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1 Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2 Novosibirsk State University, Novosibirsk, Russia3 Siberian Regional Hydrometeorological Research Institute, Novosibirsk, Russia
Sensitivity Operator Inverse Modeling Framework
for Advection-Diffusion-Reaction Models with Heterogeneous Measurement Data
Alexey Penenko 1,2, Vladimir Penenko 1,2, Elena Tsvetova 1, Alexander Gochakov 3, Elza Pyanova 1, and
Viktoriia Konopleva 1,2
vEGU, 2021
E-mail: [email protected]
•Unified approach to different measurement data including image-type measurement data
(large volume of data with unknown value w.r.t. the considered inverse modelling task):
•Pointwise and time-series concentrations data (in situ data)
•Concentration profiles (aircraft sensing, lidar profiles, etc).
•Satellite images of concentration fields.
Motivation
•The progress in parallel computations technologies: the algorithms should allow natural mapping to the parallel
computation architectures (ensemble algorithms, splitting, decomposition, etc.)
•Unified framework for inverse and data assimilation problems (briefly, inverse modeling problems) for non-
stationary advection-diffusion-reaction models in various applications. E.g.
•Air quality studies (environmental applications)
•Biological problems (morphogen theory & developmental biology)
•The progress in nonlinear ill-posed operator equation solution (different regularization methods, SVD,
convergence theory, etc.) and analysis methods.
2
Solve & Analyze in the same way
Combine various sources of data
Measurement operators
Advection-diffusion-reaction model
( )( )diag ( , , ) = ( , , ) ,ll ll
l l l l lP t t ft
r
− − + + +
yu φ φ y
0= , , = 0,l l t x
The domain
( )( )diag = , ( , ) (0, ],R
l l l l l outt T + n x
[0, ]T T =
= , ( , ) (0, ],D
l l int T xBC:
: ,Q
q
→ φ
Direct problem
operator
advection-diffusion destruction-production
= [qH +( )I φ ] I,
Inverse problem
IC:
rectangular
Given NoiseTo find
• Pointwise concentrations
• 2D images
• vertical profiles,
• etc
Model scale: 0D,1D,2D,3D 1, , cl N= - number of species
{ , , }q r y=
3
H
, :l lP
• Polynomial
• Sigmoid
functions
• Integral
Uncertainty
function
specification
Conventionally used to
evaluate misfit cost
function gradients
(2) (2) (2) (1) (1), , , ,[ , ] [ , ] ( , , , , ) ,yR
K t q qy
+ = + +h H φ h H φδ h φμ δ φr δ φH
Sensitivity relation
( ) = 0,T+Ψ
Lagrange type identity (sensitivity relation)
-divided difference operator
( ) ( ) ( )1 1( ( ) ) ( ( , , ) ) = ,l
l l l ldiag G tt
h
− − − +
2u μ φ φ Ψ
( ) ( ) ( )( )( ) ( ) ( )( ) ( )( ) ( ) ( )( )1 1 1 1( , , ) , , , , , ,G t diag P t P t diag t
= + − 2 2 2 2
φ φφ φ φ φ φ φ φ φ
+ adjoint problem
boundary conditions
(( )) [ ]m mq=φ φ
Sensitivity functions
Adjoint problem: Given find:( ) ( )
, , , 1, 2,m m
m =h φ μ
Linear parametrizations: m m
m
rr = , = , [ ]m m Rm
r
h φ h
Ψ
(2) (1) = −φ φ φ
= , 1, ,m
m
t ll m MH=
=
4
( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )( )1(2) (2) (1) (1)( , , , , ) , ; , ; ,y yK t q q t P t diag= −2 2 1 2 2 1
φ φ φ y ,y φ y ,y φ
Sensitivity Operator Construction
5
Sensitivity operator
Sensitivity relation (Lagrange type identity)
measU U
= (ξ)uGiven functions (functionals)
( )2 1 1 ( )( )[ ] [ ] [ ] [ ]UH
− = − ( ) ( ) (2) ( )φ q φ uq φ q φ q e
( )( )] [ ][ ] [Q
M
− = −(2) (1) (2) (1)(2) (1)qφ q φ q u qu qq
Image (model) to structure operator
[Dimet et al,2015]
Sensitivity operator ( ) ( )[ ][ ]
,U
R
R
MM
→
(2) (
(
1)
2) (1)
q q uq q
z z e
The inverse problem solution for any and satisfy( )
q q U
( ) ( )( )[ ] = [ , ] .PrU U UU
meas
H M H
− − + I φ q q q q q I
Parametric family of quasi-linear
operator equations
Parallel computation w.r.t.
U
To control the data considered and dimensionality
Idea: [Marchuk G. I., On the formulation of certain inverse problems, Dokl. Akad. Nauk SSSR, 156:3 (1964), 503–506], 6
Sensitivity relation based
• Coarse-fine mesh method (Sequential solution refinement with sequential adjoint problems solving) [Hasanov, A.; DuChateau, P. &
Pektas, B. An adjoint problem approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equation//
Journal of Inverse and Ill-Posed Problems, 2006 , 14 , 1-29]
• Using sensitivity function aggregates (supports, level sets, etc.): [Pudykiewicz, J. A. Application of adjoint tracer transport equations for
evaluating source parameters Atmospheric Environment, Elsevier BV, 1998, 32, 3039-3050], [Penenko, V. et al. Methods of sensitivity theory and
inverse modeling for estimation of source parameters, Future Generation Computer Systems, Elsevier BV, 2002, 18, 661-671], [Bieringer, P. E. et
al. Automated source term and wind parameter estimation for atmospheric transport and dispersion applications Atmospheric
Environment, Elsevier BV, 2015, 122, 206-219]
• Adjoint function for each measurement datum with the solution of the resulting operator equation [Marchuk G. I., On the
formulation of certain inverse problems, Dokl. Akad. Nauk SSSR, 156:3 (1964), 503–506],
• Passive tracer (linear) case: [Issartel, J.-P. Rebuilding sources of linear tracers after atmospheric concentration measurements //
Atmospheric Chemistry and Physics, Copernicus GmbH, 2003 , 3 , 2111-2125]
• Nonlinear case, pointwise sources: [Mamonov, A. V. & Tsai, Y.-H. R. Point source identification in nonlinear advection-diffusion-
reaction systems Inverse Problems, IOP Publishing, 2013 , 29 , 035009]
Cost function based
• Cost functional gradients with adjoint problem solution (single element ensemble for the discrepancy).
• Gradient computation with adjoint ensemble when adjoint is independent of direct solution [Karchevsky, A., Eurasian journal of
mathematical and computer applications, 2013 , 1 , 4-20]
• Representer method (optimality system decomposition, ensemble generated for discrepancies for each measurement data) [Bennett,
A. F. Inverse Methods in Physical Oceanography (Cambridge Monographs on Mechanics) Cambridge University Press, 1992]
Adjoint problem solution ensemblesin inverse problem algorithms
7
Adjoint ensemble construction
= | [0, ], [0, ], [0, ], ,x x y y t t mesU L ηθ θ θx y t
e• Fourier cos-basis
• Wavelets, curvlets, etc. [Dimet et al.,2015]
«a priori» approach – ensemble for the class of problems (smoothness)
=1
1( , , ) ( , , ) ( , , ), =
= ,
0,
Nc
k
t x i y j
x y t
l
C T t C X x C Y y le
l
2 cos , > 01( , , ) = .
1, = 0
t
C T t TT
«a posteriori» approach – ensemble for the considered problem
• «Adaptive basis»: chose elements of with maximal projections
[Penenko, A. V.; Nikolaev, S. V.; Golushko, S. K.; Romashenko, A. V. & Kirilova, I. A. Numerical Algorithms for Diffusion Coefficient Identification in Problems of Tissue Engineering //
Math. Biol. Bioinf., 2016 , 11 , 426-444 (In Russian)]
• «Informative basis»: use left singular vectors of the operator [ , ]Um
(0) (0)r r
U[ ] ,Pr
Umes
−(0)
ηθ θ θx y t
φ r I e
[Penenko, A.; Zubairova, U.; Mukatova, Z. & Nikolaev, S. Numerical algorithm for morphogen synthesis region identification with indirect image-type measurement data //
Journal of Bioinformatics and Computational Biology, World Scientific Pub Co Pte Lt, 2019 , 17 , 1940002]
(inverse source problem, 2D, components are measured )
Defined by the adjoint problem sources sets
8
Inversion algorithm
( )( ) ( )( ) ( )( ), [ , ] [ ,, ]]Pr [=UU
m
UU
e s
UU
a
H m Hm m − − + + − −
* **qφ q q q qq qI q q q δIq
,
= .Pr,
T T
unknowns
UUT T
mesunknowns
m mm NH I
m m m N
+
+
−
δq φ q
C+
-truncated SVD inversion parametrized by conditional number
, inversesource problem
, inversecoefficient problem
src t x y
unknowns
coeff
L N N NN
N
=
Newton-
Kantorovich
-type update
[ , ]Um m q q
( )unknownsN
Nonlinearity:
sequential increase of
the conditional number
Admissible solutions:
projection regularization
Noise:
discrepancy
principle
Optional monotonicity:
monotonic decrease of the
discrepancy
Theoretical foundations: [Issartel, J.-P., 2003], [Cheverda V.A., Kostin V.I., 1995], [Kaltenbacher et al, 2008],
[Vainikko, Veretennikov, 1986] 9
Adjoint ensemble inverse modeling framework
Inverse Problem Statement(PDE/ODE)
Adjointproblem
Lagrange-type Identity
Family of quasi-linear ill-posed operator
equations
Generalensemble
Inverse problem solution algorithm
Sensitivity function
Finite ensemble
Ill-posed nonlinear
operator equation
methods
# computation
threads &
memory available
Data assimilation mode
Sensitivity operator analysis
measUProcess model
10
«Illumination»functions
Singular spectrum
Measurement
data
specification
Uncertainty function
specification
Information quantity
Projection to the kernel
Heterogeneous Measurement Data
11
Heterogeneous measurement data
12
• «Timeseries»
• «Pointwise»
• «Integral»
• «Snapshot»
0=1
, = ( , ) ( , ) .
Nc T
l lHl
a b a t b t d dt
x x x
( )
( ) ( ) ( )( ) ( ) ( , ), 0, , , ,m m m
m measl
x t t T x l L ( ) ( ) ( ) ( )= ( , , ) ( ) ( );h C T t x x l l
− −
( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( , ), , , ,m m m m m
Tm measl
x t x t l L ( ) ( ) ( ) ( )= ( ) ( ) ( );h x x t t l l
− − −
( )
( ) ( ) ( )( ) ( ) 0( , ) , , ,
T m m m
m measl
x t dt x l L ( ) ( ) ( )= ( ) ( ).h x x l l
− −
( )
( ) ( ) ( )( ) ( ) ( , ), , 0, , ,m m m
m measlx t t l T L x ( ) ( ) ( ) ( ) ( )
= ( , , ) ( , , ) ( ) ( );x yh C X x C Y y t t l l
− −
In variational approach we sum the misfit functions
for different measurements, in the adjoint ensemble-
based approach we join the corresponding
ensembles
A. Penenko, V. Penenko, E. Tsvetova, A. Gochakov, E. Pyanova, V. Konopleva Sensitivity Operator-Based Approach to the Interpretation
of Heterogeneous Air Quality Monitoring Data // submitted
Specific measurement types
13
O3 Monitoring sites locations
(b): time series (red crosses),
pointwise measurements in
space and time (blue circles),
integrals over a time interval
(magenta triangles).
Pointwise (100) Time-series (60)
Integral (5) Snapshot (20x20)
3.5 day-long
summer
scenario,
source-term
uncertainty,
NOx-O3 cycle
NO Emission sources
«Composite»measurements
14
Composite
Concentration field
«сcontinuation» error
Uncertainty (source-term)
identification error
15
Estimating the inverse modeling result prior to solution
Inverse problem solution times measured
in direct problem solution times
Relative error vs computation time
Sensitivity operator analysis
16
( )( ) ( )( )M q q H I A q− = −
( )( ) ( )( )T T T TM MM M q q M MM H I A q+ +
− = −
«Illumination» function characteristics: [ ] T T
l l lE M M MM M+
=
( )1
NT T T T
l ll
M MM M diag M MM M+ +
=
=
Projector to the orthogonalcomplement of the sensitivity
operator kernel
Equivalent to the generalized Sensitivity functions
Issartel, J.-P. Emergence of a tracer source from air concentration measurements, a new strategy
for linear assimilation // Atmospheric Chemistry and Physics, 2005, 5, 249-273
( )T Tdiag M M M M+
Kappel, F. & Munir, M. Generalized sensitivity functions for multiple output systems // Journal of Inverse and Ill-posed Problems, 2017, 25
Pseudo-inverse
( )( )T TM MM M q q+
−
Singular spectrum decay
17
Sensitivity operator analysis«Illumination»
function Reconstructed
solution (~500 iters.)
Projection of the
«Exact»
solutions to the
orthogonal
complement
of the sensitivity
operator kernel
(~1 iter.) Reconstruction error & its
predictions by projection
Singular spectrum
Thank you for your attention!
18
The work has been supported by
Ministry of Science and Higher Education of the Russian Federation Large Scientific Project
No. 075-15-2020-787 (heterogeneous measurement systems)
Penenko A.V. Penenko V.V.
Pianova E.A.Gochakov A.V.
Tsvetova E.A.
Konopleva V.S.
Sensitivity Operator & Adjoint Ensemble References
19
1. Penenko, A. Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-
type measurements // Inverse Problems & Imaging (AIMS), 2020, 14, 757-782 doi:10.3934/ipi.2020035
2. Penenko, A. V.; Mukatova, Z. S. & Salimova, A. B. Numerical study of the coefficient identification algorithm based on ensembles of adjoint problem
solutions for a production-destruction model // International Journal of Nonlinear Sciences and Numerical Simulation, accepted doi:
10.1515/ijnsns-2019-0088
3. Penenko, A. V. & Salimova, A. B. Source Identification for the Smoluchowski Equation Using an Ensemble of Adjoint Equation Solutions // Numerical
Analysis and Applications, 2020, 13, 152-164 doi: 10.1134/s1995423920020068
4. Penenko, A.; Zubairova, U.; Mukatova, Z. & Nikolaev, S. Numerical algorithm for morphogen synthesis region identification with indirect image-type
measurement data // Journal of Bioinformatics and Computational Biology, 2019 , 17 , 1940002 doi:10.1142/s021972001940002x
5. V. V. Penenko A. V. Penenko, E. A. Tsvetova and A. V. Gochakov Methods for studying the sensitivity of atmospheric quality models and inverse
problems of geophysical hydrothermodynamics // Journal of Applied Mechanics and Technical Physics, 2019, Vol. 60, No. 2, pp. 392–399 doi:
10.1134/S0021894419020202
6. Penenko, A. V. A Newton–Kantorovich Method in Inverse Source Problems for Production-Destruction Models with Time Series-Type Measurement
Data // Numerical Analysis and Applications, 2019 , 12 , P. 51-69 doi: 10.1134/s1995423919010051
7. Penenko, A. V. Consistent Numerical Schemes for Solving Nonlinear Inverse Source Problems with Gradient-Type Algorithms and Newton–
Kantorovich Methods // Numerical Analysis and Applications, 2018 , 11 , P.73-88 doi:10.1134/s1995423918010081
8. Penenko, A. V.; Nikolaev, S. V.; Golushko, S. K.; Romashenko, A. V. & Kirilova, I. A. Numerical Algorithms for Diffusion Coefficient Identification in
Problems of Tissue Engineering // Math. Biol. Bioinf., 2016 , 11 , 426-444 doi: 10.17537/2016.11.426 (In Russian)
9. Penenko, A. On a solution of the inverse coefficient heatconduction problem with the gradient projection method // Siberian electronic
mathematical reports, 2010, 23 , 178-198. (in Russian)