some state-of-the-art inverse
and ill-posed source and boundary value problems for elliptic PDEs can be found
in [20, 39, 40]. Some of the existing optimization based strategies to solve inverse
source and boundary value problems for Laplace and Poisson equations can be found
in [24,25,26,30,32,41,42,43,44,45]. A state of the art discussion on inverse problems
is also provided in Chapter 2.
23
1.3 Proposed Approaches
In this thesis work, state observer design techniques are developed for steady-state
elliptic PDE systems. The idea is to use space as a time-like variable to develop
iterative observer algorithms to solve source and boundary data estimation problems
for elliptic PDE systems. Both early and late lumping techniques are used to develop
different variants of the proposed iterative observer algorithms.
As stated before, almost all the existing observer design techniques are focussed
on time-varying PDE systems [3, 11, 16, 18, 34, 36, 37, 38]. There has been very little
effort to develop observer-like algorithms for systems governed by steady-state elliptic
PDEs. One such example is [46], where an extra time variable is introduced to
solve steady-state heat conduction problem modeled by elliptic PDE as a parabolic
problem. However, the focus of this thesis work is to develop state observers that
sweep over the entire domain in a particular time-like direction to solve boundary
and source data estimation problems for linear elliptic equations namely Laplace and
Poisson equations. The theoretical framework is developed using functional analysis
tools and semigroup theory. The idea of using a space variable as time-like allows the
development of numerically efficient algorithms. The various algorithms, presented in
this thesis, are numerically implemented using finite difference discretization schemes.
The algorithms tackle 2D regular shaped annulus and rectangle domains and 3D
domains with two congruent parallel surfaces. The main contributions of this thesis
are also listed in the following section.
1.4 Thesis Contributions
Key contributions of this thesis report are listed below,
• Developed Iterative Observer for Boundary Estimation Problem for
Laplace Equation:
24
Iterative observer is developed using space as time-like to estimate the unknown
boundary data. The functional analysis framework and convergence of the
iterative observer are proved. The algorithm is implemented on a 2D rectangular
and annulus shaped domains.
• Provided Source Estimation and Localization Strategy for Poisson
Problem:
Source localization and estimation strategy is presented using iterative observer
algorithm for the boundary value problem for Poisson equation on regular
shaped domains.
• Designed Robust Iterative Observer for Ill-posed Boundary Estima-
tion Problem for Laplace Equation with Noisy Data:
Robust counterpart of the iterative observer is developed to tackle the inverse
and ill-posed boundary data estimation problem for Laplace equation on reg-
ular shaped domain. The proof of convergence of the developed algorithm is
provided.
• Extended Source Localization and Estimation Strategy to Inverse
Source Problems for Poisson Equation:
The source localization and estimation strategy is extended to tackle inverse
and ill-posed source problems for Poisson equation. The strategy and results
are presented on rectangular domains.
• Developed Extensions to 3D-Shaped Domains and Introduced Steady-
State Energy Field Imaging Technique:
Boundary estimation and source localization strategies are extended to 3D
shaped domains with two congruent parallel surfaces. The technique involve
dimension decomposition approach that divides down the 3D problem to a set
25
of 2D subproblems. Finally the iterative observer algorithm is implemented on
the 2D subproblems.
The various chapters are organized as follows. Some state-of-the-art literature
on the state observers and inverse and ill-posed problems is provided in Chapter 2.
An iterative observer is designed in Chapter 3 that sweeps over the regular shaped
annulus and rectangle domains to solve the boundary data estimation problems for
the Laplace equation. As discussed earlier, such boundary value problems are ill-
posed in the sense of stability. This makes the use of standard iterative observer
algorithms impossible without further investigations. Thus, a robust counterpart
of this iterative observer is developed in the second half of this chapter. Next, a
source estimation and localization strategy is developed in Chapter 4 to solve source
localization and estimation problems for the Poison equation in the 2D rectangular
domain. The strategy works well for the cases of smooth and noisy measurement
data. A dimension decomposition approach, based on iterative observer design, is
presented in Chapter 5 to solve the source and boundary data estimation problems
for 3D domains with two congruent parallel surfaces and insulated side boundary
surfaces. Theoretical and numerical results are consistently presented throughout the
document.
26
Chapter 2
State Observer Theory and Inverse Problems: Two Worlds
Apart
We must not cease from exploration and the end of all our exploring will
be to arrive where we began and to know the place for the first time.
(—T.S. Eliot)
2.1 Introduction
In the first half of this chapter, some state of the art literature on observation and
control design theory for dynamical systems is presented. Key definitions along with
some theoretical results are recalled. In the second half, some fundamental concepts
on inverse problems are discussed. The notion of ill-posedness is explained. The
literature study provided in this chapter will help to develop the framework for the
upcoming analysis and results in later chapters.
In the following, some dynamical systems concepts are presented for time evolving
finite and infinite dimensional systems.
2.2 Observation Theory for Dynamical Systems
In the last few decades, there has been a lot of focus on interdisciplinary research to
develop novel kind of solution strategies for challenging problems. With combined
efforts from diverse areas of research, new versatile and robust solution techniques
are emerging for various types of mathematical problems. One such joint venture
is the study of initial boundary value problems modeled by PDEs using dynamical
27
system techniques. In order to develop the understanding for observation problems
for linear PDE systems, it is natural to look into the theory developed for analogous
finite dimensional time varying systems.
2.2.1 Finite-Dimensional Linear Time-Invariant Systems
Given n ∈ N∗ and A ∈ Cn×n, consider the linear time invariant dynamical system,
ξ(t) = Aξ(t) +Bu(t) ∀ t ≥ 0, (2.1)
where ξ(t) ∈ Cn is the state of the system, ξ(t) represents system dynamics, u(t) is the
input and B is the input operator. For a pure observation problem, let u(t) = 0. The
above representation is called state-space representation. A state-space representation
for a dynamical system is a mathematical model of a physical system as a set of input,
output and state variables related by first-order evolution equations. “State-space”
refers to the space whose axes are the state variables. The state of the system can be
represented as a vector within that space. Now, suppose partial measurements y(t)
of the system are available through an observation operator C ∈ Cm×n, with m ∈ N∗
such that,
y(t) = Cξ(t) = CetAξ(0) ∀ t ≥ 0, (2.2)
here a natural question arises whether the observation y(t) is good enough to solve
problem (2.1) or not. Or in other words, whether the partial measurements y(t)
are good enough to provide full information about systems states ξ(t) for all t ≥
0. Answer to this question lies in the concept of observability of finite dimensional
systems.
Definition 1. Let (A,C) be the system defined by equations (2.1) and (2.2). Also
28
define the initial data to output map Ψτ ∈ L(Cn, L2([0,∞);Cm)) by setting,
(Ψτξ0) =
CetAξ0 t ∈ (0, τ),
0 t > τ.
(2.3)
System (A,C) is said to be observable if for some τ > 0, KerΨτ = 0. A necessary
and sufficient condition for observability of finite dimensional system (2.1)-(2.2) is the
so-called Kalman rank condition.
Theorem 1. [3, 11] We have,
ker Ψτ = ker
C
CA
CA2
...
CAn−1
(2.4)
In particular, system (A,C) is observable if and only if
rank
C
CA
CA2
...
CAn−1
= n. (2.5)
Proof. Proof of the above theorem is also recalled in appendix A.
Note that Kalman condition (2.5) is independent of τ . This shows in particular that
a finite-dimensional linear system (A,C) is observable in arbitrarily small time. As
will be seen later this is not true for infinite dimensional systems.
Definition 2. A matrix A ∈ Cn×n is called Hurwitz matrix if its eigenvalues are
29
located in the open left half-plane:
σ(A) ⊂ λ ∈ C|Re(λ) < 0 . (2.6)
Note that a system of the form (2.1) associated to a Hurwitz matrix A generates
stable trajectories:
limt→+∞
z(t) = 0. (2.7)
More precisely it can be proved that there exists 0 < ω < minλ∈σ(A) Re |λ| and M > 0
(depending on ω) such that
‖ξ(t)‖ ≤Me−ωt‖ξ0‖, ∀t > 0, ∀ξ0 ∈ Cn. (2.8)
2.2.1.1 State Observer
In dynamical systems theory, a state observer is a mathematical system that provides
an estimate of the internal states of a given physical system, from measurements of
the inputs and outputs of the real system.
As stated earlier, state of a linear discrete time system is assumed to satisfy equations,
ξ = Aξ,
ξ(0) = 0,
y(t) = Cξ(t).
(2.9)
The observer model of the physical system is typically derived from these equations.
Additional terms may be included in order to ensure that, on receiving successive
measurements, the model’s state converges to that of the real system. In particular,
the output of the observer may be subtracted from the real system output and then
multiplied by a matrix K, this is then added to the equations for the state of the
30
ξ
(ξ − ξ)→ 0as t→∞
y
y
+
−
System
ξ = Aξ
Model
˙ξ = Aξ +K
(y − Cξ
)K
Figure 2.1: Concept of a general Luenberger observer, dashed lines represent theauxiliary observer structure for the linear finite-dimensional time varying systems
observer to have Luenberger observer, defined by the equations below,
˙ξ = Aξ +K
(y − Cξ
). (2.10)
The above equation contains the copy of the real system plus a correction term. The
state estimation error dynamics are given by,
e = (A−KC)e, (2.11)
where e = ξ − ξ. Whenever pair (A,C) satisfies Kalman rank condition, (A −
KC) can be made Hurwitz such that continuous time error dynamics converges to
zero asymptotically. The general structure of a Luenberger observer for linear finite-
dimensional time varying systems is also shown in Figure 2.1. It is important to note
that, for time varying dynamical systems, various kinds of state observers have been
developed in literature [13,14,15,16,18,34,47,48].
31
2.2.2 Infinite-Dimensional Linear PDE Systems
At present, dynamical systems theory is one the most interdisciplinary areas of re-
search. The need for control design arises in most of the present day industrial
applications. In fact, the overlap of dynamical systems concepts applied to complex
physical systems modeled by deterministic and stochastic PDEs has produced a new
and highly enriched branch of modern mathematics. Readers can refer to some state-
of-the-art literature available in this domain [3, 4, 6, 11, 12, 38, 49, 50, 51, 52]. In the
following, only observation theory concepts for dynamical systems modeled by linear
time varying PDEs are discussed.
2.2.2.1 State-Space Representation for Second-Order Linear
PDE Systems
Many time varying physical phenomena including heat transfer and wave propagation
are modeled by second-order PDEs. Linear second-order PDEs on a two-dimensional
xt plane can be represented as,
a∂2v
∂t2+ b
∂2v
∂t∂x+ c
∂2v
∂x2+ d
∂v
∂t+ e
∂v
∂x+ fv = 0, (2.12)
where a, b, c, d, e and f are constants and a, b and c are not all zero. The above
equation can be represented as a first-order state equation in one of the variables as:
∂ξ
∂t= ξ(t) = Aξ, (2.13)
with t as time variable and state vector ξ. The state vector of such a system is an
element in an infinite-dimensional normed space. The state operator matrix A is
32
given as:
ξ(t; .) =
ξ1(t; .)
ξ2(t; .)
, A =
0 1
c
a
∂2
∂x2+e
a
∂
∂x+ f
b
a
∂
∂x+d
a
, (2.14)
with ξ1 = v and ξ2 =∂v
∂t. A dynamical system modeled by 2nd order linear PDEs
can be written as a first order system along with the initial condition as follows,
ξ = Aξ,
ξ(0) = ξ0.
(2.15)
with some boundary conditions, where “ ˙ ” represents partial derivative with respect
to variable t and A is a differential operator matrix.The available information about
the problem can be written by using an observation operator C acting on state variable
ξ such that,
y(t) = Cξ(t), (2.16)
here y is the output or observation. System dynamics are described by the partial
differential operator matrix A. Let us pose three questions before trying to find a
solution strategy to solve mathematical problem given by equations (2.15) and (2.16).
Q 1 : Solution of the first order state equation, given in equation (2.15), involves
exponential of the differential operator matrix A. What does it mean by the
exponential of A?
Q 2 : Does there exist a solution to the problem (2.15) and (2.16)?
Q 3 : What can be a particular algorithm to solve problem (2.16)?
Answers to above questions along with the design of solution algorithms is discussed
in infinite dimensional setting in the following subsections. The exponential of op-
33
erator matrix A leads to the study of semigroup generated by operator A. Naively,
a semigroup is the generalization of an exponential function to infinite dimensional
spaces. Answer to Q 2 leads to the study of system observability. First we study
observability concept for finite dimensional systems as given below. For more details,
readers are referred to works of Curtain and Zwart [3], review paper of Zuazua [5]
and Tucsnak and Weiss [11].
2.2.2.2 Exponential of Operator A
Let X be a complex Hilbert space (with inner product (., .) and corresponding norm
‖.‖) and A : D(A) → X be an unbounded operator on X. The objective is to find
the solution of the evolution problem of the form (2.15). Note that if A ∈ L(A) is a
bounded operator on X then the unique solution of (2.15) is given by,
ξ(t) = etAξ0. (2.17)
Typically A is a partial differential operator matrix and hence unbounded, the expo-
nential etA is no more defined. However, there is a class for which this notion can be
defined, leading to the concept of semigroups. For the introduction to the semigroups
readers are referred to the textbooks of Pazy [38] and Brezis [53]. In the following
strongly continuous semigroup is defined.
2.2.2.3 C0-Semigroup
C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a
generalization of the exponential function. Just as exponential functions provide so-
lutions of scalar linear constant coefficient ordinary differential equations, strongly
continuous semigroups provide solutions of linear constant coefficient ordinary differ-
ential equations in Banach spaces.
34
Definition 3. [11, 38] A family T= (Tθ)θ≥0 of operators in L(X) is a strongly
continuous semigroup on X if
1. Tθ=0 = I. (identity property)
2. Tθ+φ =TθTφ ∀θ, φ > 0. (semigroup property)
3. limθ→0,θ>0 Tθz = z ∀z ∈ X. (strong continuity property)
Intuitively, T = (Tt)t≥0 models the time evolution of the state of a process, the
state at time t ≥ 0 being described by ξ(t) =Ttξ0. The simplest but rather limited
class of C0 semigroups is given by, Tt = etA for A∈ L(X).
Definition 4. [11, 38] A strongly continuous semigroup T is called a contraction
semigroup if ‖Tt‖ ≤ 1 for all t ≥ 0.
Definition 5. [11,38] Let T be a strongly continuous semigroup. Then, the operator
A : D(A)→X defined by,
D(A) =
ξ ∈ X : lim
t→0+
Ttξ − ξt
exists
, (2.18)
Aξ = limt→0+
Ttξ − ξt
, ∀ξ ∈ D(A). (2.19)
is called infinitesimal generator (or the generator) of the semigroup. Semigroup gen-
erated by an operator A is often denoted by T= (etA)t≥0.
Definition 6. [11, 38] Let A be the generator of a strongly continuous semigroup
T= (etA)t≥0. Then for ξ ∈ D(A) and t ≥ 0, we have etAξ ∈ D(A) and
d
dtetAξ = AetAξ = etAAξ. (2.20)
Definition 7. [11, 38]
35
• An operator A: D(A)→ X is called dissipative if,
Re(Aξ, ξ) ≤ 0, ∀ξ ∈ D(A). (2.21)
• An operator A: D(A)→ X is called m-dissipative (or maximal dissipative) if it
is dissipative and I −A is surjective.
Theorem 2. [38] (Lumer-Phillips Theorem) Let A : D(A) → X be an unbounded
operator on a Hilbert space X. Then the following two assertions are equivalent.
1. A is maximally dissipative.
2. A is the generator of a contraction semigroup (Tx)x≥0, i.e. ‖Tt‖ ≤ 1 for all
t > 0.
2.2.2.4 Observability for Infinite-Dimensional PDE Systems
For infinite-dimensional PDE systems, depending on the density argument in infinite-
dimensional spaces, there are at least three observability concepts as explained in the
following. Assume that Y is a complex Hilbert space and that C∈ L(X,Y ) is an
admissible observation operator for T. Let τ > 0, and let Ψτ be the output operator
associated with (A, C) given by,
(Ψτξ0) =
CTtξ0 t ∈ (0, τ),
0 t > τ.
(2.22)
These operators are elements of L(X,L2([0,∞);Y )).
Definition 8. [11] Let time τ > 0.
• The pair (A, C) is exactly observable in time τ if Ψτ is bounded from below.
• (A, C) is approximately observable in time τ if ker Ψτ = 0.
36
• The pair (A, C) is final state observable in time τ if there exists a kτ > 0 such
that ‖Ψτξ0‖ ≥ kτ‖Tτξ0‖ for all ξ0 ∈X.
It can be seen, using density of D(A∞) in X, that the exact observability of (A, C)
in time τ is equivalent to the fact that there exists kτ > 0 such that,
∫ τ
0
‖CTtξ0‖2dt ≥ k2τ‖ξ0‖2 ∀ ξ0 ∈ D(A∞). (2.23)
A simple example of exactly observable system based on the string equation is pro-
vided in Appendix A. The Luenberger type of observer design has a similar structure
as given,
˙ξ = Aξ +K
(y − Cξ
). (2.24)
The main challenge is to design the observer gain K such that (A−KC) is dissipative.
The observer design for infinite dimensional steady-state elliptic PDE systems will be
discussed in details in chapter 3. On the other hand, for time varying PDE systems
in infinite dimensional setting, various works can be visited [15,16,34,36,37].
In the following section some state of the art literature for inverse problems is
discussed.
37
2.3 Inverse Problems
Most people, if you describe a train of events to them, will tell you what
the result would be. They can put those events together in their minds,
and argue from them that something will come to pass. There are few
people, however, who, if you told them a result, would be able to evolve
from their own inner counsciousness what the steps were which led up to
that result. This power is what I mean when I talk of reasoning backward
or analytically. (—Arthur Conan Doyle, A Study in Scarlet)
An inverse problem is the one in which cause behind a certain physical phe-
nomenon is investigated with a given mathematical model for the system. Real phys-
ical phenomena are governed by laws of nature with infinite artifacts at different
scales. Some of these phenomena like diffusion and wave propagation etc. can be
modeled fairly accurately using PDEs. However, no mathematical model for a par-
ticular physical phenomenon is perfect, further, there are always some measurement
errors due to noise and other unavoidable circumstances, all these factors make inverse
problems a challenging field of research. In the last half century, with the advent of
highly remarkable computational devices, it has been possible to model complex ini-
tial boundary value problems for PDEs using efficient numerical techniques. This has
provided a center stage to study PDE based model problems numerically, as well as
their theoretical analysis of uniqueness and stability. Almost always these problems
are ill-posed in terms of existence, uniqueness or stability. Thus, their uniqueness and
stability studies are at the heart of the development of new solution methodologies.
2.3.1 Forward and Inverse Problems
In the jargon of the theory of inverse problems, there always exists a forward or
direct problem and vice versa. A forward problem is usually a well-posed problem in
contrast to the inverse problem which is ill-posed. This chapter describes ill-posedness
of inverse problems and some state of the art solution strategies.
38
In the remaining part of this chapter, letM(x) = y be the mathematical represen-
tation of a given physical system withM be the mathematical model, x be the input
or model parameters and y be the output data measurements. The forward problem
is to find the output y from the given mathematical modelM and input parameters x.
Forward Problem:
Mathematical
Model M(x)Input x Data y
In general, the forward problem is well-posed. This means with given input x, a
stable M(x) can be found uniquely. On the other hand inverse problem is to find
out the hidden model parameters x from given mathematical model of the systemM
and output data measurements y. In terms of functional analysis, an inverse problem
is represented as a mapping between metric spaces. Many of the inverse problems
are ill-posed in terms of stability, that is, the forward map M(x) does not have a
continuous inverse.
Inverse Problem:
Mathematical
ModelData Input?
2.3.2 Ill-posedness
Definition 9. A problem is called ill-posed in the sense of Hadamard [54] if it fails
to satisfy any of the following properties,
1. There exists at least one solution to the problem. (existence)
2. There exists at most one solution to the problem. (uniqueness)
39
3. The solution continuously depends on the the data. (stability)
Definition 10. Let X and Y be normed spaces, M : X → Y be any linear or
nonlinear mapping. Equation Mx = y is called properly posed or well-posed, if the
following holds [39]:
1. For every y ∈ Y , there exists, at least one x ∈ X such thatMx = y. (existence)
2. For every y ∈ Y , there exists, at most one x ∈ X such that Mx = y. (unique-
ness)
3. The solution of the inverse problem, x continuously depends on the data; that
is, for every sequence (xn) ⊂ X with Mxn → Mx as n → ∞, it follows that
xn → x as n→∞. (stability)
2.3.3 Some Solution Strategies for the Inverse Problems
In practical applications output data measurements are always corrupted with some
noise i.e. yδ = y + ε (ε be an additive noise). Further as stated earlier if the forward
map M(x) does not have a continuous inverse then small errors due to noise can
totally destory the solution of the inverse problem. Therefore, for constructing a stable
approximation of the solution a regularization strategy is required. Regularization
means constructing an approximate continous map Πα : Y → X that inverts M
approximately. Design of such a continous reverse map Πα always require some
apriori information on the system. In a broader sense solution strategies for inverse
problems lie in following two catagories.
1. Deterministic Regularization Techniques.
2. Stochastic or Bayesian Inversion.
40
2.3.3.1 Deterministic Regularization Techniques [39]
These regularization techniques are based on acquiring approximately continuous
reverse map using some a priori information on the system in a deterministic way. A
priori information usually comes from the physics of the problem and is exterior to
mathematics. This added information assures the robustness of the new approximate
solution against the measurement errors in the data. Following are various standard
regularization techniques used,
• Tikhonov’s Regularization, the most well-known method of regularization works
on the standard least square linear regression plus an added regularization func-
tional. The functional added is capable of taking into account the a priori
information about the system [39].
• Reduction of the number of parameters, to reduce the sensibility of the criteria
to data noise.
• Introduction of constraints, equalities and inequalities inside the functional just
to reduce the solution domain to physically acceptable values.
• Filtering of the trial data using signal processing techniques (frequency filters,
modified Fourier transforms, time frequency transformations etc.) [55].
• Quasireversibility method well suited for some Cauchy problems for elliptic
partial differential equations. This method works by changing the order of
derivative of the partial differential operator in a way to obtain a well-posed
problem [56].
2.3.3.2 Stochastic or Bayesian Inversion
In this class of solution strategies, all variables are considered to be random in order
to present every uncertainty. Thus one is interested in probability density functions.
41
This function is associated with the unknowns and with the data of the problem from
which one looks for the characteristic values: average values, correlations, value of
the largest probability [57].
An important thing to note is that often during the solution process an inverse
problem boils down to a numerical optimization problem in which a numerically com-
puted solution of the inverse problem is compared to some real experimental data in
the form of a cost function. Thus the study of the numerical optimization techniques
is also at the heart of solving an inverse problem [58]. There are various numerical
optimization techniques like linear or non-linear least squares, maximum plausibility,
Monte-Carlo method, linear programming, simulated annealing, genetic or evolu-
tionary algorithms, optimal control methods etc. Detailed study of the numerical
optimization techniques can be found in standard literature [12,57,58].
2.3.4 Ill-posedness of Boundary Data Estimation Problems
for Laplace Equation
In general, source and boundary data estimation problems for elliptic PDEs are highly
sensitive to noise and are ill-posed in the sense of stability. Two simple examples are
presented in the following to highlight this fact.
2.3.4.1 On the Annulus Domain Ω1 [25]
Let Ω1 be the annulusz = reιθ : r0 < r < 1, 0 ≤ θ < 2π
such that,
4u = 0 0 ≤ θ < 2π, r0 < r < 1,
u(eiθ) = γ(eiθ) 0 ≤ θ < 2π,
∂ru(eiθ) = 0 0 ≤ θ < 2π,
(2.25)
42
for analytical solution, let us use the Fourier series expansion of solution u and the
boundary values u|r=1 = γ,
u(r, θ) =∞∑
k=−∞
uk(r)eikθ, and, γ(eiθ) =
∞∑k=−∞
γkeikθ (2.26)
Writing the Laplacian in polar co-ordinates, a family of ordinarily differential equa-
tions is obtained, and for the sequence of Fourier coefficients uk,
r∂
∂r
(r∂uk∂r
)− k2uk = 0, r0 < r < 1,
uk(1) = γk,
∂∂ruk(1) = 0.
(2.27)
It is easy to see that the solution of (2.25) can be written as Fourier series,
u(reiθ) =∞∑
k=−∞
(rk + r−k
2
)γke
ikθ, (2.28)
since r0 < r < 1, the factor r−k grows exponentially as k → ∞, similarly rk grows
exponentially as k → −∞. This implies the analytical solution is highly sensitive
to noise or high frequency components in the available data. The solution is highly
ill-posed in terms of Hadamard’s stability criteria.
2.3.4.2 On the Rectangular Domain Ω2 [59]
Let Ω2 be a rectangular domain of base length π, such that,
4u = 0 x ∈ [0, π], y > 0,
u(x, 0) = 0 x ∈ [0, π],
∂u
∂y(x, 0) = α sin(nx) x ∈ [0, π].
(2.29)
43
Above Cauchy problem has a solution,
u(x, y) =α
nsinh(ny) sin(nx), (2.30)
for any ε > 0, c > 0 and y > 0 it is possible to find α and n such that,
‖α sin(nx)‖ < ε and∥∥∥αn
sinh(ny) sin(nx)∥∥∥ > c.
Clearly the above solution does not continuously follow the data.
2.4 Conclusion
After a quick literature review of observation theory for the finite- and infinite-
dimensional linear dynamical systems, along with the notion of ill-posedness, we are
all set to develop estimation algorithms for infinite-dimensional linear steady-state
systems modeled by the elliptic equations. In the following chapter, different from
traditional optimization based techniques, we develop iterative observer and robust
iterative observer algorithm to solve ill-posed boundary data estimation problems for
the steady-state Laplace equation.
44
Chapter 3
Iterative Observers for Boundary Estimation Problems for
Laplace Equation
Every solution to every problem is simple. It’s the distance between the
two where the mystery lies. (—Derek Landy, Skulduggery Pleasant)
3.1 Introduction
In this chapter, iterative observer algorithms are introduced for boundary estimation
problems for the steady-state Laplace equation. The mathematical problems are in-
troduced on regular-shaped domains with available data on the parts of the boundary.
First, an iterative observer algorithm is developed that sweeps over the whole domain
using space as time-like. The theoretical framework using semigroup theory is devel-
oped. Then, a robust optimal iterative algorithm is presented to tackle the ill-posed
boundary estimation problems with noisy boundary data. Two types of boundaries,
rectangular and annulus, are considered to develop the framework. Further types of
domains are considered in the later on chapters.
The boundary estimation problem for Laplace equation, also known as the Cauchy
problem for the Laplace equation has been a fundamental problem of interest in many
diverse areas of science and engineering. For example non-destructive testing appli-
cations in mechanics, where we are interested in finding inside cracks from boundary
measurements [27]. Biomedical applications in finding the actual heart potential
from electrocardiogram (ECG) data collected on the body torso. Finding the ac-
tual heart potential is vital to understand the functionality of heart valves [28, 29].
45
Readers may refer to a number of existing numerical solution techniques for elliptic
Cauchy problems for further understanding of nature of the mathematical problem,
e.g. [24, 25,26,41,42,43,44,45].
3.2 Iterative Observer Design for Boundary Estimation
In this section, first, a boundary estimation problem for steady-state elliptic Laplace
equation is mathematically formulated, on a rectangle domain, and then an iterative
observer algorithm is presented to solve this problem. The theoretical results are
presented using functional analysis framework.
3.2.1 Problem Statment on a Rectangular Domain
Let Ω2 be a rectangle domain in R2 with ΓB,ΓT ,ΓL,ΓR be top, bottom, left and right
boundaries respectively as shown in Figure 3.1 such that Ω2 = Ω2∪ΓB∪ΓT ∪ΓL∪ΓR,
Ω2 = (0, a) × (0, b) and ΓB ∩ ΓT ∩ ΓL ∩ ΓR = ∅. Boundary estimation problem for
Laplace equation is defined as,
Find u(x) on ΓB:
4u =∂2u
∂x2+∂2u
∂y2= 0 in Ω2,
u = f(x) on ΓT ,
∂u
∂n= g(x) on ΓT ,
(3.1)
with homogeneous Dirichlet or Neumann side boundaries, f and g are given suffi-
ciently smooth and∂
∂nrepresents the normal derivative to the top boundary ΓT .
In the following subsection, an iterative observer is developed to solve boundary
estimation problem for Laplace equation posed on the rectangular domain. The
theoretical concepts can be extended to annulus domain as well. However, to avoid
repetition, only the rectangular case is discussed.
46
Ω2 = (0, a)× (0, b)
ΓB
ΓT
ΓL
(0, 0)
(0, b)
(a, 0)
(a, b)
ΓR
Figure 3.1: Rectangular domain Ω2.
3.2.2 Notations and Definitions
In this section, let X be a Hilbert space with inner product 〈., .〉 and corresponding
norm ‖.‖. If X and Y are two Hilbert spaces then L(X, Y ) denotes the space of linear
operators from X to Y with induced norm. Further L(X) = L(X,X). Let an infinite
dimensional linear dynamical system be presented in state space representation as,
ξ(x) = Aξ(x); y(x) = Cξ(x); (3.2)
such that “ ˙ ” represents partial derivative with respect to time-like variable x, ξ be
a state vector, A : D(A) → X be the state operator matrix, C ∈ L(X, Y ) be the
observation operator with observation space Y .
Definition 11. Let x be a time-like variable, a family T = (Tx)x≥0 of operators in
L(X) defines a strongly continuous semigroup (C0-Semigroup) on X if,
1. T0 = I, (identity property)
2. Tx+w = TxTw, ∀x,w ≥ 0, (semigroup property)
3. limx→0+ ‖Txξ − ξ‖ = 0 ∀ξ ∈ X. (strong continuity property)
Definition 12. Let x be a time-like variable, C ∈ L(X, Y ) be the observation oper-
ator. For all x > 0, let Ψx ∈ L(X,L2 ([0, x];Y )) be the output map operator for the
47
system (3.2) such that,
(Ψxξ(0)) (x) =
CTxξ(0) ∀ x ∈ [0, x],
0 ∀ x > x.
(3.3)
Definition 13. Let time-like variable x > 0 and T be the strongly continuous semi-
group on space X with the generator A : D(A) → X and C ∈ L(X, Y ) be the
observation operator. The pair (C,A) is exactly observable in x if Ψx is bounded
from below.
The above definition of exact observability of the pair (C,A) is equivalent to the fact
that there exists kx > 0 such that,
∫ x
0
‖Ψxξ(0)‖2 dx ≥ k2x ‖ξ(0)‖2 ∀ ξ(0) ∈ X. (3.4)
Definition 14. Pair (C,A) as defined above is final state observable in time-like
interval x if there exists a constant kx > 0 such that,
‖Ψxξ(0)‖ ≥ kx‖Txξ(0)‖ ∀ ξ(0) ∈ X. (3.5)
Note 1: For x → 0 and given that k0 > 0, then using strong continuity of operator
semgigroup T we can see that definitions in equation (3.4) and (3.5) converge.
Lumer-Phillips Theorem:
Let A : D(A) → X be an unbounded operator on a Hilbert space X. Then the
following two assertions are equivalent.
1. A is maximally dissipative.
2. A is the generator of a contraction semigroup (Tx)x≥0, i.e. ‖Tx‖ ≤ 1 for all
x > 0.
48
Definition 15. Let x ∈ [c, d) for all c, d ∈ R and d > c then xm, for all m ∈ Z =
0 ∪ Z+, represents x over mth iteration over the interval [c, d) .
The idea of iteration is also illustrated in Figure. 3.2 for a rectangular domain Ω2.
Further, without loss of generality, let s ∈ [0, π/4], A : D(A)→ X be an unbounded
differential operator matrix given as,
A =
0 1
− ∂2
∂s20
, (3.6)
such that,
X = H1ΓT
(0,π
4
)× L2
(0,π
4
), (3.7)
D(A) =
[f ∈ H2
(0,π
4
)∩H1
ΓT
(0,π
4
)| dfds
(0) = c2
]×H1
ΓT
(0,π
4
), (3.8)
where,
H1ΓT
(0,π
4
)=f ∈ H1
(0,π
4
)| f(0) = c1
, (3.9)
and c1, c2 are constants (coming from Cauchy data at a particular point on ΓT ) and
X is a Hilbert space with scalar product given by,
⟨ q1
q2
,
p1
p2
⟩ =
∫ π4
0
dq1
ds(s)
dp1
ds(s)ds
+
∫ π4
0
q1(s)p1(s)ds+
∫ π4
0
q2(s)p2(s)ds. (3.10)
It can be seen that D(A∞) is dense in X.
Note 2: The state operator matrix A has two positive definite operators on anti-
diagonal, this indicates that A has both positive and negative eigenvalues. As given
above, the state operator matrix does not generate a strongly continuous semigroup.
The existence of exponential of such an operator, under certain conditions, is studied
49
a1
a2
a3
a4
a1
a2
a3
a4
Figure 3.2: Idea of iterations over rectangular domain Ω2.
in the following subsections.
3.2.2.1 Change of Variables
We propose to write down the Laplace equation in rectangular coordinates as given in
system (5.1) as a first order state equation by introducing two new auxiliary variables
ξ1, ξ2 as follows, ξ1(x, y) = u(x, y),
ξ2(x, y) =∂u
∂x,
(3.11)
and the resulting equation can now be written as,
∂ξ
∂x= Aξ, (3.12)
where,
ξ =
ξ1(x, y)
ξ2(x, y)
, A =
0 1
− ∂2
∂y20
. (3.13)
Using the new state variables ξ1 and ξ2 problem (5.1) can be written in an equivalent
form as,
50
Find ξ1(x, y) on ΓB:
∂ξ
∂x= Aξ in Ω2,
Cξ(x) = ξ1(x) = f(x) on ΓT ,
∂ξ1
∂y= g(x) on ΓT ,
(3.14)
with homogeneous Dirichlet/Neumann side boundaries.
3.2.3 Observer Design
Boundary value problem as given in system of equations (3.14) has a first-order state
equation in variable x and overdetermined data is available on ΓT . Before the intro-
duction of iterative observer equations, let us assume that left-hand boundary ΓL is
connected to right-hand boundary ΓR to have the notion of infinite time-like variable
x over the rectangular domain. The reason for having such an assumption is that
we are trying to develop an observer using space as time-like and hoping that this
observer will converge asymptotically in variable x. Let m be a non-negative integer
index of iteration over the domain Ω in horizontal direction. Let xm, as given in Def-
inition 15, represents x ∈ [0, a) for the m-th iteration over the interval [0, a). After
introducing iteration index m, now an observer-like algorithm can be developed as
follows,
Main result
Theorem 3. For consistent Cauchy data, boundary value problem given in (3.15)
asymptotically (m = 1, · · · ,∞) converges to the true solution of boundary value
51
problem (3.14).
∂
∂xξ(xm, y) = Aξ(xm, y)−KC(ξ(xm, y)− ξ) in Ω2,
∂
∂yξ1(xm, y) = g(x) on ΓT ,(∂2
∂x2+
∂2
∂y2
)ξ1(xm, y) = −KC(ξ(xm, y)− ξ) on ΓB,
ξ(xm, y) |initial= ξ(xm−1, y) in Ω2,
(3.15)
where “ ˆ ” represents estimated quantity and ξ(xm, y) |initial represents a bounded
estimate over the whole domain Ω2 at the start of m-th iteration. The algorithm starts
at index m = 1, which represents first iteration. ξ(x0, y) is initial guess at the start of
the first iteration over the whole domain Ω2. Any bounded initial guess ξ(x0, y) can
be chosen. For each subsequent iteration, the result of the previous iteration is used
as initial estimate as given in the last equation in (3.15). Third equation in (3.15)
is the assumption that Laplace equation is valid on the bottom boundary and this
provides necessary boundary condition required on ΓB. C is the observation operator
such that Cξ = ξ1 |ΓT . K is the correction operator chosen in such a way that state
estimation error on ΓT given by (Cξ(xm, y) − Cξ) converges to zero asymptotically
(m = 1, · · · ,∞).
3.2.4 Preliminary Analysis
Before moving to the proof of theorem (9), we note that the solution of the first-order
equation in system (3.14) leads to the concept of semigroup generated by unbounded
differential operator matrix A. We study the exponential of A using the functional
analysis framework from section 3.2.2.
52
3.2.4.1 Existence of Exponential of A
Theorem 4. Let n ∈ Z? (set of non zero integers), for A : D(A) → X (as given in
(3.13), (3.7) and (3.8)) there exists an infinite set of orthonormal eigenvectors (Φn)
and corresponding eigenvalues (λn). Furthermore, A generates a strongly continu-
ous semigroup for vectors
p1
p2
∈ X, if and only if, the decay rate of sequence
⟨ p1
p2
,Φn
⟩is greater than the growth rate of sequence eλnx for all n ∈ Z?.
Proof. Let,
Φn(y) = ρn
αnϕn(y)
βnϕn(y)
, (3.16)
be the orthonormal set of eigenvectors of operator A and λn be the eigenvalues such
that,
AΦn = λnΦn, (3.17) βnϕn
− ∂2
∂y2(αnϕn)
= λn
αnϕn
βnϕn
.
Assuming that αn, βn do not depend on y, second equation above suggests that we
are interested in finding the eigenfunctions of Laplacian operator−∂2
∂y2. This signifies
that unknown eigenfunctions ϕn ∈ C∞. Solving two equations in (3.17) gives,
λn =βnαn, (3.18)
ϕn(y) = C1 cos (λny) , (3.19)
where αn and βn depend on n. C1 and λn are chosen such that ϕn(y) in (3.19) forms
an orthonormal basis in L2(0, π
4
), with C1 = −
√8π, αn = 1 and βn = λn = 6 − 8n.
53
Finally an orthonormal set of eigenvectors can be formed in X with respect to norm
defined by (3.10) as,
Φn(y) = ρnφn(y) = ρn
αnϕn(y)
βnϕn(y)
, (3.20)
where, |ρn| =1∣∣√2βn∣∣ > 0, is a normalization factor. Now let us try to write semigroup
generated by operator matrix A can be written as an infinite series,
∑n∈Z?
eλnx
⟨ p1(y)
p2(y)
,Φn(y)
⟩Φn(y), ∀
p1
p2
∈ X. (3.21)
For x = 0 the above infinite series is clearly convergent, whereas for x → 0+ the
limit does not exist. Further, we note that above series expression (3.21) satisfies
identity and semigroup properties as given in Definition 11. However, it lacks strong
continuity, except if we assume that the projection terms in angle brackets above
decay faster than the growth rate of eλnx. This condition true for a wide range of
analytical functions that have a finite number of non-zero projections on the basis Φn.
This also reveals a historical fact about solving Cauchy problems for steady-state heat
equation that unique and stable solutions does not exist for non-smooth data [54].
Let the Hilbert space X, as given in equations (3.7) and (3.10), be composed of two
mutually exclusive parts as
X = X1 ⊕X2, (3.22)
where X1 satisfy conditions as stated above such that A forms a strongly continuous
semigroup and X1 and X2 both make the full space X. Thus with this additional
smoothness assumption equation (3.23) represents the strongly continuous semigroup
54
generated by operator matrix A.
Tx
p1(y)
p2(y)
=∑n∈Z?
eλnx
⟨ p1(y)
p2(y)
,Φn(y)
⟩Φn(y),∀
p1
p2
∈ X1.
This implies,
Tx
p1(y)
p2(y)
=∑n∈Z?
eλnxρn
(αn
⟨dp1
dy,dϕndy
⟩L2(0,π
4 )+ αn 〈p1, ϕn〉L2(0,π
4 )
+βn 〈p2, ϕn〉L2(0,π4 )
)Φn, ∀
p1
p2
∈ X1.
(3.23)
3.2.4.2 System Observability
Proposition 1. Let T be the strongly continuous semigroup generated by operator
matrix A under the assumptions as given in theorem 4. For any arbitrarily small
ε > 0 such that if |x− x| < ε, the pair (C,A) is final state observable (and further
exactly observable using Note 1 from section 3.2.2) in time-like interval |x− x| > 0
at a particular x, where C ∈ L(X, Y ) and Y = R.
Proof. Let ξ(0) ∈ X1 be the initial guess at x = 0, given by,
ξ(0) =
ξ1(0)
ξ2(0)
=
p1(y)
p2(y)
. (3.24)
Φn(y) for n ∈ Z? be an orthonormal basis in X. Let us first prove the final state
observability condition for a general mode Φn′ with corresponding eigenvalue λn′ as
follows,
55
For all Φn′ ∈ X and n′ ∈ Z?,
‖TxΦn‖X =
∥∥∥∥∥∑n∈Z?
eλnx 〈Φn′ ,Φn〉Φn
∥∥∥∥∥X
,
= eλnx ‖Φn‖X ,
= eλnx, (3.25)
also,
‖CTxΦn‖Y =
∥∥∥∥∥∑n∈Z?
eλnx 〈Φn′ ,Φn〉 CΦn
∥∥∥∥∥Y
,
= eλnx ‖CΦn‖Y ,
= eλnx|ρn|. (3.26)
Comparing equations (3.25) and (3.26) implies,
‖CTxΦn‖Y ≥ k?1 ‖TxΦn‖X , (3.27)
where k?1 > 0, if and only if,
k?1 ≤ |ρn|, (3.28)
for a particular choice of Φn there always exists k?1 such that final state observability
condition (3.5) is satisfied.
C ∈ L(X, Y ) is a linear boundary observation operator. Now let ξ(0) =∑
n∈Z? γnΦn
where γn are projection terms whose decay rate is greater than the growth rate of
eλnx with λn as eigenvalues of A corresponding to eigenvectors Φn. Clearly∑
n∈Z? γn
and∑
n∈Z? ρn are bounded from above, hence,
‖CTxξ(0)‖Y ≥ k?2 ‖Txξ(0)‖X ∀ ξ(0) ∈ X1, (3.29)
where k?1, k?2 both are independent of x. Further, using Note 1, for arbitrarily small
56
time-like interval ε pair (C,A) is exactly observable.
3.2.5 Convergence Analysis
After establishing the concept of exponential of A, under certain conditions, and the
fact that pair (C,A) is final state and exact observable, we are all set to prove the
main result.
3.2.5.1 Proof of the Main Result
Proof. Let us define state estimation error e(xm, y) as the difference of true state
ξ(x, y) from the one estimated ξ(xm, y),
e = ξ − ξ =
e1(xm, y)
e2(xm, y)
=
ξ1(xm, y)− ξ1(x, y)
ξ2(xm, y)− ξ2(x, y)
. (3.30)
Solution of the boundary value problem (3.14) with consistent boundary data provides
u = ξ1 over the whole domain Ω2. Boundary value problem for the state estimation
error can be given by subtracting problem (3.14) from the state observer equations
(3.15) as follows,
For m ≥ 1, find e(xm, y) = (ξ(xm, y)− ξ(x, y)) ∈ Ω2:
∂
∂xe(xm, y) = (A−KC)e(xm, y) in Ω2,
∂
∂ye1(xm, y) = 0 on ΓT ,(∂2
∂x2+
∂2
∂y2
)(ξ1(xm)− h(x)
)= −KCe(xm) on ΓB,
e(xm, y) |initial= e(xm−1, y) in Ω2.
(3.31)
57
Here h(x) is the true analytical solution on ΓB using consistent Cauchy data. Further
using the assumption that Laplace equation is valid on ΓB, the above system of error
dynamic equation can also be written in an equivalent form as,
For m ≥ 1, find e(xm, y) ∈ Ω2:
∂
∂xe(xm, y) = (A−KC)e(xm, y) in Ω2\ΓT ,
∂
∂ye(xm, y) = 0 on ΓT ,
e(xm, y) |initial= e(xm−1, y) in Ω2.
(3.32)
First equation in (3.32) is a system of ODEs in variable x and solution to this system
has to do with the exponential or the semigroup generated by operator matrixA−KC.
Let us denote this semigroup with S. Then solution to above system of ODEs can be
written as,
e(xm, .) = Sxm e(x0, .) m ≥ 1, (3.33)
for a particular iteration index m, xm is x ∈ [0, a) over m-th iteration. Given that un-
der certain conditions semigroup generated by A is strongly continuous, the observer
gain K can be chosen in a way that A−KC is dissipative. Then Sxm will decay expo-
nentially and state estimation error e(xm, .), for a number of iterations over the whole
domain, asymptotically converges to zero for any bounded initial value of e(x0, .).
3.2.5.2 Existence of Observer Gain K
Let the Hilbert space X as given in equations (3.7) and (3.10) be composed of two
mutually exclusive parts as,
X = X1 ⊕X2, (3.34)
58
where X1 satisfy conditions as stated in Theorem 4 such that A forms a strongly
continuous semigroup, and X1 and X2 both make the full space X. Following theorem
provides conditions on the existence of operator gain K.
Theorem 5. Under conditions as stated in theorem 4, let A as given in equation
(3.13) be the generator of a strongly continuous semigroup, C ∈ L(X, Y ) be an
observation operator and Y = R), then the following assertions are equivalent.
1. There exists a positive definite self-adjoint operator product KC ∈ L(X) where
K ∈ L(Y,X) such that A−KC generates a maximally dissipative semigroup.
2. There exists arbitrarily small ε > 0 such that if ‖x− x‖ < ε then pair (C,A) is
exactly observable in time-like interval ε.
Proof. Given self-adjoint positive definite operator product KC ∈ L(X), let us denote
by S and T the semigroups generated, under certain conditions, by A − KC and A
respectively.
1⇒ 2 :
Assume S is dissipative, let us show the following observability inequality, that is,
there exists x, kx > 0 such that,
∫ x
0
‖CTxe0‖2 dx ≥ k2x ‖e0‖2 ∀e0 ∈ X1, (3.35)
A is densely defined so the above inequality is enough to prove exact observability
for e0 ∈ D(A). Given e0 ∈ D(A), e(x) = Sxe0 presents the unique solution of,
∂e
∂x= (A−KC)e(x),
e(0) = e0.
(3.36)
59
Multiplying first equation in (3.36) by e(x),
1
2
d
dx‖e(x)‖2 = Re
⟨∂e
∂x, e(x)
⟩,
= Re 〈(A−KC)e(x), e(x)〉 , (3.37)
in this part we assume that A−KC is m-dissipative,
d
dx‖e(x)‖2 ≤ 0, (3.38)
Let e(x) = γ(x) + ζ(x) such that γ = Txe0 is the solution of,
∂γ
∂x= Aγ(x),
γ(0) = e0,
(3.39)
and ζ is the solution of,
∂ζ
∂x= Aζ(x)−KCe(x),
ζ(0) = 0.
(3.40)
Further we have that KC is positive definite,
0 ≤ Re 〈KCγ,KCγ〉 ≤ ‖KCγ(x)‖2X . (3.41)
Combining equations (3.38) and (3.41),
d
dx‖e(x)‖2 ≤ ‖KCγ(x)‖2
X , (3.42)
60
integrating both sides,
1
2
∫ x
0
d
dx‖e(x)‖2 dx ≤ 1
2
∫ x
0
‖KCγ(x)‖2X dx,
2(‖e(x)‖2 − ‖e0‖2) ≤ ∫ x
0
‖KCγ(x)‖2X dx,
(3.43)
finally we have,
k ‖e(x)‖2X1≤
∫ x
0
‖Cγ(x)‖2Y dx, (3.44)
where k =2
‖K‖2> 0 is independent of x. For arbitrarily small time-like interval
x− 0 = x = ε, above inequality is same as observability inequality (3.35).
2⇒ 1 :
We have e(x) = γ(x) + ζ(x), where γ is the solution of open loop system (3.39)
and e(x) is the solution of closed loop feedback system (3.36), we have that,
〈(A−KC)γ(x), γ(x)〉 ≥ 〈(A−KC)e(x), e(x)〉 , (3.45)
multiplying with −1 and using inequality (3.37),
〈(KC)γ(x), γ(x)〉 ≤ −1
2
d
dx‖e(x)‖2 + 〈(A)γ(x), γ(x)〉 , (3.46)
integrating both sides from 0 to x and with a positive α > 1 such that,
∫ x
0
〈(KC)γ(x), γ(x)〉 dx ≤ α(‖e0‖2 − ‖e(x)‖2)+
∫ x
0
〈Aγ(x), γ(x)〉 dx, (3.47)
61
Further we can write,
Re 〈Aγ, γ〉 = Re
⟨∂γ
∂x, γ
⟩,
=1
2
d
dx‖e(x)− ζ(x)‖2 ,
≤ 1
2
d
dx‖e(x)‖2 +
1
2
d
dx‖ζ(x)‖2 ,
≤ d
dx‖e(x)‖2 +
1
2
d
dx‖γ(x)‖2 , (3.48)
given that, under certain conditions, A generates a strongly continuous semigroup,
there exists β > 1 such that,
Re 〈Aγ, γ〉 ≤ βd
dx‖e(x)‖2 , (3.49)
integrating both sides,
∫ x
0
Re 〈Aγ, γ〉 dx ≤ −β(‖e0‖2 − ‖e(x)‖2) , (3.50)
combining above inequality with (3.51),
∫ x
0
〈(KC)γ(x), γ(x)〉 dx ≤ (α− β)(‖e0‖2 − ‖e(x)‖2) , (3.51)
α and β can be chosen appropriately large, let us take α− β = 2, we have
∫ x
0
〈KCγ(x), γ(x)〉 dx ≤ 2(‖e0‖2 − ‖e(x)‖2) ,∫ x
0
〈Cγ(x), Cγ(x)〉 dx ≤ 2
‖K‖(‖e0‖2 − ‖e(x)‖2) ,∫ x
0
‖Cγ(x)‖2Y dx ≤
2
‖K‖(‖e0‖2 − ‖e(x)‖2) , (3.52)
62
now using observability inequality (3.35) we have,
k2 ‖e0‖2 ≤ 2
‖K‖(‖e0‖2 − ‖e(x)‖2) ,
‖e(x)‖2 ≤(
1− 1
‖K‖
)‖e0‖2 , (3.53)
where
(1− 1
‖K‖
)< 1 if ‖K‖ > 1.
In the following section, the iterative observer is implemented numerically us-
ing fictitious points on the estimated solution boundary. Numerical results are also
presented.
3.2.6 Numerical Implementation
For numerical implementation, first order state equation given in (3.14) can be dis-
cretized in variable x using forward Euler as follows,
ξ = Aξ,ξn+1 − ξn
4x= Aξn,
ξn+1 = (I + (4x)A)ξn, (3.54)
here I is the identity matrix, n is the discrete index for variable x and 4x is the step
size along x after discretization. Further, equation (3.54) is discretized in variable y
using second-order accurate centered finite difference schemes to discretize the first-
and second-order derivative terms. The Cauchy data is available on the top bound-
ary, however, for the bottom boundary ΓB there is no data available and we assume
that the Laplace equation is valid on this boundary as given in problem (5.71). Nu-
merically, this condition can be implemented using fictitious points along the inner
boundary, as explained in following section.
63
3.2.6.1 Boundary Condition on ΓB
As stated above, a first-order state equation can be thought of as an ODE with respect
to variable x. Solution of this ODE is the state ξ over the whole vertical line, that is,
(y |ΓB , y |ΓT ). This can be thought of as a 2D Laplace equation that has been split
into a series of 1D state equations. To solve this 1D state equation in variable x, an
initial condition over the whole interval (y |ΓB , y |ΓT ) and boundary conditions on ΓB
and ΓT are required. Any initial guess can be chosen as (A−KC) will be dissipative
and any initial guess dies out. Two bounday conditions on ΓT are available, that is,
the measurement data and the available Neumann boundary data on ΓT . However,
on ΓB, it is assumed that Laplace equation is satisfied. That is,
∂2u
∂x2= −∂
2u
∂y2on ΓB. (3.55)
Equation (3.55) contains a second-order derivative in variable y. To discretize this
second derivative using a second-order accurate centered finite difference discretiza-
tion scheme on ΓB, there needs to be a fictitious point further outside the boundary
ΓB as shown in figure 3.3 [60].
i=0;n
i=1;n
i=2;n
i=0;n− 1
i=1;n− 1
i=2;n− 1
i=0;n + 1
i=1;n + 1
i=2;n + 1
n
i
ΓB
ΓT
Figure 3.3: Domain Ω2 after discretization and fictitious points outside ΓB, indexi = 0 represents fictitious points.
The second equation in (3.54), after full discretization, can be written as,
(ξ2)n+1i − (ξ2)ni4x
= −(ξ1)ni+1 − 2(ξ1)ni + (ξ1)ni−1
(4y)2, (3.56)
64
here i is the discrete index and (4y) is the step size along variable y such that i = 1
on the bottom boundary ΓB. Equation (3.56) on the bottom boundary ΓB can be
written as,
(ξ2)n+11 − (ξ2)n14x
= −(ξ1)n2 − 2(ξ1)n1 + (ξ1)n0(4y)2
, (3.57)
index i = 0 represents fictitious point and taking out this fictitious point gives,
(ξ1)n0 = 2(ξ1)n1 − (ξ1)n2 −(4y)2
4x
(ξ2)n+11 − (ξ2)n1
. (3.58)
(ξ1)n1 , (ξ1)n2 , (ξ2)n1 and (ξ2)n+11 are given by the initial guess of the states over the whole
domain. The algorithm is run for a number of iterations along x by using the solution
of the previous iteration as a guess for the next until the final convergence is achieved.
In the following subsection observer is presented in semi-discrete form and fictitious
points method is used to tackle the boundary condition on ΓB.
3.2.6.2 Observer in Semi-Discrete Form
In the following, state observer presented in system of equations (5.71) is discretized
only in variable x for simplicity.
ξn+1,m = (I + (4x)A)ξn,m −KC(ξn,m − ξn) in Ω2,
∂
∂yξn,m1 = gn(x) on ΓT ,
1
24x
(ξn+1,m
1 − ξn−1,m1
)= − ∂2
∂y2ξn,m1
−KC(ξn,m − ξn) on ΓB,
ξn,m |initial= ξn,m−1 in Ω2,
(3.59)
again, here ˆ represents estimated quantity and m is the index of iteration over the
rectangular domain. ξm,n |initial represents estimate for particular value of index n at
the start of mth iteration. The algorithm starts at m = 1 and index m = 0 represents
65
the raw data over the whole mesh before start of the algorithm. At the start of the
algorithm, any initial guess can be chosen over the whole domain and then solution
of the previous iteration is used as a guess for subsequent iterations. Fictitious points
are computed using formula given in (3.58) and are used in third equation in (3.59)
to discretize the second-order derivative with respect to y on ΓB. An important point
to note here is that the true Neumann boundary condition is applied on the outer
boundary and operators A,K and C are continuous in variable y.
3.2.6.3 Algorithm Step-by-Step
• Step 1 : Initialize mesh over the whole domain Ω2 with ξm=0 = ξ0.
• Step 2 : For m = 1, start at a particular value of x,
– Compute the fictitious point value (ξ1)n,m=10 for particular value of n using
equation (3.58).
– Solve system of equations (3.59) to find estimate ξn+1,m over a particular
vertical line.
– Repeat the process of finding fictitious point from equation (3.58) and
solving system of equations (3.59) for all n until one iteration on interval
of length a on the rectangular domain shown in Figure. 3.1 is complete.
• Step 3 : Repeat Step 2 for m ≥ 2 using result of (m− 1)th iteration as a guess
for mth iteration until convergence is achieved. That is, ‖ξ1 − ξm1 ‖ΓT < ε.
66
3.2.6.4 State Estimation Error and Computation of the Ob-
server Gain
State error boundary value problem in semi-discrete form can be written as,
For m ≥ 1, find en,m = (ξn,m − ξn) ∈ Ω2:
en+1,m = (I + (4x)A−KC) (ξn,m − ξn) in Ω2\ΓT ,
∂
∂yen,m1 =
∂
∂y
(ξn,m1 − ξn1
)= 0 on ΓT ,
en,m |initial= en,m−1 in Ω2.
(3.60)
Finally the state error difference equation after full discretization can be written as,
en+1,m = (I + (4x)A−KC) en,m for m ≥ 1, (3.61)
here A,K and C are discrete versions of operators A,K and C respectively and e is
the state estimation error after full discretization. Given (I + (4x)A) and observa-
tion matrix C, gain matrix K can be computed using Ackermann’s formula for pole
placement in Matlab such that eigenvalues of (I + (4x)A−KC) are inside the unit
circle on the complex plane [61].
3.2.7 Results and Simulations
For all numerical and analytical solutions in this section, a rectangle domain Ω =
(0, a)× (0, b) with a = 2π and b = 12
is considered. To validate the observer approach
a number of examples are presented as follows.
3.2.7.1 Example 1: Homogeneous Neumann Side Boundaries
Consider the boundary value problem in a rectangular domain with homogeneous
Neumann side boundaries as shown in Figure 3.4. This problem can be solved using
67
Ω2
u = cos(2x)
uy = 0
ux = 0 ux = 0
Figure 3.4: Two dimensional rectangle domain with homogeneous Neumann sideboundaries, Example 1.
separation of variables and solution is given as,
u(x, y) =cosh(4π(y − b)/a)
cosh(4πb/a)cos(4πx/a), (3.62)
To validate the observer-based approach this analytical solution given in (3.62) along
with homogeneous Neumann boundary condition is used as Cauchy data on the top
boundary ΓT . Using this Cauchy data state observer algorithm is run for a number
of iterations to recover the unknown boundary data on the bottom boundary ΓB.
Figure 3.5 shows the comparison of exact solution and the one recovered by using
Cauchy data on ΓB and observer algorithm.
0 1 2 3 4 5 6−1
−0.5
0
0.5
1
x−axis
Solu
tion u
Comparison of exact and observer solution on ΓB
Exact solution
Observer soluton
Figure 3.5: Comparison of exact and observer constructed solution on the bottomboundary ΓB.
68
3.2.7.2 Example 2: Homogeneous Dirichlet Side Boundaries
Ω2
u = sin(2x)
uy = 0
u = 0 u = 0
Figure 3.6: Two dimensional rectangle domain with homogeneous Dirichlet sideboundaries, Example 2.
Consider the boundary value problem with homogeneous Dirichlet side boundaries
as shown in Figure 3.6. Analytical solution is given as,
u(x, y) =cosh(4π(y − b)/a)
cosh(4πb/a)sin(4πx/a), (3.63)
Now using this analytical solution along with homogeneous Neumann boundary con-
dition on the top boundary ΓB, observer algorithm is run for a number of iterations to
recover the unknown Dirichlet boundary data on ΓB. Figure 3.7 shows the comparison
of the exact and observer constructed solution on ΓB.
0 1 2 3 4 5 6−1
−0.5
0
0.5
1
x−axis
Solu
tion u
Comparison of exact and observer solution on ΓB
Exact solution
Observer soluton
Figure 3.7: Comparison of exact and observer constructed solution on the bottomboundary ΓB.
69
3.2.7.3 Example 3: Linear Combinations of Example 1 and
2
It is easy to see that any linear combination of above two example problems can be
solved using the observer-based technique. In other words any Dirichlet boundary
data on ΓB that can be represented as a trigonometric Fourier series can be recov-
ered using the observer-based approach given homogeneous Dirichlet, Neumann or
Robin kind of side boundaries. The requirement of such homogeneous side bound-
aries suggest that there are no active sources on the side boundaries which is indeed
the case for many applications like electrocardiography (ECG) where the objective is
to find a heart electric potential which is deep inside the body from the ECG data
available only on a limited part of body torso [28, 29]. The observer-based approach
is the preferred technique in cases where there is no information available on the side
boundaries. Figure 3.8 compares the exact solutions in different test cases to the one
obtained by using the observer. Numerical solution was achieved using homogeneous
Neumann boundaries on ΓL,ΓR and ΓT and non zero Dirichlet data on ΓB. The
observer solution was constructed using only the Cauchy data on ΓT .
The design of a dynamical systems inspired technique like observer for a steady-
state boundary value problem is challenging and the idea to use one of the space
variables as a time-like variable has not been considered before. Different from stan-
dard approaches to tackle this problem, an iterative observer is constructed in infinite
dimensional setting on a rectangle domain without introducing an extra time vari-
able. Laplace equation is presented as a first-order state equation with state operator
matrix. Conditions for the existence of exponential generated by this state operator
matrix are provided. Further the conditions for the existence of observer gain are
detailed. Numerical results are provided for different example test cases.
In the following section, first, it is established that boundary data estimation
problems for Laplace equation are highly sensitive to noise in the available boundary
70
0 1 2 3 4 5 6
−1.5
−1
−0.5
0
0.5
1
1.5
x−axis
Solu
tion u
Comparison of exact and observer solution on ΓB
Exact solution
Observer soluton
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
x−axis
Solu
tion u
Comparison of exact and observer solution on ΓB
Exact solution
Observer soluton
0 1 2 3 4 5 6
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x−axis
So
lution u
Comparison of exact and observer solution on ΓB
Exact solution
Observer soluton
0 1 2 3 4 5 6
−1
−0.5
0
0.5
1
1.5
x−axis
So
lution u
Comparison of exact and observer solution on ΓB
Exact solution
Observer soluton
Figure 3.8: Comparison of exact and observer constructed solution on the bottomboundary ΓB.
71
data. Further, to tackle such ill-posed problems, a robust iterative counterpart of the
iterative observer is developed.
3.3 Robust Iterative Algorithm for Boundary Estimation
In this section, a robust iterative observer is developed to solve ill-posed boundary
data estimation problem for Laplace equation, on an annulus domain, with noisy
boundary data. The problem is formulated on an annulus domain and an optimal
iterative algorithm is developed to solve the problem.
3.3.1 Problem Statement on an Annulus Domain
Let Ω1 be an annulus domain in R2 with two boundaries (Γin,Γout) and a hole as
shown in Figure 3.9 such that Ω1 = Ω1 ∪Γin ∪Γout and Γin ∩Γout = ∅. The boundary
estimation problem for Laplace equation in polar co-ordinates over Ω1 is given as,
Find u(θ) ∈ Γin :
4u = r2∂
2u
∂r2+ r
∂u
∂r+∂2u
∂θ2= 0 in Ω1,
∂u
∂r= g(θ) on Γout,
u(θ) = f(θ) on Γout,
(3.64)
where g = g + ω and f = f + υ represent perturbed Cauchy data with ω(η1, σ1)
and υ(η2, σ2) as additive white Gaussian noise with mean values η1, η2 and variances
σ1, σ2 respectively.
3.3.2 Problem Reformulation
In the subsection, we propose to rewrite this ill-posed boundary value problem in
state-space-like representation using one of the space variables as a time-like variable.
72
Ω1
Γin
Γout
Figure 3.9: Annulus domain Ω1 with inner boundary Γin and outer boundary Γout.
Let us introduce two auxiliary variables ξ1(r, θ) and ξ2(r, θ) such that,
ξ1(r, θ) = u(r, θ),
ξ2(r, θ) =∂u
∂θ.
(3.65)
Further, rewriting the Laplace equation as a first-order state equation gives,
ξ = Aξ, (3.66)
with,
ξ(r, θ) =
ξ1(r, θ)
ξ2(r, θ)
, (3.67)
A =
0 1
−r2 ∂2
∂r2− r ∂
∂r0
, (3.68)
where “ ˙ ” represents partial derivative with respect to θ. Now problem (3.64) can
be reformulated as,
73
Find ξ1(r, θ) ∈ Γin : ξ = Aξ in Ω1,
∂ξ1
∂r= g(r, θ) on Γout,
ξ1(r, θ) = f(r, θ) on Γout.
(3.69)
It is important to reemphasize that an analytical solution of the above problem exists
for smooth Cauchy data (f, g) only, and arbitrarily small noise in the Cauchy data
can destroy the solution [21,25].
In the following, a well-posed forward problem is presented. Solution of this well-
posed forward problem will be used to test the accuracy of the proposed optimal
algorithm.
Find ξ1 |Γout= f(r, θ) :
ξ = Aξ in Ω1,
∂ξ1
∂r= g(r, θ) on Γout,
ξ1(r, θ) = h(r, θ) on Γin,
(3.70)
where g and h are noise free smooth boundary conditions. Solution of the above
forward problem provides Cauchy data (f, g) on Γout. Now the inverse problem is
to estimate h(r, θ) using noise-free (f, g) and noisy (f , g) Cauchy data. In the
following sections problem (3.69) is discretized using finite difference discretization
and an optimal estimator is developed.
3.3.3 Derivation of Optimal MSE Minimizer Algorithm
Before proper algorithm formulation, let us discretize problem (3.69) using finite
difference discretization.
74
3.3.3.1 Numerical Discretization
Forward Euler-like approximation is used to discretize first derivative along θ as,
∂ξ
∂θ≈ 1
4θ[ξn+1 − ξn
], (3.71)
where n is discrete positive integer index for variable θ and 4θ is a small step size.
State equation, after discretization in variable θ, can be written as,
ξn+1 = (I + (4θ)A) ξn, (3.72)
here I is 2× 2 identity matrix. Second equation in (3.72) can be written as,
(ξ2)n+1 = (ξ2)n +4θ(−r2 ∂
2
∂r2− r ∂
∂r
)(ξ1)n , (3.73)
Next to discretize in variable r, following second-order accurate finite difference dis-
cretization approximations are used to discretize first and second order derivative in
above equation.
∂
∂r(ξ1)n ≈ 1
2(4r)[(ξ1)ni+1 − (ξ1)ni−1
], (3.74)
∂2
∂r2(ξ1)n ≈ 1
(4r)2
[(ξ1)ni+1 − 2 (ξ1)ni + (ξ1)ni−1
], (3.75)
i is the discrete index for variable r, i = 1 on Γin and (4r) is a reasonably small step
size along r. Now from (3.73),
1
4θ[(ξ2)n+1
i − (ξ2)ni]
=−r2
i
(4r)2
[(ξ1)ni+1 − 2(ξ1)ni + (ξ1)ni−1
]+−ri
2 (4r)[(ξ1)ni+1 − (ξ1)ni−1
],
(3.76)
75
rewriting above equation on inner boundary Γin, that is, at i = 1 gives,
1
4θ[(ξ2)n+1
1 − (ξ2)n1]
=−r2
1
(4r)2 [(ξ1)n2 − 2(ξ1)n1 + (ξ1)n0 ] +−r1
2 (4r)[(ξ1)n2 − (ξ1)n0 ] .
(3.77)
As state earlier, index i = 1 represents point on Γin and thus i = 0 represents fictitious
points outside domain Ω1 and close to inner boundary Γin, as shown in Figure 3.10.
Taking out fictitious points (ξ1)n0 from the above equation gives,
(ξ1)n0 =1
r21 −
4r2r1
[−(r2
1 +(4r)
2r1
)(ξ1)n2 + 2r2
1(ξ1)n1
]− 1
r21 −
4r2r1
[(4r)2
4θ
(ξ2)n+11 − (ξ2)n1
].
(3.78)
It is important to reemphasize here that our objective is to develop an optimal al-
gorithm which runs iteratively using θ as a time-like variable. Right-hand side of
the equation (3.73) only depends on the variable r. Thus fictitious point (ξ1)n0 for a
particular n can be found uniquely as it depends on (ξ2)n+11 . The idea of assuming
fictitious points on the inner boundary can be understood as an assumption that
Laplace equation is satisfied on the inner boundary. That is, we are trying to solve a
2D problem line by line assuming that,
∂2ξ1
∂θ2= −
(r∂ξ1
∂r+ r2∂
2ξ1
∂r2
)on Γin, (3.79)
which is an applicable boundary condition on Γin in the current state space-like
setup. At the start of the algorithm, (ξ1)n1 , (ξ1)n2 , (ξ2)n1 and (ξ2)n+11 come from the
initial guess over the domain Ω1. These fictitious points are used in optimal estimator
equations to tackle the boundary condition on the inner boundary, whereas Neumann
boundary condition g is used to tackle the boundary condition on outer boundary.
76
Γout
Γini = 0, n
i = 0, n + 1
i = 0, n− 1
n
i
Figure 3.10: Fictitious points close to inner boundary Γin, with index i = 0.
3.3.3.2 Optimal Estimator Derivation
After full numerical discretization the state equation can be written as,
ξn+1 = (I + (4θ)A) ξn in Ω1, (3.80)
along with Neumann boundary condition∂ξ1
∂r= g(r, θ) on Γout and boundary condi-
tion given by equation (3.79) on Γin. A is the discrete version of A and I is an identity
matrix. Both A and I are square matrices of the same size with dimension depending
on number of discretization points along r. Next Dirichlet boundary condition on
Γout can be written as a measurement or observation equation in discrete form as,
yn = Cξn + υn = fn + υn, (3.81)
where C is a discrete observation matrix such that Cξn = (ξ1)n |Γout , fn is the Dirichlet
boundary condition at θn = n × 4θ along with measurement noise υn. An impor-
tant condition to be satisfied for a system in state-space-like representation (3.80)
and (3.81) is the system observability. Observability is equivalent to the existence of
solution in linear dynamical theory [6, 62]. The Kalman rank condition for observ-
77
ability is given in Theorem 1 and also highlighted in the following. Observability is
the measure of how well the internal states can be recovered from the measurements
of output (observations). Let α be a positive integer constant, numerically system
(C,A′), with C as discrete observation matrix and A′ as discrete state operator matrix
of size α× α, is called observable if,
rank(T ) = rank
C
CA′
CA′2
...
CA′α−1
= α. (3.82)
In our case A′ = (I + (4θ)A). Let the covariance matrices of process and measure-
ment noise be stationary over θ and are given by,
Q = E[ωn(ωn)T
], (3.83)
R = E[υn(υn)T
], (3.84)
where Q is of the size of (I + (4θ)A) and R is a 1 × 1 matrix. Let ξn be the
estimate of true state ξn. The difference of estimated and true state can be written
as en =(ξn − ξn
). It is important to consider the ability of the estimator to predict
the states over a period of time-like variable θ, hence a feasible metric is the expected
value of positive definite error functional(ξn − ξn
)2
, given by,
ε(θ) = E[(en)2
]= E
[en(en)T
]= P n, (3.85)
where P n is the error covariance matrix at θ = n×4θ.
Proposition 1. Let pair (C,A′), with A′ = (I + (4θ)A), be observable as given
78
in equation (3.82) and n be a discrete non-negative integer index for variable θ. As
n → ∞ (that is, as θ → ∞, representing iterations over annulus domain), follow-
ing two step algorithm minimizes the mean square error functional given in equation
(3.85).
Prediction-step
¯ξn = A′ξn−1 in Ω1,
P n = A′P n−1A′T +Q,
¯ξn=0 = P n=0 = 0,
(3.86)
with Neumann boundary condition g on Γout and fictitious points
close to Γin given by equation (3.78), taking care of boundary
conditions.
Correction-step ξn =
¯ξn +Kn
(yn − C ¯
ξn),
Kn = P nCT (CP nCT +R)−1,
P n = (I −KnC)P n,
(3.87)
where ξn is an estimate of ξn,¯ξn is the prior-estimate of ξn, P n is the prior-estimate
of state error covariance matrix P n given by equation (3.85). Q and R as given in
equations (3.83) and (3.84) and Kn is the gain matrix.
Remark 1. Before going to the proof, it is important to remark that the idea of
iterations (n→∞) replaces the asymptotic time in standard Kalman filter algorithm.
79
Proof. Let us re-write equation (3.85) as,
P n = E[(ξn − ξn)(ξn − ξn)T
]. (3.88)
Let us assume the prior estimate of ξn is called¯ξn and was obtained by the knowledge
of the system (prediction). The correction equation can be written using the prior
estimate with measurement data as follows,
ξn =¯ξn +Kn
(yn − C ¯
ξn). (3.89)
Kn is the gain matrix which will be derived in a moment. The term(yn − C ¯
ξn)
is
the innovation or measurement residual. Substituting equation (3.81) into (3.89),
ξn =¯ξn +Kn
(Cξn + υn − C ¯
ξn). (3.90)
Now substituting equation (3.90) into (3.88) gives,
P n = E[ (
(I −KnC)(ξn − ¯ξn)−Knυn
) ((I −KnC)(ξn − ¯
ξn)−Knυn)T]
, (3.91)
taking the error of prior estimate(ξn − ¯
ξn)
as uncorrelated to measurement noise
gives,
P n = (I −KnC) E[(ξn − ¯
ξn)(ξn − ¯ξn)T
](I −KnC)T +Kn E
[υn(υn)T
](Kn)T ,
(3.92)
now substituting equations (3.84) and (3.88) into above equation yields,
P n = (I −KnC)P n(I −KnC)T +KnR(Kn)T . (3.93)
80
Above equation is the error covariance update equation and the diagonal contains
mean-square error as shown,
P nn =
E[en−1(en−1)T ] E[en(en−1)T ] E[en+1(en−1)T ]
E[en−1(en)T ] E[en(en)T ] E[en+1(en)T ]
E[en−1(en+1)T ] E[en(en+1)T ] E[en+1(en+1)T ]
, (3.94)
the trace of above matrix is the mean-square error and it is to be minimized with
respect to Kn. Rewriting equation (3.93) gives,
P n = P n −KnCP n − P nCT (Kn)T +Kn(CP nCT +R)(Kn)T , (3.95)
using the fact that trace of a matrix is equal to trace of its transpose, it can be seen
that,
T [P n] = T [P n]− 2T [KnCP n] + T [Kn(CP nCT +R)(Kn)T ], (3.96)
where T [P n] is the trace of covariance matrix P n. Differentiating with respect to Kn,
d
dKnT [P n] = −2CP n + 2Kn(CP nCT +R), (3.97)
setting the derivative equal to zero gives,
(CP n)T = Kn(CP nCT +R), (3.98)
solving for Kn gives,
Kn = P nCT (CP nCT +R)−1. (3.99)
Above equation is the optimal estimator gain equation, which minimizes the mean
81
square state estimation error. Substituting above equation into equation (3.93) yields,
P n = P n − P nCT (CP nCT +R)−1CP n,
= P n −KnCP n,
= (I −KnC)P n, (3.100)
which is the update equation for state error covariance matrix with optimal gain
Kn. Equations (3.89), (3.99) and (3.100) develop an estimate of variable xn. State
prediction is achieved using state equation,
¯ξn+1 = A′ξn, (3.101)
along with Neumann boundary condition g on Γout and equation (3.79) on Γin. To
complete the recursion it is important to find an equation which projects state error
covariance matrix into next θ-step, θ + 1. This is achieved by forming an expression
for the prior error (prediction error).
en+1 = ξn+1 − ¯ξn+1,
= A′en + ωn. (3.102)
Now extending equation (3.85) to n+ 1 gives,
P n+1 = E[en+1(en+1)T
],
= E[(A′en + ωn)(A′en + ωn)T
], (3.103)
en and ωn are uncorrelated as en is the error accumulated in previous n steps and ωn
82
is the process error for n-th step. This implies,
P n+1 = E[A′en(A′en)T
]+ E
[ωn(ωn)T
],
= A′P nA′T
+Q. (3.104)
This has completed the optimal estimator recursive loop.
3.3.4 Numerical Results
For all numerical examples presented in this section an annular domain with r ∈
[0.5, 1] is considered. Number of states ξi are chosen such that Kalman rank condition
given in equation (3.82) is satisfied. Pair (C,A) is observable for Nr ≤ 8, where Nr
is the number of discretization points along r. The results presented in this section
were obtained in just 3 to 4 iterations over the domain Ω1. Running the algorithm
for more iterations has no effect on results. The algorithm was tested for various
cases with smooth, non-smooth and noisy Cauchy data. Numerical results on a
(Nr ×Nθ) = (8× 2000) grid are summarized in following two subsections, where Nθ
is the number of discretization points along θ.
3.3.4.1 For Smooth Data
Figure. 3.13 shows the solution obtained by solving well-posed problem (3.70) with
h = sin(θ) + sin(3θ) and g = 0. Now (f, g) |Γout obtained by solving problem (3.70) is
used as Cauchy data to find unknown h on the inner boundary Γin. Figure. 3.12 shows
the solution obtained by the robust iterative observer. Figure. 3.14 compares the true
h = sin(θ) + sin(3θ) with the one obtained by the algorithm on Γin. For smooth data
case process and measurement noise co-variance was taken σ1 = σ2 ≈ 10−7 ∼ 10−6,
that is, putting more confidence into the process and also assuming very small error
83
in data with η1 = η2 = 0.
Figure 3.11: True solution or solutionobtained by solving the problem (3.70)with h = sin(θ) + sin(3θ) and g = 0over Ω1
Figure 3.12: Solution obtained fromoptimal iterative algorithm over Ω1,after a number of iterations in the di-rection of time-like variable θ
Figure 3.13: Difference between thetrue and optimal observer algorithmsolutions over Ω1
0 1 2 3 4 5 6
−1.5
−1
−0.5
0
0.5
1
1.5
θ−axis
So
lutio
n ξ
1 o
n Γ
in
Analytical Soln. vs. Robust Iterative Obsv. Soln. on Γin
Analytical Solution
Robust Obsv. Soluton
Figure 3.14: On Γin, comparison oftrue boundary h = sin(θ) + sin(3θ) tothe one recovered by optimal iterativealgorithm using Cauchy data from Γout
3.3.4.2 For Non-smooth Data
Figure. 3.15 shows the numerical solution over the whole domain Ω1 with a non-
smooth pulse signal h applied on Γin and the Neumann zero boundary condition (g =
0) on Γout. Figure. 3.16 shows the recovered solution over Ω1 obtained by using the
Cauchy data on Γout. Again as smooth data case, the process and measurement noise
84
covariance was assumed to be very small σ1 = σ2 ≈ 10−7 ∼ 10−6 and η1 = η2 = 0.
Figure. 3.17 shows the error over the whole domain. Figure. 3.18 shows the signal
recovery on the unknown data boundary Γin. It can be seen that this algorithm is
well-suited for edge detection of such a pulse-shaped signal.
Figure 3.15: Numerical solution overΩ1 obtained by solving the problem(3.70) with pulse shaped h on Γin andg = 0 on Γout
Figure 3.16: Solution obtained fromoptimal iterative algorithm over Ω1,after a number of iterations in the di-rection of time-like variable θ
Figure 3.17: Difference between thetrue and the recovered solutions overΩ1
0 1 2 3 4 5 6
−3
−2
−1
0
1
2
3
θ−axis
So
lutio
n ξ
1 o
n Γ
in
Analytical Soln. vs. Robust Iterative Obsv. Soln. on Γin
Analytical Solution
Robust Obsv. Soluton
Figure 3.18: On Γin, comparison of thetrue pulse-shaped boundary signal tothe one recovered by optimal iterativealgorithm using only the Cauchy datafrom Γout
85
3.3.4.3 For Noisy Data
In this section, robust iterative observer, which is also an optimal iterative algorithm,
is studied for the case of noisy Cauchy data. First, problem (3.70) is solved using
h = sin(θ) on Γin and Neumann zero boundary condition (g = 0) on Γout. Solution of
problem (3.70) provided f on Γout. Next an additive white Gausian noise with η2 = 0
and variance σ2 = 10−3 was added to f . This noisy f along with Neumann zero
(g = 0) was used as Cauchy data to solve the inverse problem. Process covariance σ1
was assumed to be ≈ 10−4.
Figure. 3.19 shows the noisy Dirichlet data on Γout. Figure. 3.20 compares the
true h = sin(θ) with the one recovered by the algorithm on the Γin using the noisy
measurement Dirichlet data and the homogeneous Neumann boundary data from
Γout. Figure. 3.21 shows the comparison of percentage relative error in the measure-
ment Dirichlet data on the outer boundary Γout with the percentage relative error in
the recovered solution on the inner boundary Γin (recovered solution obtained using
noisy Dirichlet measurement data and homogeneous Neumann on Γout). During all
these measurements, process noise variance σ1 was taken as 10−4, whereas, measure-
ment noise variance σ2 was adjusted according to the relative error in the Dirichlet
measurement data and η1 = η2 = 0. Percentage relative error is given by,
% Relative Error =1
‖h‖2
‖h− hrecovered‖2 × 100, (3.105)
where ‖.‖ is the Eucledian 2-norm. It is obvious from Figure. 3.20 that error in
the recovered solution is reduced with smaller and smaller error in Cauchy data.
However, for arbitrarily small noise in Cauchy data still there’s around 2% relative
error in solution, this might come from numerical discretization error.
86
0 1 2 3 4 5 6
−1
−0.5
0
0.5
1
θ−axis
Am
plit
ud
e
Noisy data on Γout
Noisy data on Γout
Figure 3.19: Noisy data on Γout Figure 3.20: On Γin: Comparison oftrue h = sin(θ) to the one recovered byrobust iterative algorithm using noisyCauchy data with measurement noisevariance σ2 = 10−3
3.4 Conclusion
An iterative observer and its robust iterative optimal counterpart are presented to
solve boundary data estimation problems for the Laplace equation. The algorithms
are developed using one of the space variables as time-like to solve steady-state bound-
ary data estimation problems. Stable and efficient numerical results for multiple test
cases are also presented. The iterative observer algorithms reflect the possibility of
considering a steady-state problem from a dynamical theory perspective by using
one of the space variables as time. Successful implementation of the algorithm and
promising results show a step forward in the direction of using dynamical systems’ in-
spired algorithms to solve steady-state problems modeled by time independent PDEs
and without introducing a particular notion of time. In the following chapter, the
iterative algorithms are used to solve source localization and estimation problems for
the Poisson equation.
87
Figure 3.21: Percentage relative error in the Dirichlet measurement data on Γout vs.percentage relative error in the recovered solution on Γin
88
Chapter 4
Iterative Observer-based Approach for Source Localization
and Estimation for Poisson Equation
Imagination is more important than knowledge. For knowledge is limited
to all we now know and understand, while imagination embraces the entire
world, and all there ever will be to know and understand.
(—Albert Einstein)
4.1 Introduction
Source localization and identification problems for Poisson equation are frequently
used to model physical phenomena in various engineering disciplines. Just to name a
few, heat source localization and identification problem [63], recovery and determina-
tion of cracks [64], [65]. Source identification problems in electromagnetic theory [66].
Electroencephalography (EEG) and Magnetoencephalography (MEG) problems to
monitor electric and magnetic activity in human brain [67]. Some classical source
estimation and localization problems can also be found in [68] and [69].
Various methods and strategies have been presented in the literature to tackle
such problems. For example [70] presents a Green’s functions based method to iden-
tify the unknown point sources. The technique involves the study of the method of
fundamental solutions and requires analytical skills to deal with Green’s functions. A
method based on Poisson integrals for solving source seeking task is proposed in [30].
The method involves a sensor-equipped vehicle to provide pointwise measurements of
quantity emitted by the source. A numerical method is presented by [71]. The draw-
89
back of this numerical method is that it needs prior approximate positions of point
sources which is not the case in many of the physical applications. Another work
based on a weighted residual approach using harmonic functions is presented by [32].
Again the algorithm works for the cases where an estimated localized region is avail-
able for the point sources. Tikhonov regularization based methods to solve inverse
source problem for Poisson equation are presented in [23]. The method presented is
useful for distributed source estimation only.
In this chapter, iterative observer-based method is developed to localize and es-
timate sources in a system governed by Poisson equation. Both smooth and noisy
observation data cases are tackled. The strategy developed in this chapter highlights
the fact that dynamical systems’ inspired algorithms can be used for source localiza-
tion/estimation problems for Poisson equation.
4.2 Iterative Observer-based Strategy for Point Source Lo-
calization
In this section, a method based on iterative observer design is presented to solve
point source localization problem for Poisson equation with given boundary data.
The procedure involves solutions of multiple boundary estimation sub problems using
the available overdetermined Dirichlet and Neumann data from different parts of the
boundary. An optimal weighted sum of these solution profiles localizes point sources
inside the domain. A method to compute these weights is also provided. Numerical
results are presented using finite differences in a rectangular domain.
90
4.2.1 Problem Statement
Let Ω2 be a bounded rectangular domain in R2 as shown in Figure 4.1 and ∂Ω2 =
Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4 be the boundary of Ω2. Consider the Poisson equation,
4u = f in Ω2, (4.1)
with Laplacian operator 4 =∂2
∂x2+
∂2
∂y2, along with Neumann boundary condition,
∂u
∂n= h on ∂Ω2, (4.2)
where∂
∂nrepresents normal derivative to the boundary. Let us assume that,
u = g on ∂Ω2, (4.3)
is obtained by solving boundary value problem (4.1) and (4.2) with known f and
h. Existence and uniqueness for the solution of problem (4.1),(4.2) is well-known
for consistent f and h, that is the case when f ∈ L2(Ω2) and h ∈ L2(∂Ω2) such
that g ∈ H12 (∂Ω2). Further, for this particular choice of f , h and g, the solution
u ∈ H1(Ω2). However, because of the steady-state diffusive nature of system, problem
of finding the unknown source f from observed g and h, is not obvious. Let us assume
that the steady-state potential field u is generated by a number of distinct point
sources inside the domain:
f(x, y) =N∑k=1
Ckδ(x− xk, y − yk), (x, y) ∈ Ω2, (4.4)
where δ’s represent Dirac delta point sources and scalars Ck are the corresponding
magnitudes.
91
Ω2
Γ1
Γ2
Γ3 Γ4
y
x
Figure 4.1: Left: Rectangular domain Ω2 with ∂Ω2 = Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4, Right:Co-ordinate axis.
The objective here is to find locations (xk, yk) for all k inside Ω2 from available overde-
termined data g and h on ∂Ω2.
In the following section, the boundary estimation subproblems are presented,
which can be solve using the iterative observer presented in chapter 3. Solutions
to these sub problems will then help to solve the source localization problem for
Poisson equation in a later section.
4.2.2 Preliminary Analysis and Results
Let us observe that the available non-homogeneous boundary data as given in equation
(4.2) and (4.3) has four components on various parts of ∂Ω2:
g = ∪4i=1 gi|Γi , (4.5)
h = ∪4i=1 hi|Γi , (4.6)
Let us propose the boundary estimation problem:
Find ξ1 on Γ2:
∂ξ
∂x= Aξ in Ω2,
ξ1 = Cξ = g1 on Γ1,
∂ξ1
∂n= h1 on Γ1,
∂ξ1
∂nξ1 = 0 on Γ3 ∪ Γ4,
(4.7)
where ξ = (ξ1 ξ2)T represents state vector, C ∈ L(X,R) is the boundary observation
92
operator and operator matrix A : D(A)→ X:
X = H1Γ1
(α)× L2 (α) , (4.8)
D(A) =[H2 (α) ∩H1
Γ1(α)]×H1
Γ1(α) , (4.9)
where α is in the interval (y |Γ1 , y |Γ2) and without loss of generality let α ∈ [0, π/4],
and
H1Γ1
(α) =
p1 ∈ H1 (α) | dp1
dy|Γ1= h1
, (4.10)
here X is a Hilbert space with scalar product given by,
⟨ q1
q2
,
p1
p2
⟩ =
∫α
dq1
dy(y)
dp1
dy(y)dy
+
∫α
q1(y)p1(y)dy +
∫α
q2(y)p2(y)dy. (4.11)
The first equation in (4.7) represents Laplace equation,
4S1 = 0 on Ω2, (4.12)
with two auxiliary variables ξ1 = S1 and ξ2 =∂S1
∂x:
ξ =
ξ1
ξ2
, A =
0 1
− ∂2
∂y20
. (4.13)
Let us introduce the idea of iterations using time-like variable over the domain Ω2 as
follows. Let x be a variable defined over the interval [c, d) for all c, d ∈ R and d > c
then x[m], for all m ∈ Z = 0 ∪ Z+, represents x over mth iteration over the interval
[c, d). The idea of iteration is also shown in Figure. 3.2.
93
4.2.2.1 Iterative Observer Algorithm:
Theorem 6. Let ξ1 be the estimate of ξ1, then solution ξ1 of boundary value problem
(4.14) asymptotically (m→∞) converges to the solution of boundary value problem
(4.7) over Ω2:
∂
∂xξ(x[m]) = Aξ(x[m])−KC(ξ(x[m])− ξ) in Ω2,
∂
∂yξ1(x[m]) = h1 on Γ1,
∂2
∂y2ξ1(x[m]) = − ∂2
∂x2ξ1(x[m])−KC(ξ(x[m])− ξ) on Γ2,
ξ(x[m]) |initial= ξ(x[m−1]) in Ω2,
(4.14)
where ‘ ˆ ’ represents estimated quantity and ξ(x[m]) |initial represents the estimate
over the whole domain Ω2 at the start of mth iteration. Observer starts at index
m = 1 which represents first iteration. ξ(x[m=0]) is initial guess at the start of the
first iteration over the whole domain Ω2.
The proof of Theorem 6 is along similar lines as given in section 3.2.5. The
third equation in (4.14) is the assumption that Laplace equation is valid on the top
boundary and this provides necessary boundary condition required on Γ2. C is the
observation operator such that Cξ = ξ1 |Γ1 . K is the correction operator chosen in
such a way that state estimation error on Γ1 given by (ξ− ξ) exponentially converges
to zero over Ω2. Conditions for the existence of observer gain K are given in chapter
3. In the following section, a strategy to localize point sources inside domain Ω2 is
presented to solve problem (4.1) with known boundary conditions (4.2), (4.3) and
unknown point sources given by (4.4) by exploiting the iterative observer design.
94
4.2.3 Point Source Localization Strategy
The main theoretical result of this section is presented in form of a theorem as given
below.
4.2.3.1 Main Result
Theorem 7. For all i in integer set 1, 2, 3, 4, let Si be the solution over Ω2 obtained
using iterative observer (4.14) with Si = gi|Γi and∂
∂nSi = hi|Γi . Let for all i, wi be
the solution obtained by solving boundary value problem for Laplace equation,
4wi = 0 in Ω2,
wi = 1 on Γi,
wi = 0 on ∂Ω2\Γi,
(4.15)
then,
u =1
2
[4∑i=1
Siwi
]in Ω2, (4.16)
f =4∑i=1
∇Si.∇wi in Ω2, (4.17)
where u and f solve the boundary value problem for Poisson equation given by equa-
tions (4.1), (4.2) and (4.3).
Proof. We have Si and wi satisfy Laplace’s equation for all i ∈ 1, 2, 3, 4. From
equation (4.16) we can write,
∇u =1
2
4∑i=1
(Si∇wi + wi∇Si) , (4.18)
∇.(∇u) = ∇.
(1
2
4∑i=1
(Si∇wi + wi∇Si)
), (4.19)
95
which gives,
f = 4u =4∑i=1
∇Si.∇wi. (4.20)
thus we have that u as given in equation (4.16) satisfies Poisson equation 4u = f up
to an additive constant. Now using the properties of wi, Si we have to show that u
and f as given by equations (4.16) and (4.17) fully satisfy boundary value problem
given by equations (4.1), (4.2) and (4.3). We can write,
∫Ω2
[4∑i1
wi4Si
]dΩ2 = 0, (4.21)
using Green’s first identity,
∫∂Ω2
[4∑i=1
wi∂Si∂n
]d∂Ω2 =
∫Ω2
[4∑i=1
∇wi.∇Si
]dΩ2, (4.22)
We have wi = 1 on Γi and zero elsewhere on boundary. This gives,
∫∂Ω2
[4∑i=1
wi∂Si∂n
]d∂Ω2 =
4∑i=1
[∫Γi
∂Si∂n
d∂Ω2
]=
∫∂Ω2
h dΩ2, (4.23)
applying divergence theorem we have,
∫∂Ω2
h dΩ2 =
∫Ω2
∇.∇u dΩ2 =
∫Ω2
f dΩ2. (4.24)
Combining equations (4.22), (4.23) and (4.24),
∫Ω2
f dΩ2 =
∫Ω2
[4∑i=1
∇wi.∇ui
]dΩ2, (4.25)
above relation is true for all sizes of rectangular domains Ω2, thus we have,
f =4∑i=1
∇Si.∇wi. Q.E.D. (4.26)
96
Remark 2. The mathematical result presented in Theorem 7 is valid for all sizes of
rectangular domains.
4.2.3.2 Point Source Localization as a Two-step Strategy
The main theoretical result presented in the previous section can be used to localize
unknown point sources inside the domain. The process is illustrated as a two-step
strategy.
Step 1 : For all i ∈ 1, 2, 3, 4, compute iterative observer solutions Si over Ω2,
using Si = gi and∂Si∂n
= hi on Γi and iterative observer.
Step 2 : For all Si, compute corresponding weight solutions wi over Ω2. Find the
estimate of u using weighted sum as given in equation (4.16). This estimate of u
approximates the true solution u over the whole domain Ω2. The approximation
is fairly accurate for f ∈ L2(Ω2). However, for Dirac delta point sources local
minima or maxima provide their locations inside Ω2.
In the following section numerical simulation results are presented to locate un-
known point sources.
4.2.4 Numerical Simulations
4.2.4.1 Graphical Illustration of the Process
In this subsection, the point source localization strategy using iterative observer is
illustrated graphically with the help of MatLab1 simulations for the simple case of
a single point source in the middle of square domain of dimension [0, 1] × [0, 1] as
shown in Figure 4.2. It is important to note that any size of rectangular domain can
be considered. More complicated scenarios are presented in the next subsection.
1MatLab is a trademark of The MathWorks, Inc.
97
Figure 4.2: Numerical simulation result for the Poisson equation over domain Ω2 witha point source in the middle using homogeneous Neumann boundary data.
Step 1:
Figure 4.2 represents a point source in the middle of a square domain with homogenous
Neumann boundary data using finite difference discretization schemes on a 400× 400
uniform grid. This particular case under consideration is symmetric from all sides
hence would be easier for the illustrative purposes. Dirichlet boundary data can be
extracted from this solution profile. The objective is to recover the location of point
source inside this domain. Figure 4.3 shows g1 on bottom boundary. Because of
symmetry, g2, g3 and g4 on remaining parts of the boundary would be the same. Next
this Dirichlet data g1 is used to estimate S1 = ξ1 on opposite boundary Γ2 as given
in problem (4.7). Figure 4.4 shows g1 and estimated S1 = ξ1 on Γ2. It is important
to note that maximum number of states while solving problem (4.7) after full dis-
cretization depends on discrete observability condition also known as Kalman rank
condition. In this particular case 10 states were considered. Similarly for g2, g3 and
g4, estimated S1 = ξ1 on the corresponding opposite boundaries would be the same
because of the symmetry. Top plot in Figure 4.5 represents the full solution plot S1
over the whole domain Ω2 obtained using finite difference discretization on a 400×400
uniform grid with g1 on Γ1, corresponding S1 = ξ1 on opposite boundary Γ2 and in-
98
sulated (homogeneous Neumman) side boundaries. Similar the remaining three plots
in Figure 4.5 (from 2nd to 4th) represent S2, S3, S4 obtained by solving problem (4.7)
with g2, g3, g4 on Γ2,Γ3,Γ4 respectively and corresponding estimated ξ1 on respective
opposite boundary and insulated (homogeneous Neumann) side boundaries.
0 0.2 0.4 0.6 0.8 1
−6
−4
−2
0Dirichlet data on the boundary
Γi
Am
plit
ude
Figure 4.3: Dirichlet data g1 on bot-tom boundary Γ1. Because of symme-try g2|Γ2, g3|Γ3 and g4|Γ4 would be sim-ilar.
0 0.2 0.4 0.6 0.8 1−60
−50
−40
−30
−20
−10
0
Am
plit
ude
Dirichlet data and recovered signal on opposite boudaries
Dirichelt data g1 on Γ
1
Estimated S1 on opposite boundary
Figure 4.4: Dirichlet data g1|Γ1 and es-timated S1 = ξ1 on opposite boundaryΓ2 using iterative observer. Becauseof symmetry, qualitatively similar pro-files for other three cases.
Step 2:
Figure 4.6 represents the weight profiles w1, w2, w3 and w4 corresponding to solution
profiles shown in Figure 4.5. It is important to note that these weights are normalized.
Next these weights are applied to corresponding solution profiles by point to point
multiplication to obtain resultant weighted profiles. Finally these weighted solution
profiles are added to obtain weighted sum as shown in Figure 4.7. The minimum
point where the marker is placed represents the location of the point source.
4.2.5 Further Simulation Results
In this section different scenarios are presented to reflect the accuracy of the scheme to
localize point sources. All the numerical results were obtained using finite difference
discretization schemes on a square domain. The mesh size was 400 × 400 for all
99
Figure 4.5: Top plot: Solution profile S1 over Ω2 obtained by solving problem (4.7)with g1 on Γ1 and corresponding S1 = ξ1 on opposite boundary Γ2 and insulated(homogeneous Neumann) side boundaries.From 2nd to 4th: Plots for S2, S3 and S4 obtained using similar procedure.
100
Figure 4.6: From top to bottom: Weight profiles w1, w2, w3 and w4 over Ω2 corre-sponding to solution profiles S1, S2, S3, S4 respectively, as shown in Figure 4.5.
Figure 4.7: Weighted sum as given in equation (4.16) over Ω2. Marker in the middlerepresents the minima, where the negative point source is located.
101
Figure 4.8: Top: Non-centered point source inside Ω2. Bottom: Weighted sum withmarker representing minimum point and the location of point source.
simulation plots except for the Iterative observer algorithm where because of discrete
observability condition only 10 states were considered. The issue of limited number
of observable states after numerical discretization for distributed parameter systems
is well-known, [6,72]. Figure 4.8 represents the solution of the forward problem with
a non-centered point sources and the weighted sum obtained by applying two step
strategy. It can be seen that point source is very well localized. Figure 4.9 represents
the case with two point sources. The algorithm presented here works well for point
sources that are well separated, however, in case where multiple point sources are in
close vicinity, it not so easy to distinguish as shown in Figure 4.10.
4.3 Robust Iterative Algorithm-based Strategy for Inverse
Source Localization Problem
Source localization problem for Poisson equation with available noisy boundary data
is well-known to be highly sensitive to noise. The problem is ill-posed and lacks
to fulfill Hadamards stability criteria for well-posedness as presented in Chapter 2.
In this section, robust iterative observer algorithm along with the available noisy
boundary data from the Poisson problem is used to localize point sources inside a
102
Figure 4.9: Top: Two opposite polarity well seperated point sources in Ω2. Bot-tom: Weighted sum with markers representing minimum and maximum points andlocations of two point sources.
Figure 4.10: Top: Three closely located point sources in Ω2. Bottom: Weighted sumwith minima locating approximate position of point sources.
103
rectangular domain. The algorithm is inspired from Kalman filter design and is
implemented on a rectangular domain using finite difference discretization schemes.
Numerical implementation along with simulation results is detailed.
4.3.1 Problem Formulation
Let Ω2 be a bounded rectangular domain in R2 as shown in Figure 4.1 and ∂Ω2 =
Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4 be the boundary of Ω2. Let us consider the Poisson equation,
4u = f in Ω2, (4.27)
with Laplacian operator 4, along with Neumann boundary condition,
∂u
∂n= h on ∂Ω2, (4.28)
where ∂n represents normal derivative to the boundary. Existence and uniqueness for
the solution of problem (4.27),(4.28) is well-known for consistent f and h, that is the
case when f ∈ L2(Ω2) and h ∈ L2(∂Ω2) and the solution u in this case belongs to
H1(Ω2). Let the solution of boundary value problem (4.27),(4.28) gives,
u = g on ∂Ω2, (4.29)
Now let us assume that the steady-state potential field u is generated by a number
of distinct point sources inside the domain Ω2:
f(x, y) =N∑k=1
Ckδ(x− xk, y − yk), (x, y) ∈ Ω2, (4.30)
where δ(x − x, y − y), k = 1, · · · , N represent Dirac delta point sources localized
at (xk, yk) and scalars Ck, k = 1, · · · , N are the corresponding magnitudes. In this
section we are interested in unknown source localization problem from noisy bound-
ary measurements. We suppose that the measurement g and Neumann boundary h
104
are corrupted with additive white Gaussian noise ε1 and ε2 respectively, such that
ε1(η1, σ1) and ε2(η2, σ2) with mean values η1, η2 and variances σ1, σ2.
The objective, in this section, is to find locations (xk, yk) for all k = 1, · · · , N inside
Ω2 from available noisy measurement data g and known boundary h on ∂Ω2.
As discussed for the smooth data case, in the first part of this chapter, the source
localization problem for Poisson equation presented above boils down to a weighted
sum of solutions of multiple boundary estimation problems for Laplace equation. This
mathematical result is also presented in section 4.2.3. However, in the following sec-
tion, a robust iterative observer design is presented to solve the boundary estimation
problem for steady-state Laplace equation.
4.3.2 Robust Iterative Observer Design
In this section, we present robust iterative observer design for boundary estimation
for Laplace equation which helps to solve source localization problem for Poisson
equation. Let us observe that noisy boundary measurement g and Neumann boundary
h has four components on four different parts of boundary ∂Ω2,
g = g1|Γ1 ∪ g2|Γ2 ∪ g3|Γ3 ∪ g4|Γ4 , (4.31)
h = h1|Γ1 ∪ h2|Γ2 ∪ h3|Γ3 ∪ h4|Γ4 . (4.32)
Let us propose following boundary estimation problem for Laplace equation,
Find steady-state potential field S1 on Γ2:
4S1 = 0 in Ω2,
S1 = g1 on Γ1,
∂S1
∂n= h1 on Γ1,
∂S1
∂n= 0 on Γ3 ∪ Γ4.
(4.33)
105
Boundary value problem (4.33) is also known as Cauchy problem for Laplace equation.
Let us propose to reformulate this problem in a control-familiar form as following,
Find ξ1 on Γ2:
∂ξ
∂x= Aξ in Ω2,
ξ1 = Cξ = g1 on Γ1,
∂ξ1
∂n= h1 on Γ1,
∂ξ1
∂n= 0 on Γ3 ∪ Γ4,
(4.34)
with observation or measurement operator C and two auxiliary variables ξ1 = S1 and
ξ2 =∂S1
∂x:
ξ =
ξ1
ξ2
, A =
0 1
− ∂2
∂y20
, (4.35)
where A : D(A) → X as given in (4.8), (4.9) and (4.11). In chapter 3, a Kalman
filter-like robust optimal algorithm is already presented to solve ill-posed boundary
estimation problem for Laplace equation on an annulus domain. Here this algorithm
design is adapted to the rectangular domain using rectangular coordinates and with
the idea of iterations over Ω2 in the direction of time-like variable.
4.3.2.1 Numerical Discretization
Let us introduce Forward Euler-like approximation to discretize first derivative along
x as,∂ξ
∂x≈ 1
4x[ξn+1 − ξn
], (4.36)
where n is discrete positive integer index for variable x and 4x is a small step size.
State equation, after discretization in variable x, can be written as,
ξn+1 = (I + (4x)A) ξn, (4.37)
here I is 2× 2 identity matrix. Second equation in (4.37) can be written as,
106
(ξ2)n+1 = (ξ2)n +4x(− ∂2
∂y2
)(ξ1)n , (4.38)
to discretize second-order derivative term in above equation we use following second-
order accurate centered finite difference discretization formula,
∂2
∂y2(ξ1)n ≈ 1
(4y)2
[(ξ1)ni+1 − 2 (ξ1)ni + (ξ1)ni−1
], (4.39)
here i is discrete index for variable y, i = 1 on Γ2 and (4y) is a reasonably small step
size along y. Now from (4.38),
(ξ2)n+1i = (ξ2)ni −
4x(4y)2
[(ξ1)ni+1 − 2(ξ1)ni + (ξ1)ni−1
], (4.40)
rewriting above equation on Γ2, that is, at i = 1 gives,
(ξ2)n+11 = (ξ2)n1 −
4x(4y)2 [(ξ1)n2 − 2(ξ1)n1 + (ξ1)n0 ] , (4.41)
index i = 1 represents points on boundary Γ2, and index i = 0 represent fictitious
points outside the domain Ω2 along the boundary Γ2. Fictitious points are shown in
Figure 4.11. The equation for the fictitious points can be written as,
(ξ1)n0 = −(4y)2
4x[(ξ2)n+1
1 − (ξ2)n1]
+ 2(ξ1)n1 − (ξ1)n2 . (4.42)
Further at a particular index n fictitious point (ξ1)n0 can be determined uniquely as
it depends on (ξ2)n+11 . In each iteration at a particular n the fictitious point value
guess is determined using equation (4.42). This guess is then used in as a boundary
data to improve estimation on Γ2. Theoretically, this is equivalent to the assumption
that Laplace equation is satisfied on Γ2,
∂2u
∂x2= −∂
2u
∂y2on Γ2, (4.43)
107
which is possible as we are solving 2D Laplace equation as a first order state equation.
i=0;n
i=1;n
i=2;n
i=0;n− 1
i=1;n− 1
i=2;n− 1
i=0;n + 1
i=1;n + 1
i=2;n + 1
n
i
Γ1
Γ2
Figure 4.11: Left: Domain Ω2 after discretization and fictitious points outside Γ2,index i = 0 represents fictitious points (in blue).
4.3.2.2 Robust Iterative Observer Derivation
After full discretization equation (4.37) can be written as,
ξn+1 = (I + (4x)A) ξn, (4.44)
where A is discrete operator matrix. Further the output equation as given in (4.34)
can be written as,
ξn1 = Cξn + εn1 = gn1 on Γ1, (4.45)
where C is the discrete observation operator matrix. An important notion for exis-
tence of solution of state-space like system represented by equations (4.44) and (4.45)
is the discrete observability condition also called Kalman rank condition [6, 72]. The
system is observable if observability matrix formed by pair (C,A′) is full rank, where
A′ = I+(4x)A. Observability condition puts a restriction on the size of operator ma-
trix A, in other words on the possible number of discretization points in y-direction.
Let ε1 and ε2 be the measurement and process noise respectively such that their
covariance matrices be stationary over x and given by,
R = E[εn1 (εn1 )T
], (4.46)
Q = E[εn2 (εn2 )T
], (4.47)
108
here of R is 1 × 1 and Q is the size of A′. Let ξn be the estimate of true state ξn
and en represents the difference of true and estimated states at n. The goal is to
develop an iterative observer that is capable to predict true states over a period of
time-like variable x. For this, let positive definite error functional (en)2 = (ξn − ξn)2
be a feasible metric given by
ε(x) = E[(en)2
]= E
[en(en)T
]= P n, (4.48)
where P n is the error covariance matrix at x = n ×4x. The idea of iteration over
the rectangular domain can be introduced as following, let x be a variable defined in
the interval [c, d) for all c, d ∈ R and d > c then xm for all m ∈ N represents x over
mth iteration over the interval [c, d) .
In a similar way, we introduce idea of iterations over the rectangular domain
Ω2 in the direction of time-like variable. The right-end boundary is assumed to be
connected with the left end boundary to have time-like notion of the space variable.
Robust Iterative Observer Equations:
Prediction-Step:
¯ξn,m = A′ξn−1,m in Ω2,
P n,m = A′P n−1,mA′T +Q,
¯ξn=0,m=0 = P n=0,m=0 = 0,
(4.49)
with Neumann boundary condition h on Γ1 and fictitious points
close to Γ2 given by equation (4.42), taking care of boundary
conditions.
109
Correction-Step:ξn,m =
¯ξn,m +Kn,m
(yn − C ¯
ξn,m),
Kn,m = P n,mCT (CP n,mCT +R)−1,
P n,m = (I −Kn,mC)P n,m,
(4.50)
where ξn,m is an estimate of ξn,m,¯ξn,m is the prior-estimate of ξn,m, P n,m is the prior-
estimate of state error covariance matrix P n,m given by equation (4.48). Q and R as
given in equations (4.47) and (4.46) and Kn,m is the gain matrix.
At the start of iterative observer algorithm, (ξ1)n1 , (ξ1)n2 , (ξ2)n1 and (ξ2)n+11 given
in equation (4.42) come from the initial guess over the domain Ω2. The fictitious
points from equation (4.42) are then used in robust observer equations to tackle
boundary condition on Γ2, whereas Neumann boundary condition h is used to tackle
the boundary condition on Γ1.
Proposition 2. Let pair (C,A′) be observable and m be a discrete non-negative inte-
ger index of iteration for variable x. As m→∞ (that is for arbitrarily large number
of iterations over rectangular domain Ω2), the two-step algorithm given by equations
(4.49) and (4.50) minimizes the mean square error functional given in equation (4.48).
Proof. (Above proposition and the following proof are also detailed chapter 3 for
boundary estimation problem for Laplace equation on the annulus domain Ω1)
Let us re-write equation (4.48) as,
P n,m = E[(ξn,m − ξn,m)(ξn,m − ξn,m)T
]. (4.51)
Let us assume the prior estimate of ξn,m is called¯ξn,m and was obtained by the
knowledge of the system (prediction). The correction equation can be written using
the prior estimate with measurement data as follows,
ξn,m =¯ξn,m +Kn,m
(yn,m − C ¯
ξn,m). (4.52)
110
Kn,m is the gain matrix which will be derived in a moment. The term(yn − C ¯
ξn,m)
is the measurement residual. Substituting equation (4.45) into (4.52),
ξn,m =¯ξn,m +Kn,m
(Cξn,m + εn1 − C
¯ξn,m
). (4.53)
ε1, ε2 remain the same for particular n over all iterations m ∈ N. Now substituting
equation (4.53) into (4.51) gives,
P n,m = E[(
(I −Kn,mC)(ξn,m − ¯ξn,m)−Kn,mεn1
)(
(I −Kn,mC)(ξn,m − ¯ξn,m)−Kn,mεn1
)T], (4.54)
taking the error of prior estimate(ξn,m − ¯
ξn,m)
as uncorrelated to measurement noise
gives,
P n,m = (I −Kn,mC) E[(ξn,m − ¯
ξn,m)(ξn,m − ¯ξn,m)T
](I −Kn,mC)T +Kn,m E
[εn1 (εn1 )T
](Kn,m)T , (4.55)
now substituting equations (4.46) and (4.51) into above equation yields,
P n,m = (I −Kn,mC)P n,m(I −Kn,mC)T +Kn,mR(Kn,m)T . (4.56)
Above equation is the error covariance update equation and the diagonal of P n,m
contains mean squared error. This mean squared error has to be minimized with
respect to Kn,m. Re-writing equation (4.56) gives,
P n,m = P n,m −Kn,mCP n,m − P n,mCT (Kn,m)T
+Kn,m(CP n,mCT +R)(Kn,m)T , (4.57)
using the fact that trace of a matrix is equal to trace of its transpose, it can be seen
111
that,
T [P n,m] = T [P n,m]− 2T [Kn,mCP n,m]
+ T [Kn,m(CP n,mCT +R)(Kn,m)T ], (4.58)
where T [P n,m] is the trace of covariance matrix P n,m. Differentiating with respect to
Kn,m,
d
dKn,mT [P n,m] = −2CP n,m + 2Kn,m(CP n,mCT +R), (4.59)
setting derivative equal to zero gives,
(CP n,m)T = Kn,m(CP n,mCT +R), (4.60)
solving for Kn,m gives,
Kn,m = P n,mCT (CP n,mCT +R)−1. (4.61)
The above equation is the optimal estimator gain equation which minimizes the mean
square state estimation error. Substituting above equation into equation (4.56) yields,
P n,m = P n,m − P n,mCT (CP n,mCT +R)−1CP n,m,
= P n,m −Kn,mCP n,m,
= (I −Kn,mC)P n,m, (4.62)
which is the update equation for state error covariance matrix with optimal gain
Kn,m. Equations (4.52), (4.61) and (4.62) develop an estimate of variable xn,m. State
prediction is achieved using state equation,
¯ξn+1,m = A′ξn,m, (4.63)
112
along with Neumann boundary condition g on Γout and equation (4.43) on Γin. To
complete the recursion it is important to find an equation which projects state error
covariance matrix into next x-step, x+ 1. This is achieved by forming an expression
for the prior error (prediction error).
en+1,m = ξn+1,m − ¯ξn+1,m,
= A′en,m + εn2 . (4.64)
Now extending equation (4.48) to n+ 1 gives,
P n+1,m = E[en+1,m(en+1,m)T
],
= E[(A′en,m + εn2 )(A′en,m + εn2 )T
], (4.65)
en,m and εn2 are uncorrelated as en,m is the error accumulated in previous n steps and
εn2 is the process error for n-th step. This implies,
P n+1 = E[A′en,m(A′en,m)T
]+ E
[εn2 (εn2 )T
],
= A′P n,mA′T
+Q. (4.66)
This has completed the iterative observer loop.
4.3.3 Two-step Process for Source Localization
The theoretical result presented previously can be used to devise a two-step source
localization method, similar to section 4.2.3.2, as follows,
Step 1 : For all i compute robust iterative observer solutions Si over Ω2, as given
in Proposition 1, using Si = gi and∂Si∂n
= hi on Γi.
Step 2 : For all i, compute weight profiles ωi by solving well-posed boundary value
problem (4.15). Combine Si’s and ωi’s using equation (4.16) to have an estimate
113
of Poisson problem solution u over Ω2. The position of Dirac delta sources inside
the domain is provided by the maxima or minima of estimated u.
4.3.4 Numerical Results
For all numerical simulation results a square domain Ω2 of dimension [0, 1]× [0, 1] was
considered. For robust iterative observer implementation, number of state variables
were chosen such that pair (C,A′) fulfills discrete Kalman rank condition. Robust iter-
ative observer provided unknown boundary estimate for Cauchy problem for Laplace
equation using part of Cauchy data from Poisson problem. Cauchy data on Γi along
with this estimated boundary data on opposite boundary and Neumann zero side
boundaries were used to obtain Si for all i ∈ 1, 2, 3, 4 on a 400× 400 grid. Weight
profiles wi were also computed on the same grid size by numerically solving system
(4.15). All numerical results were obtained using MatLab2. Fig. 4.12 represents the
solution of problem (4.27),(4.28) over Ω2 with f = δ(0.5, 0.5) and h = 0 and using
centered finite difference discretization scheme on a 400 × 400 uniform grid. The
solution obtained on the boundary is then corrupted with additive white Gaussian
noise with η1 = 0 and σ1 = 5× 10−4. This noisy Dirichlet boundary data is used as a
measurement along with homogeneous Neumann boundary to recover the unknown
point source location. Fig. 4.13 represents noisy measurement data. Fig. 4.14 shows
noisy measurement on Γi and iterative observer boundary estimate on the opposite
boundary with η2 = 0 and σ2 = 0.1. From top to bottom Fig. 4.15 shows S1, S2, S3
and S4 respectively. Similarly from top to bottom Fig. 4.16 shows weight profiles
w1, w2, w3 and w4 respectively. Fig. 4.17 represents weighted sum obtained using
equation (4.16). This weighted sum gives estimate of solution u. The minimum point
represented with white marker locates the point source inside Ω2. Fig. 4.18 repre-
sents the localization of point sources for the case where two opposite polarity point
2MatLab is a trademark of The Mathworks Inc.
114
Figure 4.12: Square domain Ω2 with a point source in the middle using homogeneousNeumann boundary data.
sources are located inside domain Ω2. The white markers in the bottom figure rep-
resent approximate locations of point sources. The measurement data was corrupted
with additive white Gaussian noise with η1 = 0 and σ1 = 5× 10−4.
Figure 4.13: Noisy measurement datag with σ1 = 5× 10−4, because of sym-metry of the special case under consid-eration (one point source in the mid-dle of the domain), qualitatively simi-lar profiles on all parts of ∂Ω2.
Figure 4.14: Comparison of noisy mea-surement data g on Γi and robust it-erative observer solution on the oppo-site boundary. σ1 = 5 × 10−4 andσ2 = 1 × 10−1. Because of symme-try, qualitatively similar profiles on allparts of ∂Ω2.
115
Figure 4.15: Top figure: Solution profile S1 with measurement data g1 on Γ1, insulatedside boundaries and boundary estimate obtained using robust iterative observer onΓ2. From 2nd to 4th: Plots for S2, S3 and S4 respectively.
116
Figure 4.16: From top to bottom: Weight profiles w1, w2, w3 and w4 respectively,obtained by solving boundary value problem (4.15) for i ∈ 1, 2, 3, 4.
Figure 4.17: Weighted sum obtained using equation (4.16). Minimum point repre-sented with white marker in the middle provides the location of point source.
117
Figure 4.18: Top: Solution of problem (4.27),(4.28) with two point sources of oppositepolarity and homogeneous Neumann boundary on ∂Ω2. Below: Weighted sum ob-tained using equation (4.16). Minimum and maximum points represented with whitemarkers provide locations of point sources.
4.4 Distributed Potential Field Estimation for Poisson Equa-
tion
The iterative observer algorithms presented in chapter 3 can also be used for steady-
state distributed potential field estimation problem for the Poisson equation. The
distributed potential field estimation over the whole domain from boundary mea-
surements is a challenging problem. The numerical simulation results obtained using
the two-step strategy highlight the significance of observer-based algorithms. In the
following section the potential field estimation problem is formulated.
4.4.1 Problem Formulation
Let Ω2 be a rectangular domain with boundary ∂Ω2 = Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4 as shown
in Figure 3.1. Let us consider the steady-state diffusion model, with a source term,
known as Poisson equation,
4u = f in Ω2, (4.67)
118
where 4 is the Laplacian operator, u represents the steady-state potential field and
f is the source term.
The object is to estimate the unknown steady-state potential field u from the available
Cauchy data on the boundary ∂Ω2, given by,
u = g on ∂Ω2,
∂u
∂n= h on ∂Ω2.
(4.68)
The two-step solution strategy as presented in section 4.2.3 is used to estimation
steady-state distributed potential field u. In the following section, to avoid repetition,
only numerical simulation results are presented.
4.4.2 Numerical Results
Various simulation results for steady-state distributed potential field estimation are
presented. All the simulations are done using finite difference discretization schemes
on a square domain of dimension 1×1. The two-step strategy as graphically illustrated
in the previous section is used for estimation. Figure 4.20 compares the weighted sum
obtained using two-step solution strategy to the exact numerical simulation result over
Ω2 (exact solution obtained with known f and Neumann boundary data) for various
test cases. As obvious from the results, the two-step solution strategy can well recover
the distributed potential field u using only boundary data.
4.5 Conclusion
An iterative observer-based technique is presented to solve source localization and
estimation problems for Poisson equation on rectangular domains. The idea of using
dynamical system like algorithm, using space variable as time-like, has not been
studied in the literature before. The two-step strategy using robust iterative observer
119
Figure 4.19: Steady-state potentialfield u over Ω, obtained by numeri-cally solving Poisson equation (4.67)for various f with homogeneous Neu-mann side boundaries.
Figure 4.20: Weighted sum obtainedfrom equation (4.16), which providesestimate to the steady-state potentialfield u over Ω2 for various test casesshown in Figure 4.19.
120
algorithm happens to be robust to noise for well separated point sources, further the
distributed potential field estimation is fairly good.
In the following chapter, a dimension decomposition approach is presented to
tackle 3D domains with two congruent parallel surfaces. Once the 3D problem is
decomposed into 2D sub problems, then the iterative observer-based methods are
used to solve the sub problems.
121
Chapter 5
Dimension Decomposition Approach for 3D Domains
Walls are tricky. Sometimes it’s not about breaking them down by force,
but about finding the weak spots and gently nudging.
(—My Kindred Spirit)
5.1 Introduction
In this chapter, a dimension decomposition approach is presented which helps to
solve boundary and source data estimation problems on three-dimensional domains
with two congruent parallel surfaces using iterative observer algorithms presented in
previous chapters. First of all, a boundary estimation problem for Laplace equation
on a 3D domain is presented in state-space-like representation and then the system
observability is studied. Based on the special observability result, the 3D problem is
subdivided into a number of 2D subproblems over rectangular cross-sections. These
subproblems can be independently solved using previously presented iterative ob-
servers. In the later half of this chapter, this dimension decomposition approach is
used to estimate unknown sources for Poisson problem in 3D.
5.2 Boundary Estimation Problem for Laplace Equation in
3D
In this section, boundary estimation problem for Laplace equation is formulated in a
control familiar form on a 3D domain. Using one of space variables, boundary value
problem is written as a forward evolving system in state-space-like representation.
122
Problem formulation and theoretical analysis is presented in the following.
5.2.1 Problem Formulation
Let Ω3 ∈ R3 with two congruent parallel surfaces ΓB,ΓT ∈ R2 as shown in Figure 5.1.
Cauchy data is given on one of the two surfaces and objective is to find the solution on
the opposite boundary surface. Boundary estimation problem for Laplace equation
can be given as,
Find u on ΓT :
4u = 0 in Ω3,
u = f on ΓB,
∂u
∂n= g on ΓB,
∂u
∂n= 0 on ∂Ω3\ ΓB ∪ ΓT ,
(5.1)
here∂
∂nrepresents normal derivative to the boundary surface. 4 is the Laplacian
operator in rectangular co-ordinates such that 4 =∂2
∂x21
+∂2
∂x22
+∂2
∂x23
.
5.2.1.1 Change of Variables
As proposed for a two-dimensional problem, we rewrite Laplace equation given in
(5.1), in rectangular coordinates, as a first-order state equation by introducing two
new auxiliary state variables ξ1 and ξ2 as follows,
ξ1(x1, x2, x3) = u(x1, x2, x3),
ξ2(x1, x2, x3) =∂u
∂x1
,
(5.2)
123
Ω3
ΓB
ΓT
x1 − plane
Figure 5.1: Left: Domain Ω3 with two congruent parallel surfaces ΓB and ΓT (ΓB andΓT Lipschitz continuous); Right: Plane containing time-like co-ordinate x1.
where x1, x2 and x3 represent rectangular coordinates with x1x2 plane parallel to the
two congruent parallel surfaces ΓB,ΓT , by introducing these auxiliary variables the
resulting Laplace equation can now be written as,
∂ξ
∂x1
= Aξ, (5.3)
where,
ξ =
ξ1
ξ2
; A =
0 1
− 4x2,x3
0
; − 4x2,x3
= − ∂2
∂x22− ∂2
∂x32. (5.4)
Note 3: The state operator matrix A has two positive definite operators on anti-
diagonal, this clearly shows that A has both positive and negative eigenvalues. As
given above, the state operator matrix does not generate a strongly continuous semi-
group. The existence of exponential of such an operator, under certain conditions, is
studied in the following subsections.
Let A : D(A) → X, as given in equation (5.4), be defined on a rectangular cross-
124
section ω of Ω3 (parallel to x2x3 plane, as shown in Figure 5.2) such that,
ω =
x ∈ ω : x =
x2
x3
∈ [a2, b2]
[a3, b3]
;
, (5.5)
where b2 > a2, b3 > a3, a2, a3, b2, b3 ∈ R+, ω = ω ∪ Γl∪r∪t∪b, Γb = ω ∩ ΓB and
Γt = ω ∩ ΓT .
X = H1Γb
(ω)× L2 (ω) , (5.6)
D(A) =
[f ∈ H2 (ω) ∩H1
Γb(ω) :
df
ds|Γb = c2
]×H1
Γb(ω) , (5.7)
where,
H1Γb
(ω) =f ∈ H1 (ω) : f |Γb = c1
, (5.8)
and c1, c2 are constants (coming from Cauchy data at a particular point on Γb) and
X is a Hilbert space with scalar product given by,
⟨ q1
q2
,
p1
p2
⟩ =
∫ω
∇q1(x).∇p1(x)dω +
∫ω
q1(x).p1(x)dω +
∫ω
q2(x).p2(x)dω.
(5.9)
here x as given in equation (5.5). It can be seen that D(A∞) is dense in X. ξ1 and ξ2
x2
x3
ω
Γb
Γt
Γl
(a3, a2)
(a3, b2)
(b3, a2)
(b3, a3)
Γr
Figure 5.2: Left: Cross-sectional plane ω of Ω3; Right: x2x3 plane orientation (ingray);
are called state variables and using these new variables, problem (5.1) can be written
125
in an equivalent form as,
Find ξ1 on ΓT :
∂ξ
∂x1
= Aξ in Ω3,
Cξ = ξ1 = f on ΓB,
∂ξ1
∂n= g on ΓB,
∂ξ1
∂n= 0 on ∂Ω3\ ΓB ∪ ΓT ,
(5.10)
where C : X → Y is the boundary observation operator and Y be the output space
given as,
Y =
(f1)|Γb :
f1
f2
∈ X, . (5.11)
Y forms a Hilbert space with respect to the norm,
〈q1(x3), p1(x3)〉 =
∫Γb
q1(x3)p1(x3)dx3. (5.12)
Boundary value problem (5.10) has a first order state equation in variable x1 and
overdetermined data is available on ΓB. The solution of this first order state equation
leads to the study of semigroup generated by unbounded differential operator matrix
A. Further, we study observability for the pair (C,A) in infinite-dimensional setting.
5.2.2 Theoretical Analysis and Results
5.2.2.1 Existence of Exponential of A
Theorem 8. Let m,n ∈ Z? be the non-zero set of integers, for A : D(A) → X (as
given in (5.4), (5.6) and (5.7)) there exists an infinite set of orthonormal eigenvectors
(Φmn) and corresponding eigenvalues (λmn). Furthermore, A generates a strongly
126
continuous semigroup for vectors
p1
p2
∈ X, if and only if, the decay rate of the
sequence
⟨ p1
p2
,Φmn
⟩is greater than the growth rate of the sequence eλmnx1 for
all m,n ∈ Z?.
Proof. Let m,n ∈ Z? set of all integers such that,
Φmn(x2, x3) = ρmn
αmnϕmn(x2, x3)
βmnϕmn(x2, x3)
, (5.13)
be the infinite set of orthonormal eigenvectors of operator A and λmn be the eigen-
values such that,
AΦmn = λmnΦmn, (5.14) βmnϕmn
− 4x2,x3
(αmnϕmn)
= λmn
αmnϕmn
βmnϕmn
.
Assuming that αmn, βmn do not depend on x2, x3. First equation in (5.14) gives,
λmn =βmnαmn
. (5.15)
Second equation above suggests that we are interested in finding the eigen-pairs of
the Laplacian operator − 4x2,x3
over the domain ω. Here ω is a cross-sectional view
of Ω3 parallel to x2x3 plane. Without loss of generality, let us assume that ω =
(0, a1)× (0, a2) for a1, a2 ∈ R+ as shown in Figure 5.3.
127
x2
x3
ω = (0, a1)× (0, a2)
Γb
Γt
Γl Γr
Figure 5.3: Left: Cross-sectional plane ω of Ω3 parallel to x2x3 plane; Right: x2x3
plane orientation (in gray);
The eigenvalue problem can be given as,
− 4x2,x3
ϕmn = λ2mnϕmn in ω,
ϕmn = f |Γb on Γb,
∂
∂nϕmn = g|Γb on Γb,
∂
∂nϕmn = 0 on Γl ∪ Γr,
(5.16)
Sign of λ2mn is positive from the well-known fact that − 4
x2,x3
is a positive definite
operator, further this can also be verified using Rayleigh quotient test. There can be
multiple set of eigenbasis functions and corresponding eigenvalues for the eigenvalue
problem in above form. Again, without loss of generality, let us assume that g|Γb = 0
and a1 = a2 = π/4 and compute an infinite set of eigenpairs as follows,
ϕmn(x2, x3) = −π2
64cos((6− 8m)x2) cos(4nx3), (5.17)
βmn = λmn = ±√
(6− 8m)2 + (4n)2, (5.18)
ϕmn for m,n ∈ Z form an orthonormal basis in L2(ω).
Finally, an orthonormal set of eigenvectors can be formed in X with respect to
128
norm defined by (5.9) as,
Φmn(x2, x3) = ρmnφmn(x2, x3) = ρmn
αmnϕmn(x2, x3)
βmnϕmn(x2, x3)
, (5.19)
where ρmn is a non-zero normalization factor, given as,
ρ2mn =
4096
π
(16384(6− 8m)2n2
1 + 16(6− 8m)2n2 (1 + (6− 8m)2 + 16n2)
). (5.20)
Also ρmn is observed to be a fast decaying sequence and fundamental modes (ρ21,1 =
ρ21,−1) ≥ ρ2
mn for all m,n ∈ Z?. Now let us try to write semigroup generated by
operator matrix A as an infinite series,
∑m,n∈Z?
eλmnx1
⟨ p1(x2, x3)
p2(x2, x3)
,Φmn(x2, x3)
⟩Φmn(x2, x3), ∀
p1
p2
∈ X.(5.21)
For x1 = 0 the above infinite series is clearly convergent, whereas for x1 → 0+ the
limit does not exist. Further we note that above series expression (5.21) satisfies
identity and semigroup properties as given in Definition 11, however, it lacks strong
continuity, except if we assume that the projection terms in angle brackets above are
decaying at a rate faster than the growth rate of eλmnx1 . Now with the introduction of
this assumption the limit x1 → 0+ exists. There is a large class of smooth analytical
functions that satisfy this condition. Once again, this reveals a historically known
fact about solving Cauchy problems for steady-state heat equation that unique and
stable solutions do not exist for non-smooth data [39, 54]. Let the Hilbert space X,
as given in equations (3.7) and (3.10), be composed of two mutually exclusive parts
as
X = X1 ⊕X2, (5.22)
129
where X1 satisfy conditions as stated above such that A forms a strongly continuous
semigroup and X1 and X2 both make the full space X. Thus with this additional
smoothness assumption, equation (5.23) represents the strongly continuous semigroup
generated by operator matrix A.
Wx1
p1
p2
=∑
m,n∈Z?eλmnx1
⟨ p1
p2
,Φmn
⟩Φmn, ∀
p1
p2
∈ X1, (5.23)
where X = X1 ⊕X2 such that equation (5.23) generates a strongly continuous semi-
group and the inner product in (5.23) is defined by equation (5.9).
5.2.2.2 System Observability
Proposition 2. Let W be the strongly continuous semigroup generated by operator
matrix A under the assumptions as given in theorem (8). For any arbitrarily small
ε > 0 such that if |x1 − x1| < ε, the pair (C,A) is final state observable (hence exactly
observable using Note 1 in time-like interval |x1 − x1| > 0 at a particular x1, where
C ∈ L(X, Y ).
Proof. Let ξ(0) be the initial guess at x1 = 0, given by,
ξ(0) =
ξ1(0)
ξ2(0)
=
p1(x2, x3)
p2(x2, x3)
. (5.24)
Φmn(x2, x3) for m,n ∈ Z? be an orthonormal basis in X. Let us first prove the final
state observability condition for a general mode Φm′n′ with corresponding eigenvalue
130
λm′n′ as follows,
‖WxΦmn‖X =
∥∥∥∥∥ ∑m,n∈Z?
eλmnx1 〈Φm′n′ ,Φmn〉Φmn
∥∥∥∥∥X
, ∀ Φm′n′ ∈ X, m′, n′ ∈ Z?,
= eλmnx1 ‖Φmn‖X ,
= eλmnx1 , (5.25)
also,
‖CWx1Φmn‖Y =
∥∥∥∥∥ ∑m,n∈Z?
eλmnx1 〈Φm′n′ ,Φmn〉 CΦmn
∥∥∥∥∥Y
, ∀ Φm′n′ ∈ X, m′, n′ ∈ Z?,
= eλmnx1 ‖CΦmn‖Y ,
= eλmnx1|ρmn| ‖cos(4nx3)‖Y ,
=π
8eλmnx1|ρmn|. (5.26)
Comparing equations (5.25) and (5.26) implies,
‖CWx1Φmn‖Y ≥ k1 ‖Wx1Φmn‖X , (5.27)
if and only if,
k1 ≤π
8|ρmn|, (5.28)
for a particular choice of Φmn there always exists k1 such that final state observability
condition (5.27) is satisfied.
C ∈ L(X, Y ) is a linear boundary observation operator. Now let ξ(0) =∑
m,n∈Z? γmnΦmn
where γmn are projection terms whose decay rate is greater than the growth rate of
eλmnx1 with λmn as eigenvalues of A corresponding to eigenvectors Φmn. Clearly,∑m,n∈Z? γmn and
∑m,n∈Z? ρmn are bounded from above, hence,
‖CWx1ξ(0)‖Y ≥ k2 ‖Wx1ξ(0)‖X , ∀ ξ(0) ∈ X1, (5.29)
131
where k1, k2 both are independent of x1. Further using Note 1 for arbitrarily small ε
pair (C,A) is exactly observable.
This special exact and final state observability result for arbitrarily small ε suggests
that to recover solution, on a line (ω ∩ Γ2) at a particular x1, we do not require any
other measurement from adjacent lines except the line (ω ∩ Γ1). This implies we can
decompose the solution along time-like variable x1. Further for each x1 solution can
be computed in parallel.
5.2.3 Dimension Decomposition
Observability result given in Proposition 2 suggests that problem (5.10) can be solved
on a particular cross-section ω independent of any help from the adjacent cross-
sections. This, in other words, suggests that, in this particular setting (domain Ω3
with two congruent parallel surfaces as given in Figure. 5.1), diffusion along time-like
variable x1 is zero and, on a particular ω (as shown in Figure. 5.3), problem (5.10)
boils down to,
For all ω ∈ Ω3, find u(x3) on Γt:
4x2,x3
u =∂2u
∂x22
+∂2u
∂x23
= 0 in ω,
u = fsub
on Γb,
∂u
∂n= g
subon Γb,
∂u
∂n= 0 on Γl ∩ Γr,
(5.30)
132
where Γt = ω ∩ Γ2, Γb = ω ∩ Γ1, fsub
= f |Γb , gsub
= g |Γb . Let us further introduce two
auxiliary variables ζ1, ζ2 as follows,
ζ1(x2, x3) = u(x2, x3),
ζ2(x2, x3) =∂u
∂x3
,
(5.31)
x2 and x3 are rectangular co-ordinates, by introducing these auxiliary variables the
resulting Laplace equation can now be written as,
∂ζ
∂x3
= Asubζ, (5.32)
where,
ζ =
ζ1
ζ2
; Asub =
0 1
−4x2
0
; −4x2
= − ∂2
∂x22. (5.33)
ζ1 and ζ2 are called new state variables and using these variables, problem (5.66) can
be written in equivalent form as,
For all ω ∈ Ω3, find ζ1(x2, x3) on Γt:
∂ζ
∂x3
= Asubξ in ω,
Csubζ(x3) = ζ1(x3) = f
subon Γb,
∂ζ1
∂x2
= gsub
on Γb,
∂ζ1
∂x3
= 0 on Γl∪r,
(5.34)
where Csub is the observation operator such that Csubζ = ζ1 |Γb . A boundary value
problem in this form has a first-order state equation in variable x3 and overdetermined
data is available on Γb. Before the introduction of iterative observer equations, let us
133
assume that left-hand boundary Γl is connected to right-hand boundary Γr to have
the notion of infinite time-like variable x3 over ω. The reason for having such an
assumption is that we are trying to develop an observer using space as time-like and
hoping that this observer will converge asymptotically in time-like variable x3. Let
m be a non-negative integer index of iteration over the domain ω in x3-direction. Let
x[m]3 , as given in Definition (15), represents x3 ∈ [a3, b3] for the m-th iteration over
the interval [a3, b3].
5.2.4 Observer Design for the Subproblem
Theorem 9. For consistent Cauchy data boundary value problem (5.35) asymptoti-
cally (m→∞) converges to the true solution of boundary value problem (5.34).
For all ω ∈ Ω3,
∂
∂x3
ζ(x2, x[m]3 ) = A
subζ(x2, x
[m]3 )−K C
sub(ζ(x2, x
[m]3 )− ζ) in ω,
∂
∂x2
ζ1(x2, x[m]3 ) = g
subon Γb,
∂2
∂x22
ζ1(x2, x[m]3 ) = − ∂2
∂x23
ζ1(x2, x[m]3 )−K C
sub(ζ(x2, x
[m]3 )− ζ) on Γt,
ζ(x2, x[m]3 ) |initial= ζ(x2, x
[m−1]3 ) in ω,
(5.35)
where ”ˆ” represents estimated quantity and ζ(x2, x[m]3 ) |initial represents the estimate
over the whole domain ω at the start of m-th iteration. Observer starts at index m = 1
which represents first iteration. ζ(x2, x[m=0]3 ) is initial guess at the start of the first
iteration over the whole domain Ω3. Any value of initial guess ξ(x2, x[m=0]3 ) can be
chosen at the start of first iteration. For each subsequent iteration, result of the
previous iteration is used as initial estimate as given in the last equation in (5.35).
Third equation in (5.35) is the assumption that Laplace equation is valid on the top
boundary and this provides necessary boundary condition required on Γt. Csub
is the
134
observation operator such that Csubζ = ζ1 |Γb . K is the correction operator chosen in
such a way that state estimation error on Γb given by ( Csubζ − C
subζ)) converges to zero
exponentially.
Proof. Semigroup generated by Asub
, observability study for pair ( Csub, Asub
) and proof of
theorem (9) is provided in chapter 3.
5.2.5 Numerical Implementation and Simulation Results
For the numerical simulations, a rectangular prism Ω3 is considered such that,
Ω3 =
x ∈ Ω3 : x =
x1
x2
x3
∈
[0, 2π][0, π
4
][0, 2π]
;
. (5.36)
A well-posed boundary value problem is solved using the method of separation of
variables to find analytical solution. The surface potential u = cos(x1) cos(x3) is
applied at Γ2 (x1-x3 plane at x2 = π/4). Homogeneous Neumann boundary∂u
∂n= 0
is considered on all other boundary surfaces. This gives analytical solution over the
whole domain as,
u(x1, x2, x3) =cosh(
√2x2)
cosh(√
2π4)
cos(x1) cos(x3). (5.37)
Analytical solution on Cauchy surface Γ1 (x1-x3 plane at x2 = 0) along with homoge-
neous Neumann boundary condition∂u
∂x2
= 0 on Γ1 is used as Cauchy data. Observer
algorithm is run for a number of iterations over all ω using parallel processing. The
analytical solution on Γ1 and Γ2 are shown in Figure 5.4 and 5.5 respectively. The
solution recovered using the iterative observer and divide and conquer approach is
presented in Figure 5.6. The difference between the analytical and recovered solution
on Γ2 is shown in Figure 5.7.
135
Figure 5.4: Analytical solution u on Γ1 Figure 5.5: Analytical solution u on Γ2
Figure 5.6: Recovered solution on Γ2
using observer algorithmFigure 5.7: Difference of the analyti-cal solution and the one recovered byobserver algorithm on Γ2
5.3 Point Source Localization Problem for Poisson Equation
in 3D
In this section, an iterative observer-based method is developed to solve point source
localization problem for Poisson equation in a 3D rectangular prism with available
boundary data. The technique requires a weighted sum of solutions of multiple bound-
ary data estimation problems for Laplace equation over 3D domain. The solution of
each of these boundary estimation problems involves writing down the mathematical
problem in state-space-like representation using one of the space variables as time-like.
First system observability result for 3D boundary estimation problem is recalled in
136
an infinite dimensional setting. Then, based on the observability result, the boundary
estimation problem is then decomposed into a set of independent 2D sub-problems.
These 2D problems are then solved using an iterative observer to obtain the solution.
Theoretical results are provided. The method is implemented numerically using fi-
nite difference discretization schemes. Numerical illustrations along with simulation
results are provided.
5.3.1 Problem Formulation
Let Ω4 be a rectangular prism in R3 as shown in Figure 5.8 and ∂Ω4 = ∪6i=1Γi be the
boundary of Ω4. Let us consider Poisson equation,
4u = f in Ω4, (5.38)
with Laplacian operator 4 =∂2
∂x2+
∂2
∂y2+
∂2
∂z2. Let the steady-state potential field
u is generated by a number of distinct point sources inside the rectangular prism Ω4:
f(x, y, z) =N∑k=1
Ckδ(x− xk, y − yk, z − zk) (x, y, z) ∈ Ω4, (5.39)
where δ(x− x, y− y, z− z), k = 1, · · · , N represent Dirac delta point sources localized
at (xk, yk, zk) and scalars Ck, k = 1, · · · , N are the corresponding magnitudes.
The objective here is to localize point sources δ’s for all k = 1, · · · , N from available
Cauchy data on the boundary ∂Ω4 given by,
u = g on ∂Ω4,
∂u
∂n= h on ∂Ω4.
(5.40)
In the following section a strategy to localize point sources is presented.
137
x
y
z
Γ2
Γ1
Figure 5.8: Rectangular prism Ω4 with six boundary surfaces, ∂Ω4 = ∪6i=1Γi. Surfaces
Γ1, . . . ,Γ6 represent bottom, top, and side surfaces respectively.
5.3.2 Point Source Localization as Boundary Estimation Prob-
lem
First of all, let us consider that available Cauchy data as given in equation (5.40) has
various components on different surfaces of the Ω4 such that,
g = ∪6i=1gi|Γi , (5.41)
h = ∪6i=1hi|Γi . (5.42)
In this section, a mathematical result is presented that boils down the source lo-
calization problem given in section 5.3.1 to a weighted sum of following boundary
estimation problems for Laplace equation in 3D.
For i ∈ 1, . . . , 6, find steady-state potential field Si on Γi,opp:
4Si = 0 in Ω4,
Si = gi on Γi,
∂Si∂n
= hi on Γi,
∂Si∂n
= 0 on Γi,adjs,
(5.43)
where Γi,opp is the surface opposite to Γi and Γi,adjs are the surfaces connected to Γi.
138
Theorem 10. For all i in integer set 1, . . . , 6, let Si be the steady-state potential
field over Ω4 obtained by solving boundary estimation problem (5.43). Let for all i,
vi be the solution obtained by solving boundary value problem for Laplace equation,
4vi = 0 in Ω4,
vi = 1 on Γi,
vi = 0 on ∂Ω4\Γi,
(5.44)
then,
u =1
2
[6∑i=1
Sivi
]in Ω4, (5.45)
f =6∑i=1
∇Si.∇vi in Ω4, (5.46)
where u and f solve the boundary value problem for Poisson equation given by equa-
tions (5.38) and (5.40).
Proof. We have Si and vi satisfying Laplace equation for all i ∈ 1, . . . , 6. From
equation (5.45) we can write,
∇u =1
2
6∑i=1
(Si∇vi + vi∇Si) , (5.47)
∇.(∇u) = ∇.
(1
2
6∑i=1
(Si∇vi + vi∇Si)
), (5.48)
which gives,f = 4u =
6∑i=1
∇Si.∇vi. (5.49)
thus we have that u as given in equation (5.45) satisfies Poisson equation 4u = f up
to an additive constant. Now using the properties of vi, Si we have to show that u
and f as given by equations (5.45) and (5.46) satisfy boundary value problem given
139
by equations (5.38) and (5.40). We can write,
∫Ω4
[6∑i=1
vi4Si
]dΩ4 = 0, (5.50)
using Green’s first identity,
∫∂Ω4
[6∑i=1
vi∂Si∂n
]d∂Ω4 =
∫Ω4
[6∑i=1
∇vi.∇Si
]dΩ4, (5.51)
We have vi = 1 on Γi and zero elsewhere on boundary. This gives,
∫∂Ω4
[6∑i=1
vi∂Si∂n
]d∂Ω4 =
6∑i=1
[∫Γi
∂Si∂n
d∂Ω4
]=
∫∂Ω4
h dΩ4, (5.52)
applying divergence theorem we have,
∫∂Ω4
h dΩ4 =
∫Ω4
∇.∇u dΩ4 =
∫Ω4
f dΩ4. (5.53)
Combining equations (5.51), (5.52) and (5.53),
∫Ω4
f dΩ4 =
∫Ω4
[6∑i=1
∇vi.∇Si
]dΩ4, (5.54)
above relation is true for all sizes of 3D rectangular prisms Ω4, thus we have,
f =6∑i=1
∇Si.∇vi. (5.55)
Remark 3. The mathematical result presented in Theorem 10 is valid for all sizes
of 3D rectangular prisms.
140
5.3.3 Two-step Process for Source Localization:
Based on above result, a two-step source localization technique can be developed,
Step 1 : For all i, solve boundary estimation problem (5.43) to find steady-state
potential field Si over Ω4.
Step 2 : For all i, compute weight profiles vi by solving well-posed boundary value
problem (5.44). Combine Si’s and vi’s using equation (5.45) to have an estimate
of Poisson problem solution u over Ω4. The position of Dirac delta sources inside
the domain is provided by the maxima or minima of estimated u.
In the following section, the divide and conquer approach and iterative observer al-
gorithm are used to solve boundary estimation problem (5.43).
5.3.4 Boundary Estimation Problem for Laplace Equation
In this section, the divide and conquer approach, based on iterative observer design,
is used to solve boundary estimation problem for Laplace equation. The technique
is presented to solve problem (5.43) for index i = 1, however, the theoretical results
and implementation technique remain the same for all i ∈ 1, . . . , 6.
5.3.5 Preliminary Theoretical Results
Let us introduce two auxiliary variables ξ1 = S1 and ξ2 =∂S1
∂xto rewrite Laplace
equation in problem (5.43) as,
∂ξ
∂x= Aξ, (5.56)
where,
ξ =
ξ1
ξ2
; A =
0 1
−4y,z
0
; −4y,z
= − ∂2
∂y2− ∂2
∂z2. (5.57)
141
Let A : D(A) → X be defined on a rectangular cross-section ω of Ω4, parallel to yz
plane, as shown in Figure 5.9 such that,
ω =
α ∈ ω : α =
y
z
∈ [a2, b2]
[a3, b3]
;
, (5.58)
where b2 > a2, b3 > a3, a2, a3, b2, b3 ∈ R+, ω = ω ∪ Γl∪r∪t∪b, Γb = ω ∩ Γ1 and
Γt = ω ∩ Γ2.
X = H1Γb
(ω)× L2 (ω) , (5.59)
D(A) =
[f ∈ H2 (ω) ∩H1
Γb(ω) :
df
ds|Γb = c2
]×H1
Γb(ω) , (5.60)
where,H1
Γb(ω) =
f ∈ H1 (ω) : f |Γb = c1
, (5.61)
and c1, c2 are constants (coming from Cauchy data at a particular point on Γb) and
X is a Hilbert space with scalar product given by,
⟨ q1
q2
,
p1
p2
⟩ =
∫ω
∇q1(α).∇p1(α)dω
+
∫ω
q1(α).p1(α)dω +
∫ω
q2(α).p2(α)dω. (5.62)
here α as given in equation (5.58). It can be seen that D(A∞) is dense in X. Problem
(5.43) for index i = 1 can be written in control familiar form using auxiliary variables
as,
142
x
y
z
ωω
Γt
Γb
Γl Γr
Figure 5.9: Rectangular cross-section ω at a particular value of x in yz-plane insideΩ4.
Find S1 on Γ1,opp (:= Γ2):
∂ξ
∂x= Aξ in Ω4,
Cξ = ξ1 = g1 on Γ1,
∂ξ1
∂n= h1 on Γ1,
∂ξ1
∂n= 0 on Γ1,adjs,
(5.63)
where C : X → Y is the boundary observation operator and Y be the output space
given as,
Y =
(f1)|Γb :
f1
f2
∈ X, . (5.64)
Y forms a Hilbert space with respect to the norm,
〈q1(y), p1(y)〉 =
∫Γb
q1(y)p1(y)dy. (5.65)
Boundary value problem (5.63) has a first order state equation in variable x and
overdetermined data is available on Γ1. The solution of this first-order state equation
leads to the study of semigroup generated by unbounded differential operator matrix
A. Further we study observability for the pair (C,A) in infinite dimensional setting.
143
Proposition 3. Let m,n ∈ Z? be the non-zero set of integers, for A : D(A) → X
(as given in (5.57), (5.59) and (5.60)) there exists an infinite set of orthonormal
eigenvectors (Φmn) and corresponding eigenvalues (λmn). Furthermore, A generates
a strongly continuous semigroup for vectors
p1
p2
∈ X, if and only if, the decay
rate of the sequence
⟨ p1
p2
,Φmn
⟩is greater than the growth rate of the sequence
eλmnx for all m,n ∈ Z?.
Proof. Proof of above proposition is similar to the result presented in Theorem 8.
5.3.6 Observability Result
Theorem 11. Let W be the strongly continuous semigroup generated by operator
matrix A under the assumptions as given in Proposition 1. For any arbitrarily small
ε > 0 such that if |x− x| < ε, the pair (C,A) is exact and final state observable in
time-like interval |x− x| > 0 at a particular x, where C ∈ L(X, Y ).
Proof. The proof of above theorem is similar to Proposition 2.
This special exact and final state observability result for arbitrarily small interval ε
suggests that to recover solution, on a line (ω∩Γ2) at a particular x, we do not require
any other measurement from adjacent lines except the line (ω ∩ Γ1). This implies we
can decompose the solution along time-like variable x. Further for each x solution
can be computed in parallel.
5.3.7 Dimension Decomposition
Observability result given in Theorem 2 suggests that problem (5.63) can be solved on
a particular cross-section ω independent of any help from the adjacent cross-sections.
This, in other words, suggests that diffusion along time-like variable x is zero and, on
a particular ω (as shown in Figure 5.9), problem (5.63) boils down to,
144
For all ω ∈ Ω4, find u(y) on Γt:
4y,zu =
∂2u
∂y2+∂2u
∂z2= 0 in ω,
u = g1sub
on Γb,
∂u
∂n= h1
subon Γb,
∂u
∂n= 0 on Γl ∩ Γr,
(5.66)
where g1sub
= g1 |Γb , h1sub
= h1 |Γb . Let us further introduce two auxiliary variables ζ1, ζ2
as follows, ζ1(y, z) = u,
ζ2(y, z) =∂u
∂y,
(5.67)
further by introducing these auxiliary variables the resulting Laplace equation can
now be written as,
∂ζ
∂y= A
subζ, (5.68)
where,
ζ =
ζ1
ζ2
; Asub
=
0 1
−4z
0
; −4z
= − ∂2
∂z2. (5.69)
ζ1 and ζ2 are called new state variables and using these variables, problem (5.66) can
be written in equivalent form as,
For all ω ∈ Ω4, find ζ1(y, z) on Γt:
∂ζ
∂y= A
subξ in ω,
Csubζ(y) = ζ1(y) = g1
subon Γb,
∂ζ1
∂z= h1
subon Γb,
∂ζ1
∂y= 0 on Γl∪r.
(5.70)
145
where Csub
is the observation operator such that Csubζ = ζ1 |Γb . Boundary value problem
in this form has a first order state equation in variable y and overdetermined data
is available on Γb. Before the introduction of iterative observer equations, let us
assume that left-hand boundary Γl is connected to right-hand boundary Γr to have
the notion of infinite time-like variable y over ω as shown in Figure 5.11. The reason
for having such an assumption is that we present an observer using space as time-like
with asymptotic convergence in time-like variable y. Let m be a non-negative integer
index of iteration over the domain ω in y-direction. Let y[m] represents y ∈ [a2, b2] for
the m-th iteration over the interval [a2, b2].
5.3.8 Observer Design for the Subproblem
For all ω ∈ Ω4:
∂
∂yζ(y[m], z) = A
subζ(y[m], z)−K C
sub(ζ(y[m], z)− ζ) in ω,
∂
∂zζ1(y[m], z) = h1
subon Γb,
∂2
∂z2ζ1(y[m], z) = − ∂2
∂y2ζ1(y[m], z)
−K Csub
(ζ(y[m], z)− ζ) on Γt,
ζ(y[m], z) |initial= ζ(y[m−1], z) in ω,
(5.71)
where“ˆ”represents estimated quantity and ζ(y[m], z) |initial represents the estimate
over the whole domain ω at the start of m-th iteration.
Observer starts at index m = 1 which represents first iteration. ζ(y[m=0], z) is initial
guess at the start of the first iteration over the whole domain Ω4. Any value of initial
guess ξ(z, y[m=0]) can be chosen at the start of first iteration. For each subsequent
iteration, result of the previous iteration is used as initial estimate as given in the
146
θ1
θ2
θ3
θ4
θ1
θ2
θ3
θ4
ω
Figure 5.10: Idea of iterations over rectangular cross-section ω.
last equation in (5.71). The third equation in (5.71) is the assumption that Laplace
equation is valid on the top boundary and this provides necessary boundary condition
required on Γt. Csub
is the observation operator such that Csubζ = ζ1 |Γb= g1
sub|Γb . K is
the correction operator chosen in such a way that state estimation error on Γb given
by ( Csubζ − C
subζ) converges to zero asymptotically.
Theorem 12. For consistent Cauchy data boundary value problem (5.71) asymptot-
ically (m→∞) converges to the true solution of boundary value problem (5.70).
Proof. Study of existence of exponential of Asub
under certain conditions, observability
study for pair ( Csub, Asub
) and proof of Theorem 12 are similar to the results presented
in Chapter 3.
5.3.9 Numerical Implementation and Results
In this section, numerical implementation and simulation results are presented. For
illustrative purposes, a unit cube is considered and finite difference discretization
schemes are used to implement two-step strategy for source localization. Figure 5.11
represents three orthogonal cross-sectional planes inside Ω4 which are used for 3D
visualization of different simulation plots. Figure 5.12 represents the solution of Pois-
son equation in the unit cube with homogeneous Neumann boundary data on ∂Ω4
and one point source in the middle of the domain. The solution is presented on a
147
x
y
z
Figure 5.11: Domain Ω4 with three orthogonal cross-sectional planes.
coarse 20× 20× 20 uniform grid. Solution over a particular cross-section ω is shown
in Figure 5.13 on a uniform 200 × 200 uniform grid. Because of symmetry solution
over all the three orthogonal cross-sectional planes look alike.
Now to localize this particular point source from boundary data it is required
to solve boundary estimation problem (5.43) for all i ∈ 1, . . . , 6 using dimension
decomposition approach presented in section 5.3.4. For this, using the special ob-
servability result for dimension decomposition presented in section 5.3.4, a particular
cross-section ω is considered. Dirichlet data g1sub
on Γb is extracted from ω shown in
Figure 5.13. This Dirichlet boundary data on Γb along with homogeneous Neumann
boundary data is used in iterative observer algorithm (5.71) to estimate boundary
data on the opposite boundary Γt as shown in Figure 5.14. Similarly, the iterative
observer solution is computed on all cross-sections parallel to ω to make up the full
solution profile S1 over Ω4. Similarly S2, . . . , S6 can be computed. Figure 5.15 shows
solution plot for S2 over Ω4.
Well-posed boundary value problem (5.44) is solved numerically for all i ∈ 1, . . . , 6
to compute weight profiles v1, . . . , v6. Figure 5.16 shows a particular weight profile v2
over Ω4. Finally the weighted sum given in equation (5.45) is computed numerically.
Figure 5.17 shows the weighted sum over Ω4. The local maxima in the center locate
the point source.
148
Figure 5.12: A single point source inthe middle of a 1 × 1 × 1 cube on auniform 20× 20× 20 grid.
Figure 5.13: Cross-sectional plane ωfrom Figure 5.12 on a 200 × 200 uni-form grid.
5.4 Conclusion
The divide and conquer approach presented in this chapter helps to solve boundary
estimation problem for the Laplace equation, in a particular 3D domain, as a num-
ber of independent 2D subproblems. The subproblems can be solved simultaneously,
using parallel implementation, which makes this algorithm advantageous compared
to the techniques in literature. Most of the previously existing techniques are op-
timization based methods and require the solution of the mathematical problem in
3D for a number of times with some cost criteria to obtain the convergence. Further
this demonstration of control-based algorithm solving a steady-state estimation prob-
lem in 3D highlights the potential that dynamical systems inspired algorithms can
be potentially advantageous and inspiring to develop solution techniques for steady-
state PDE problems. Further, the dimension decomposition approach presented in
this chapter is extended to source localization and estimation problems for Poisson
equation. Numerical simulation results highlight the usefulness of the proposed algo-
rithms.
149
0 0.2 0.4 0.6 0.8 1
−50
−40
−30
−20
−10
0
Am
plit
ude
Iterative observer solution
Dirichelt data g1sub
on Γb
Estimated S1 on opposite boundary
Figure 5.14: Iterative observer solution from equation (5.71) with h1,sub = 0 andg1,sub extracted from Figure 5.13 on Γb and recovered boundary data on oppositeboundary. Similarly iterative observer solution can be computed on cross-sectionalplanes parallel to ω.
Figure 5.15: Solution profile S2 on a20 × 20 × 20 uniform grid, obtainedby solving boundary estimation prob-lem (5.43) for i = 2 using iterative ob-server.
Figure 5.16: Weight profile v2 obtainedby solving boundary estimation prob-lem (5.44) for i = 2 on a uniform20 × 20 × 20 grid. (only two orthog-onal cross-sectional planes displayed)
150
Figure 5.17: Weighted sum computed using equation (5.45). Local maxima in thecenter locate the position of the point source.
151
Chapter 6
Concluding Remarks
Sometimes when I consider what tremendous consequences come from little
things... I am tempted to think... There are no little things.
(—Bruce Barton)
In this chapter, concluding discussions are provided about the iterative observer
algorithms, for source and boundary data estimation problems, presented in this
thesis. The algorithms tackle time independent steady-state PDE systems with dy-
namical system inspired algorithms using space variable as time-like. The resultant
algorithms are shown to be robust to noise and at the same time easier to implement.
6.1 Summary of the Thesis Work
An iterative observer algorithm is presented to solve boundary data estimation prob-
lem for Laplace equation in chapter 3. The observer algorithm works for smooth data
case using one of the space variables as time. The algorithm sweeps over the whole
domain for a number of iterations to obtain convergence. Numerically accurate and
stable results are presented. However, the partially available boundary data is usually
corrupted with noise in most of the practical scenarios. Further, this boundary data
estimation problem with available noisy boundary data is highly ill-posed in the sense
of stability. An optimal iterative algorithm is presented that tackles this particular
ill-posed scenario.
The iterative observer algorithm is then used to develop a strategy for source
localization and estimation for a system governed by the Poisson equation in chapter
152
4. This strategy is detailed along with simulation results for both noisy and noise-
free cases. Several test scenarios are considered. Finally an approach to tackle higher
dimensional problems is provided. The method provides a way to divide and conquer
a three dimensional problem (three dimensional domain with two congruent parallel
surfaces) to a set of two dimensional independent subproblems. These subproblems
are then used using iterative observer algorithm. Similarly, a source estimation and
localization strategy is developed for the Poisson equation problem.
The algorithms presented in this thesis are formulated as a state space like system,
that is, by writing a second-order steady-state elliptic PDE system as a first order
evolutionary ODE system. The numerical implementation is done using finite differ-
ences and is fairly simple compared to the existing optimization based techniques.
The main advantage is that the problem is formulated on a sub domain, that is,
rather than solving a three dimensional problem over three dimensional domain for
several times, a number of two dimensional problems are solved for a number of times
to obtain convergence.
6.2 Future Research Directions
The work presented in this thesis proposal can be extended in the following directions.
6.2.1 Extensions to Arbitrary Shaped Domains
The idea of using space variable as time-like put some restrictions on the shape of the
domain under consideration. The algorithms presented in this thesis work are applied
two dimensional regular shaped domains and three dimensional domains with two
congruent parallel surfaces. The extension of these algorithms to other arbitrarily
shaped domains need further investigation. Similarly, the source localization and
estimation strategy has been applied to similar kind of two and three dimensional
domains.
153
Ω
Γ∗
Γ∗
Ω
Ω Γ∗
Ω
Γ∗
Figure 6.1: Domains under considera-tion for the boundary estimation prob-lem for Laplace equation with Γ∗, theunknown data boundary.
Ω∗
∗
*Ω
Ω
*
*
Figure 6.2: Domains under consider-ation for the source localization andestimation problems for Poisson equa-tion.
6.2.2 Iterative Observer Applications to Other Steady-State
PDE Systems
The algorithms presented in this document tackle Laplace and Poisson types of linear
systems. The results presented in this thesis are first steps towards the development
of dynamical systems inspired algorithms for steady-state PDEs, thus extending the
existing concepts of control theory towards all kinds of PDE systems. The results
presented in this document are promising. However, further investigations need to be
done to tackle other steady-state linear and non linear PDE systems.
6.2.3 Steady-State Energy Field Imaging Technique
Well-known tomographic imaging techniques in radiology, geophysics, biology, arche-
ology and materials science use penetrating waves to construct the the object’s form,
especially inner visual representation. Almost all of these techniques are sensitive to
the mass density variations inside the object. For example, X-ray imaging in biomed-
ical works on the principle that high energy X-rays pass through different parts of a
patient’s body, and depending on the attenuations caused by mass density variations
154
(tissues and bones), a different intensity of X-ray will come out of the body from the
opposite end. The X-rays coming out are recorded on a film, which then provides the
information about mass density variations inside the body.
The source estimation strategy presented in Chapter 5 can be extended and used
as a steady-state energy field imaging technique in an n-D rectangular prism. The
method does not use wave penetration, rather it uses steady-state diffusive model to
recover distributed potential field inside a homogenous medium. The technique is
not sensitive to mass density variations, however, it can well recover the steady-state
forced fields inside the medium. Some preliminary numerical results are presented, in
a 3D rectangular prism domain, for the steady-state diffusion model with distributed
sources.
6.2.3.1 Initial Simulation Results
In this section, numerical implementation is detailed and initial simulation results
are presented. For illustrative purposes, a unit cube is considered and finite differ-
ence discretization schemes are used to implement two-step strategy for steady-state
potential field imaging. As shown in Chapter 5, Fig. 5.11 represents three orthogo-
nal cross-sectional planes inside Ω4 which are used for 3D visualization of different
simulation plots. Fig. 6.3 represents the solution of Poisson equation in the unit
cube with homogeneous Neumann boundary data on ∂Ω4 and a distributed sources
f = exp(−(2.5(x−0.5))2−(2.5(y−0.5))2−(2.5(z−0.5))2) centered inside the domain.
This particular choice for f is for illustrative purposes as it is centered inside the unit
cube domain. The solution is presented on a coarse 50 × 50 × 50 uniform grid. A
cross-sectional view of this numerical solution is also shown in Fig. 6.4 at x = 0.5.
Because of the qualitative similarity of this particular example, the cross-sectional
view will look alike at y = 0.5 and z = 0.5. The distributed potential field inside the
Ω4, as shown in Fig. 6.3, is to be recovered applying the imaging technique, presented
155
here, using only the boundary data with added noise.
Now, to apply the steady-state potential field imaging technique, the Dirichlet
boundary data g is extracted from the solution presented in Fig. 6.3 over ∂Ω4.
Based on the dimension decomposition approach, and the special observability result
presented in chapter 5, each particular cross-section ω is considered independent of
adjacent cross-sections. The Dirichlet data is extracted for each particular ω. This
Dirichlet data is then corrupted with additive white Gaussian noise with η1 = 0 and
σ1 = 5 × 10−3. This noisy Dirichlet data g1sub
at Γb = ω ∩ Γ1 is shown in blue in
Fig. 6.5. The data on the remaining boundaries of ω|x=0.5 will look alike because
of qualitative similarity. The solution of boundary value problem (5.70) is obtained
by applying optimal iterative algorithm (4.49), (4.50) and is shown in Fig. 6.5 in
black. The optimal iterative algorithm recovers the boundary estimate at the opposite
boundary Γt by solving the boundary estimation problem for the Laplace equation
(5.70). The corresponding solution profile S1 at ω|x=0.5 is shown in Fig. 6.6. Similarly
the solution profile S1 is computed on all cross-sections parallel to ω|x=0.5. The
full tomographic image profile S1 (solution of problem 5.43 for i = 1) over Ω4 is
shown in Fig. 6.7, this solution is obtained by stacking up all the S1 solutions over
all the cross-sections parallel to ω|x = 0.5. The weight profile v1, corresponding to
tomographic image S1 over Ω4, is obtained by numerically solving boundary value
problem (5.44), for i = 1, as shown in Fig. 6.8. In a similar way S2, . . . , S6 and
v2, . . . , v6 are computed numerically over Ω4. Finally, steady-state potential field
image is generated using equation (5.45) as shown in Fig. 6.9. A cross-sectional view
at ω|x=0.5 is also shown in Fig. 6.10. The algorithm recovers the distributed steady-
state potential field reasonably well, Fig.s 6.3 and 6.4 can be compared with Fig.s 6.9
and 6.10 respectively.
156
Figure 6.3: Numerical solution of thePoisson equation 4u = f over Ω4 withf = exp(−(2.5(x − 0.5))2 − (2.5(y −0.5))2 − (2.5(z − 0.5))2) and homoge-neous Neumann boundary data at ∂Ω4.The solution is computed over a 50 ×50 × 50 uniform grid using 2nd orderaccurate centered finite difference dis-cretization schemes.
Figure 6.4: A particular cross-sectionalview of the numerical solution pre-sented in Fig. 6.3, at x = 0.5.
0 0.2 0.4 0.6 0.8 1
−30
−20
−10
0
Am
plit
ude
Dirichlet data and recovered signal on opposite boundaries
Noisy Dirichlet data on Γb
Estimated S1 on opposite boundary
Figure 6.5: (In blue): Dirichlet data, extracted from Fig. 6.4 at Γb = ω∩Γ1, corruptedwith added white Gaussian noise with η1 = 0 and σ1 = 5×10−3; (In black) Numericalsolution of boundary estimation problem (5.66), obtained by using optimal iterativealgorithm (4.49), (4.50) at cross-section ω|x=0.5 at Γt = ω ∩ Γ2.
157
Figure 6.6: Full solution profile overcross-section ω|x=0.5 obtained by usingfinite difference discretization schemesand the estimated boundary data fromFig. 6.5
Figure 6.7: Tomographic image S1 overΩ4 obtained by solving boundary esti-mation problem for Laplace equation(5.43) using dimension decompositionapproach and optimal iterative algo-rithm.
Figure 6.8: Weight profile v1 obtained by numerically solving boundary value problem(5.44) for index i = 1.
158
Figure 6.9: Weighted sum obtainedfrom equation (5.45) by combiningtomographic image profiles S1, . . . , S6
and corresponding weights v1, . . . , v6.The simulation result shows recovery ofthe distributed potential field presentedin Fig. 6.3, using only the noise cor-rupted boundary data.
Figure 6.10: A cross-sectional view ofweighted sum over ω|x=0.5. The resultabove to be compared with the cross-sectional view given in Fig. 6.4.
159
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APPENDICES
A Appendix :
Proof of Some Theorems
Theorem 1
Proof. If Ψτξ0 = 0, then the analytic function y(t) = CetAξ0 vanishes identically on
[0, τ ]. In particular, CAkξ0 = y(k)(0) = 0 for all k ≥ 0 which shows that ξ0 is also in
kernel of the matrix appearing in the right-hand side of (2.4).
Conversely, if CAkξ0 = 0 for all 0 ≤ k ≤ n − 1, then CAkξ0 = 0 for all k ≥ 0,
since by Cayley-Hamilton theorem, the powers Ak for k ≥ n are linear combinations
of Ak for 0 ≤ k ≤ n− 1. Consequently, y(t) = CetAξ0 =∑∞
k=0
tk
k!CAkξ0 = 0 for all t
and thus ξ0 ∈ KerΨτ . The fact that (2.4) provides characterization for observability
follows immediately from (2.5).
A.1 An Example of Exactly Observable System based on
String Equation [11]
In this section a semigroup, associated with the equations modelling the vibration of
an elastic string of length π fixed at both ends, is constructed. Initial boundary value
166
problem for one dimensional wave equation can be written as,
∂2w
∂t2(x, t) =
∂2w
∂x2(x, t) x ∈ (0, π), t ≥ 0,
w(0, t) = 0, w(π, t) = 0 t ∈ [0,∞),
w(x, 0) = f(x),∂w
∂t(x, 0) = g(x) x ∈ (0, π).
(A.1)
by setting,
ξ(t) =
w(., t)
∂w
∂t(., t)
, (A.2)
system of equations (A.1) can be written as,
ξ(t) = Aξ(t) ∀t ≥ 0, ξ(0) =
f
g
. (A.3)
A.1.1 Semigroup Generated by A
Let us denoteX= H10 (0, π)×L2(0, π), which is a Hilbert space with the scalar product,
⟨ f1
g1
,
f2
g2
⟩ =
∫ π
0
df1
dx(x)
df2
dx(x)dx+
∫ π
0
g1(x)g2(x)dx. (A.4)
Define, A: D(A)→ X by
D(A) = [H2(0, π) ∩H10 (0, π)]×H1
0 (0, π), (A.5)
A
f
g
=
g
d2f
dx2
∀
f
g
∈ D(A). (A.6)
Let us denote by Z∗ the set of non-zero integers. For n ∈ Z∗, denote φn(x) =√2
πsin(nx). Family of functions (φn)n∈N is an orthonormal basis in L2[0, π]. This
167
implies that the family, defined by
Φn =1√2
1
ιnφn
φn
∀n ∈ Z∗, (A.7)
is an orthonormal basis in X with respect to the norm (A.4). The vectors φn from
(A.5) are eigenvectors ofA and the corresponding eigenvalues are λ = ιn, with n ∈ Z∗.
Operator A generates a semigroup T on X. T is given by
Tt
f
g
=∑n∈Z∗
eιnt
⟨ f
g
, φn
⟩φn ∀
f
g
∈ X. (A.8)
From the above relation it follows that,
Tt
f
g
=1√2
∑n∈Z∗
eιnt
(ι
n
⟨df
dx,dφndx
⟩L2[0,π]
+ 〈g, φn〉L2[0,π]
)φn. (A.9)
A.1.2 Observability
Proposition 4. Let X= H10 (0, π) × L2(0, π), and let A be the operator defined by
(A.6). Denote Y= C and consider the observation operator C∈ L(X,Y ) defined by,
C
f
g
=df
dx(0) ∀
f
g
∈ D(A). (A.10)
Then the pair (A, C) is exactly observable in any time τ ≥ 2π.
Proof. By using formulas (A.7) and (A.9), we have that, for all
f
g
∈ D(A),
CTt
f
g
=1√2
∑n∈Z∗
eιnt
(⟨df
dx, ψn
⟩L2[0,π]
− ι 〈g, φn〉L2[0,π]
), (A.11)
168
where ψn(x) =
√2
πcos(nx) for all n ∈ Z. The above formula and orthogonality of
family (eιnt)n∈Z∗ in L2[0, 2π] imply that,
∫ 2π
0
∣∣∣∣∣∣∣CTt f
g
∣∣∣∣∣∣∣2
dt =∑n∈Z∗
∣∣∣∣∣⟨df
dx, ψn
⟩L2[0,π]
− ι 〈g, φn〉L2[0,π]
∣∣∣∣∣2
. (A.12)
Since φ−n = −φn and ψ−n = −ψn, from (A.12) it follows that,
∫ 2π
0
∣∣∣∣∣∣∣CTt f
g
∣∣∣∣∣∣∣2
dt = 2∑n∈Z∗
∣∣∣∣∣⟨df
dx, ψn
⟩L2[0,π]
∣∣∣∣∣2
+∣∣∣〈g, φn〉L2[0,π]
∣∣∣2 . (A.13)
The above relation together with the fact that (ψn)n≥0 and (φn)n≥0 are orthonormal
basis in L2[0, π],
⇒ ∫ 2π
0
∣∣∣∣∣∣∣CTt f
g
∣∣∣∣∣∣∣2
dt = 2
∥∥∥∥∥∥∥ f
g
∥∥∥∥∥∥∥
2
∀
f
g
∈ D(A). (A.14)
Hence C is an admissible operator for semigroup T and pair (A, C) is exactly observ-
able in any time τ ≥ 2π.
B List of Papers
B.1 Journal Papers
J1: M.U. Majeed and T.M. Laleg-Kirati, “A dimension decomposition approach
based on iterative observer design for an elliptic Cauchy problem”, 2016. (archive
pre-print , under-review)
J2: M.U. Majeed and T.M. Laleg-Kirati, “Iterative Observer for Boundary Estima-
169
tion for Elliptic Equations”, 2015. (archive pre-print , under-review)
J3: M.U. Majeed and T.M. Laleg-Kirati, “Iterative Observers for Distributed Source
Estimation for Poisson Equation”, 2016. (submitted, under-review)
J4: M.U. Majeed and T.M. Laleg-Kirati, “An Optimal Iterative Algorithm Based
Technique for Steady-State Potential Field Imaging in 3D”, 2017. (under-
preparation)
B.2 Reviewed Conference Papers & Proceedings
C1: M.U. Majeed and T.M. Laleg-Kirati, “Iterative Observer Based Method for
Source Localization Problem for Poisson Equation in 3D”, The 2017 American
Control Conference (ACC 2017), Seattle, WA, USA, 2017.
C2: M.U. Majeed∗ and T.M. Laleg-Kirati, “Robust Iterative Observer for Source
Localization for Poisson Equation”, 55th Conference on Decision and Control
(CDC 2016), Las Vegas, NV, USA, 2016.
C3: M.U. Majeed∗ and T.M. Laleg-Kirati, “Localization of Point Sources for Poisson
Equation using State Observers”, 2nd IFAC Workshop on Control of Systems
Governed by Partial Differential Equations (CPDE’16), Bertinoro Italy, 2016.
(online link)
C4: M.U. Majeed and T.M. Laleg-Kirati∗, “An optimal iterative algorithm to solve
Cauchy problem for Laplace equation”, 3rd International Conference on Control
Engineering and Information Technology (CEIT), Tlemcen Algeria, 2015.
(online link) (Best Paper Award)
C5: M.U. Majeed∗ and T.M. Laleg-Kirati, “Boundary Estimation Problem for An
Infinite Dimensional Elliptic Cauchy Problem”, SIAM Conference on Control
170
and Its Applications (CT15), Paris France, 2015.
C6: M.U. Majeed and T.M. Laleg-Kirati∗, “Two-step observer approach to solve
Cauchy problem for Laplace equation”, (PICOF’14) Inverse Problems, Control
and Shape Optimization, Hammamet Tunisia, 2014.
C7: M.U. Majeed∗ and T.M. Laleg-Kirati, “Cauchy Problem for Laplace Equation
on a Square Domain using Observers”, 8th International Conference on Inverse
Problems in Engineering (ICIPE), Krakow Poland, 2014.
C8: M.U. Majeed, C. Zayane-Aissa∗ and T.M. Laleg-Kirati, “Cauchy Problem for
the Laplace’s Equation: An Observer based Approach”, The 3rd International
Conference on Systems and Control (ICSC’13), Algiers Algeria, 2013.
(online link)
( “∗” : author who presented in the corresponding conference)
B.3 Talks & Presentations
P1: Cyberphysical Systems Laboratory (CPSLab), New York University, Abu Dhabi,
UAE, July 2017.
P2: The 2017 American Control Conference (ACC), Seattle, WA, USA, May 2017.
P3: Mobile Sensors Lab, University of California Berkeley (UC Berkeley), CA, USA,
January 2017.
P4: 55th Conference on Decision and Control (CDC), Las Vegas, NV, USA, Decem-
ber 2016.
P5: 2nd IFAC Workshop on Control of Systems Governed by Partial Differential
Equations (CPDE16), Bertinoro, Italy, June 2016.
171
P6: Cymer Center for Control Systems and Dynamics, University of California, San
Diego (UCSD), CA, USA, January 2016.
P7: SIAM Conference on Control and Its Applications (CT15), Paris, France, July
2015.
P8: Inverse Problems - from Theory to Applications (IPTA2014), Bristol, U.K. Au-
gust 2014.
P9: 8th International Conference on Inverse Problems in Engineering (ICIPE), Krakow,
Poland, May 2014.
P10: Winter Enrichment Program (WEP), King Abdullah University of Science and
Technology (KAUST), KSA, January 2014.
P11: Franco-German Summer School, Inverse Problems and Partial Differential Equa-
tions, Bremen, Germany, October 2013.
P12: Applied Inverse Problems Conference, Korean Advanced Institute of Science
and Technology (KAIST) Daejeon, South Korea, July 2013.