Department Doctoral Seminar Supervisor: Prof. D. W. Zingg

Introduction Traceability Monolithic Homotopy Results Matrix-Free Conclusions

Efficient Homotopy Continuation Algorithms with Application toComputational Fluid Dynamics

David A. Brown

Department Doctoral SeminarSupervisor: Prof. D. W. Zingg

Committee: Prof. C. P. T. Groth, Prof. P. B. NairNovember 12, 2015

1/38


Algorithms for Solving Nonlinear Systems of Equations

Problem definition: solve R (q) = 0 for q, R : RN → RN

Newton’s method has high (quadratic) convergence rate

Newton’s method will not converge without a suitable initial guess

A second globally convergent algorithm should be used to globalizeNewton’s method

Newton’s method

∇R (q)∆q = −R (q)

Pseudo-Transient Continuation(

1

∆tI +∇R (q)

)

∆q = −R (q)

We solve the linear systems inexactly using an iterative solver

2/38


CFD Flow Solver Example

ONERA M6 wing

Inviscid

Ma = 0.7

AoA = 3◦

PTC =Pseudo-TransientContinuation

0 100 200 300 400 50010

−12

10−10

10−8

10−6

10−4

10−2

PTC Newton

‖R(q)‖

CPU Time

3/38



ONERA M6 wing

Inviscid

Ma = 0.7

AoA = 3◦


0 100 200 300 400 50010

−12

10−10

10−8

10−6

10−4

10−2

PTC Newton

‖R(q)‖

CPU Time

RealityFantasy

3/38



ONERA M6 wing

Inviscid

Ma = 0.7

AoA = 3◦


0 100 200 300 400 50010

−12

10−10

10−8

10−6

10−4

10−2

PTC Newton

‖R(q)‖

CPU Time

RealityFantasy

There is not much room for improvement with PTC

3/38


Convex Homotopy Continuation

Homotopy: A continuous deformation

Convex Homotopy

H (q, λk) = (1− λk )R (q) + λkG (q) = 0

k ∈ [0, p] , λk ∈ R, λ0 = 1, λp = 0, λk+1 < λk

R : RN → RN is the flow residual

G : RN → RN is the homotopy system

H (q, λ) defines a homotopy from the solution of G (q) = 0 to thesolution of R (q) = 0 (under certain conditions)

Homotopy Continuation

Gradually reduce λ from 1 to 0 while solving H (q, λ) = 0 to obtain anestimate for the solution to the problem R (q) = 0

Note that the deformation can also be interpreted as a curve in RN

4/38


Homotopy Design

The following properties are desired:

R and G should be continuous and invertible

➩ Necessary condition for the homotopy to exist

The curve should be regular (invertible Jacobian for all λ)

➩ Regular curves do not contain bifurcations

The Jacobian should be well-conditioned

➩ Reduced cost of the linear solves

The curve should exhibit modest curvature

➩ The curve will be easier to trace numerically

5/38


Some Candidate Homotopies

Convex Homotopy

H (q, λ) = (1− λ)R (q) + λG (q) = 0

G can be of the form T (q− q0) where T is a diagonal matrix

G can be a numerical dissipation operator, which is of the form ofan undivided second difference; to make this operator nonsingular,we apply pseudo-boundary conditions

far-field boundary conditionsflow-imitative boundary conditions

Global Homotopy

H (q, λ) = R (q)− λR (q0) = 0

6/38


An Example of a Homotopy Deformation

NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦

Convex homotopy using dissipation operator with far-field boundary conditions

7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38



NACA 0012 airfoil

Transonic

RANS

Re=4.76× 107

Ma = 0.8395

AoA = 3.06◦


7/38


The Predictor-Corrector Curve-Tracing Algorithm

01λ

Cur

ve v

alue

s

Corrector phase: Based on Newton’s methodPredictor phase: Based on the tangent vector

Curves with higher curvature are more difficult to trace 8/38


Traceability Metrics

How can we study the traceability of implicitly-deifned curves which existin higher-dimensional real space?

Surrogate curves: Calculate functionals (e.g. CL and CD) alongthe curveWatch movies (You already saw one)Curvature:

Parametrized curve c (s) = [q (s) ;λ (s)]Arclength parametrization: c (s) · c (s) ≡ 1Partial curvature κq ≡

√q (s) · q (s)

Taylor series around some s0:

q (s0 +∆s) = q (s0) + ∆sq (s0) +1

2∆s

2q (s0) +O

(

∆s3)

If the curve-tracing algorithm is based on the tangent, then theleading error term is

√(1

2∆s2

)2

q (s) · q (s) =1

2∆s

2κq

9/38


Traceability - Inviscid Subsonic

NACA 0012

15,000 nodes

Ma = 0.3

AoA = 1◦

00.51−0.5

0

0.5

Cd

λ00.51

0

0.2

0.4

Cl

λ

0 0.5 10

500

1000

1500

s/stot

κqs2 to

t

00.5110

0

105

1010

λ

κr

“Diss - ff”“Diss - flow”“Diag”Global

κq uses an arclength parametrizationκr uses a λ parametrization

10/38


Traceability in Different Coordinates

NACA 0012

15,000 nodes

Ma = 0.3

AoA = 1◦

00.510

0.1

0.2

λ

Cl

00.510

0.1

0.2

λ

Cd

00.510

10

20

λ

s

00.510

5

10

λ

κq

µ = 10µ = 1µ = 0.1

0 0.5 10

1000

2000

s/stot

κqs2 to

t

00.5110

0

105

1010

λ

κr

Change of coordinates: λ← λµ

1−λ+µλ

11/38


Comparison of some Homotopies

Traceability:

➩ Global > Dissipation > Diagonal

Conditioning:

➩ Diagonal > Dissipation > Global

Success rate:

➩ Global: inviscid subsonic only➩ Diagonal: inviscid subsonic and transonic, turbulent subsonic➩ Dissipation: suitable for all flows

12/38


Motivation

Predictor-corrector methods are the standard in the literature

To ensure that the algorithm converges, over-solving often occurs inthe corrector phase

The linear systems in both phases have the same matrix

Efficiency can be improved by combining the two phases

13/38


Dynamic Inverse

Let qs (λ) be the (unique) solution to F (q (λ) , λ) = 0

Consider an ODE of the form q (λ) + F (q (λ) , λ) = 0

Under certain conditions, the ODE converges to qs (λ)

If the ODE does not converge to qs (λ) then it might be possible tofind F∗ such that the modified ODE q (λ) + F∗F (q (λ) , λ) = 0

does

Even if the original ODE does converge, the modified ODE can havea better convergence rate

F∗ is called the dynamic inverse of F

14/38


Continuous Monolithic Homotopy Algorithm

The continuous monolithic homotopy algorithm is an ODE:

−q (λ) = −γH∗H (q, λ)︸︷︷︸

Corrector

+ E (q, λ)︸︷︷︸

Predictor

H∗ : RN+1 → RN is the dynamic inverse

γ > 0, γ ∈ R is a free parameter

15/38



01λ

Cur

ve v

alue

s

CurrentTarget

−γH∗H (q, λ) corrects the current tracking error

E : RN+1 → RN corrects for tracking error due to evolution of the

curve

16/38



01

γH∗H E

λ

Cur

ve v

alue

s

CurrentTarget



curve

16/38



01

γH∗H E

λ

Cur

ve v

alue

s

CurrentTarget



curve

16/38


Choosing E and H∗

Recall the continuous monolithic homotopy algorithm:

−q (λ) = −γH∗H (q, λ) + E (q, λ)

Objective

Construct E , H∗, and γ such that the above ODE converges to thehomotopy curve and can be integrated efficiently

Recall the convex homotopy:

H (q, λ) = (1− λ)R (q) + λG (q) = 0

17/38


Choosing E and H∗


−q (λ) = −γH∗H (q, λ) + E (q, λ)

Objective



H (q, λ) = (1− λ)R (q) + λG (q) = 0

Construction of E

An expression for E can be taken as an approximation to the vectortangent to the curve.

In the context of convex homotopy:

E (q, λ) = [∇qH (q (λ) , λ)]−1

[G (q)−R (q)]

17/38


Choosing E and H∗


−q (λ) = −γH∗H (q, λ) + E (q, λ)

Objective



H (q, λ) = (1− λ)R (q) + λG (q) = 0

Construction of H∗

H∗ = − [∇qH (q, λ)]−1

is a dynamic inverse as long as q is close tothe curve

This will give a Newton-like update for the corrector component

17/38


Discrete Monolithic Homotopy Algorithm

Continuous version

−q (λ) = [∇qH (q, λ)]−1 [−γH (q, λ) + G (q)−R (q)]

Discrete version

qk+1 = qk +∆λk

Predictor−−−−−−−−−−−−−−−−−Predictor︷︸︸︷

[∇qH (qk , λk)]−1 [γkH (qk , λk)

︸︷︷︸Corrector

︷︸︸︷

− G (qk) +R (qk)]

18/38



Continuous version

−q (λ) = [∇qH (q, λ)]−1 [−γH (q, λ) + G (q)−R (q)]

Discrete version

qk+1 = qk +∆λk




︷︸︸︷

− G (qk) +R (qk)]

The continuous version converges to the homotopy curve withconvergence rate Ke

−(γβ−ω)(λ0−λ), where

β depends on the corrector qualityω depends on the predictor qualityγ is a free parameter that can be chosen by the user

18/38



Continuous version

−q (λ) = [∇qH (q, λ)]−1 [−γH (q, λ) + G (q)−R (q)]

Discrete version

qk+1 = qk +∆λk




︷︸︸︷

− G (qk) +R (qk)]

The continuous version converges to the homotopy curve withconvergence rate Ke

−(γβ−ω)(λ0−λ), where

β depends on the corrector qualityω depends on the predictor qualityγ is a free parameter that can be chosen by the user

Large γ will give high convergence rate for the continuous case

Large γ is not always ideal for the discrete case

18/38


The Effect of γ on Curve-Tracing

01λ

Cur

ve v

alue

s

19/38



01λ

Cur

ve v

alue

s

γ = 1γ = 10γ = 100

19/38



01λ

Cur

ve v

alue

s

γ = 1γ = 10γ = 100

19/38



01λ

Cur

ve v

alue

s

γ = 1γ = 10γ = 100

For given ∆λ, there is an ideal γIt turns out that γ = 1

|∆λ| is ideal

We can also use γ = 1|∆λ∗|

19/38


Automatic Step-length Adjustment

Step-length |∆λ| is adjusted automatically after each update to tryto achieve ‖∆q‖ = ‖∆q‖

tar

The target value ‖∆q‖tar

is set according to ‖∆q‖tar

= ‖∆q‖0‖∆q‖0 is determined indirectly by the user by adjusting the initialstep size |∆λ|0

Users’ Guide

The user adjusts the parameter |∆λ|0Smaller |∆λ|0 will improve curve tracing accuracy but will result inmore iterations

20/38


Effect of |∆λ|0 in a Practical Example

Euler

ONERA M6 wing

Ma=0.5

AoA=6◦

00.510

0.1

0.2

0.3

0.4

0.5

λ

CL

Exact|∆λ0| = 0.1

00.510

0.1

0.2

0.3

0.4

0.5

λ

Exact|∆λ0| = 0.15

00.510

0.1

0.2

0.3

0.4

0.5

λ

Exact|∆λ0| = 0.2

00.510

0.1

0.2

0.3

0.4

0.5

λ

Exact|∆λ0| = 0.25

(17 iterations) (11 iterations) (10 iterations) (9 iterations)

21/38


A Numerical Study of Stability

00.510

0.1

0.2

0.3

0.4

0.5

λ

CL

Exact|∆λ|= 0.1|∆λ|= 0.05|∆λ|= 0.02

00.510

0.01

0.02

0.03

0.04

λ

Errorin

CL

00.5110

−6

10−4

10−2

100

λ

‖H(q

,λ)‖

τl = 0.01

00.510

0.1

0.2

0.3

0.4

0.5

λ

CL

00.510

0.01

0.02

0.03

0.04

λ

Errorin

CL

00.5110

−6

10−4

10−2

100

λ‖H

(q,λ

)‖

τl = 0.001

ONERA M6, inviscid, Ma=0.5, AoA=6◦

22/38


Comparison to Implicit Time Marching

Implicit Euler (Derivation)

qn =−R (qn)

R (qn) ≈ R (qn−1) +∇R (qn−1)∆q

⇒1

∆t∆q =−R (qn−1)−∇R (qn−1)∆q

⇒ ∆q =−

(1

∆tI +∇R (qn−1)

)−1

R (qn−1)

Monolithic Homotopy

−q (λ) =

Corrector︷︸︸︷

H∗γH (q, λ)+

Predictor︷︸︸︷

[∇qH (q, λ)]−1

[G (q)−R (q)]

H∗ ← − [∇qH (q, λ)]−1

23/38


Performance Study for an Inviscid Case

ONERA M6 wing

1.9 million nodes

32 processors

Ma = 0.6

AoA = 3◦

0 200 400 60010

−6

10−4

10−2

100

102

Time (s)

%Errorin

CL

PTCCHC - PCCHC - MH

00.5110

−4

10−3

10−2

λ

‖H(q,λ

)‖

Glossary

PTC Pseudo-transient continuationCHC-PC Convex homotopy continuation - predictor-correctorCHC-MH Convex homotopy continuation - monolithic homotopy

24/38


Performance Study for a RANS Case

ONERA M6 wing

2.3 million nodes

192 processors

Ma = 0.75

AoA = 1.5◦

Re = 1.172× 107

0 500 100010

−6

10−4

10−2

100

102

Time (s)

%Errorin

CL

PTCCHC - PCCHC - MH

00.5110

−5

10−4

10−3

10−2

λ

‖H(q,λ

)‖

Glossary


25/38


Performance Study for another RANS Case

ONERA M6 wing

36 million nodes

1024 processors

Ma = 0.8

AoA = 3◦

Re = 1× 107

0 1 2x 10

4

10−6

10−4

10−2

100

102

Time (s)

%Errorin

CL

PTCCHC - PCCHC - MH

00.5110

−8

10−6

10−4

10−2

100

λ

‖H(q,λ

)‖

Glossary


26/38


Performance Study for an Inviscid Case

ONERA M6 wing

1.9 million nodes

32 processors

H-C topology

2e+04 α = 0°

PTC CHC − PC CHC − MH

1e+04 α = 3°

2e+04

Equ

ival

ent R

esid

ual E

valu

atio

ns

α = 6°

4e+04 α = 9°

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

4e+04

Ma

α = 12°

Compared to PTC: → CHC-PC was about 12% faster→ CHC-MH was about 23% faster

Glossary


27/38


Performance Summary

Success Rate Rel. CPU TimeFlow Nodes Blocks PTC PC MH PTC PC MH

Inviscid 3D 2× 106 32 32/40 32/40 33/40 1.00 0.88 0.77Inviscid 3D 2× 107 256 21/40 27/40 27/40 1.00 0.85 0.66Laminar 3D 2× 106 48 28/40 30/40 31/40 1.00 0.91 0.60RANS 2D 2× 104 8 31/32 32/32 32/32 1.00 0.73 0.55RANS 3D 2× 106 192 16/16 16/16 16/16 1.00 1.14 0.64RANS 3D 4× 107 1024 13/16 13/16 14/16 1.00 3.33 1.00

3D results - ONERA M6 wing2D results - NACA 0012 airfoil

Summary

Usually: MH > PC > PTC

In all cases: MH > PC, MH ≥ PTC

Stability concerns can reduce the competitiveness of the MHalgorithm in some cases

28/38


Matrix-Free Monolithic Homotopy

Monolithic homotopy

−q (λ) = H∗γH (q, λ) + E

Two-Stage Formulation

H∗ is set to a diagonal matrix T

E is estimated using finite-differencing E ≈ 1λk−λk−1

∆q

The two stages can be kept separate to improve the accuracy of E :

qk+ 12= qk +

(

λk+ 12− λk

)

γT H,

qk+1 = qk+ 12+ λk+1−λk

λk−λk−1

(

qk+ 12− qk− 1

2

)

This method is more accurate but unstable

Single-Stage Formulation

This method is stable but less accurate 29/38


Matrix-Free Monolithic Homotopy

Monolithic homotopy

−q (λ) = H∗γH (q, λ) + E

Two-Stage Formulation

This method is more accurate but unstable

Single-Stage Formulation

H∗ is set to a diagonal matrix T

E is set to zero

The following ODE is produced:

q = −γT H (q, λ)

This can be integrated using explicit Euler

This method is stable but less accurate

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Performance Comparison

00.510

0.05

0.1

0.15

0.2

0.25

λ

Cl

|∆λ| = 1×10−4

00.510

0.05

0.1

0.15

0.2

0.25

λ

Cl

|∆λ| = 1×10−5

00.510

0.05

0.1

0.15

0.2

0.25

λ

Cl

|∆λ| = 1×10−6

00.510

0.05

0.1

0.15

0.2

0.25

λ

Cl

ExactSingle-StageTwo-Stage

|∆λ| = 1×10−7

30/38


Algorithm Augmentations

Let γ = href/ (λk+1 − λk )

Augmentations to the Single-Stage Algorithm

q = −γT H (q, λ)

This method is stable but inaccurate

Can use RK4 instead of Euler for time integration

This allows for larger href for improved accuracy at increased CPUcost

Augmentations to the Two-Stage Algorithm

qk+ 12= qk +

(

λk+ 12− λk

)

γT H,

qk+1 = qk+ 12+ λk+1−λk

λk−λk−1

(

qk+ 12− qk− 1

2

)

This method is accurate but unstable

An explicit filter can be used to improve stability at the cost ofaccuracy 31/38


Performance Comparison

00.510

0.05

0.1

0.15

0.2

0.25

λ

Cl

|∆λ| = 1×10−4

00.510

0.05

0.1

0.15

0.2

0.25

λ

Cl

|∆λ| = 1×10−5

00.510

0.05

0.1

0.15

0.2

0.25

λ

Cl

|∆λ| = 1×10−6

00.510

0.05

0.1

0.15

0.2

0.25

λ

Cl

Exact1-Stage1-Stage, RK42-Stage, Filter

|∆λ| = 1×10−7

32/38


Efficiency of the Augmented Matrix-Free Algorithms

101

102

103

104

105

10−3

10−2

10−1

Time (s)

FinalC

lerror

1-Stage1-Stage, RK42-Stage, Filter

33/38


Conclusions (Boring Part)

The new MH algorithm is more efficient than the predictor-correctoralgorithm

The MH algorithm is usually more efficient than PTC

Stability can become an issue for MH in some cases if the linearsystem is not solved accurately enough

An expensive solution to this problem is to solve the linear systemmore accurately

The matrix-free MH algorithm is inhererently unstable but can bestabilized using a filter

34/38


Contributions (Boring Part 2)

Constructed several new homotopies based on CFD codes to whichcontinuation algorithms could be applied

Developed new tools for efficiently calculating quantities relevant tohomotopies for large sparse systems

Developed a methodology for quantifying the suitability ofhomotopies for continuation algorithms

Developed a new class of continuation algorithm, monolithichomotopy continuation

Demonstrated the effectiveness of the monolithic homotopyalgorithms numerically

35/38


Homotopy Equations Hall of Fame

t =τ

‖τ‖, τ =

(z

−1

)

, z = [∇qH (q, λ)]−1 ∂

∂λH (q, λ)

−q (λ) = [∇qH (q, λ)]−1 [−γH (q, λ) + G (q)−R (q)]

∑

∗

n!∏n

j=1 j!mjmj !

∇∑n

j=1 mjH (c (s))

n∏

j=1

[

c(j) (s)]mj

︸︷︷︸

wn+∇H(c(s))c(n)(s)

= 0

q (λ∗) =

∑k+1i=k+1−p Kb (λ

∗, λi )q (λi )∑k+1

i=k+1−p Kb (λ∗, λi ), Kb (λ

∗, λi ) = exp

(

−(λ∗ − λi )

2

2b2

)

∗ Summation is over all positive {m1, . . . ,mn} such that∑n

j=1 jmj = n

36/38


Special Credits

Professor Zingg

Michal Osusky

Howard Buckley

P. David Boom

Jason Hicken

Chris Lee

Many others

37/38


End of PhD

Questions?

38/38

Documents

Department Doctoral Seminar Supervisor: Prof. D. W. Zingg