numerical optimization of partial di erential equationsbprotas/rouen2018_ecole_part3_v2.pdf ·...

Optimal Open–Loop ControlInverse Problem of Vortex Reconstruction

Geometry Optimization in Heat Transfer

Numerical Optimizationof Partial Differential Equations

Part III: applications

Bartosz Protas

Department of Mathematics & StatisticsMcMaster University, Hamilton, Ontario, CanadaURL: http://www.math.mcmaster.ca/bprotas

Rencontres Normandes sur les aspects theoriques etnumeriques des EDP

5–9 November 2018, Rouen

B. Protas Numerical Optimization of PDEs

https://ms.mcmaster.ca/~bprotas/



Optimal Open–Loop ControlPDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults

Inverse Problem of Vortex ReconstructionEuler System & Inverse FormulationSolution ApproachResults

Geometry Optimization in Heat TransferMotivation & Mathematical ModelOptimization ProblemResults




PDE–Constrained OptimizationDetermination of the Gradient ∇J via Adjoint SystemResults

PART I

Optimal Open–Loop Control via

Adjoint–Based Optimization





Motivation — Applications of Flow Control

I Wake Hazard

I Fluid–Structure Interaction





Statement of the Problem (I)

I Flow Domain

infV

V(t)τ

(t)ϕ

Y

X

Ω

O

Γ

D

I Assumptions:I viscous, incompressible flowI plane, infinite domainI Re = 150





Statement of the Problem (II)

I Find ϕopt = argminϕ J (ϕ) , where

J (ϕ) =1

2

∫ T

0

[power related to

the drag force

]+

[power needed tocontrol the flow

]dt

=1

2

∫ T

0

∮Γ0

[p(ϕ)n− µn ·D(v(ϕ))] · [ϕ (ez × r) + v∞] dσdt

I Subject to:[

∂v∂t + (v ·∇)v − µ∆v + ∇p

∇ · v

]=

[00

]in Ω× (0,T ),

v = 0 at t = 0,

v = ϕoptτ on Γ





Abstract Framework (I)

I Constrained optimization problem min(x ,ϕ)J (x , ϕ)

S(x(ϕ), ϕ) = 0

I Equivalent unconstrained optimization problem (note thatx = x(ϕ) )

minϕJ (x(ϕ), ϕ) = min

ϕJ (ϕ)

I First–Order Optimality Conditions (U - Hilbert space ofcontrols)

∀ϕ′∈U J ′(ϕ;ϕ′) =(∇J , ϕ′

)U = 0,

with the Gateaux differentialJ ′(ϕ;ϕ′) = limε→0

1ε [J (ϕ+ εϕ′)− J (ϕ)].





Abstract Framework (II)

I Minimization of J (ϕ) with a descent algorithm in U=⇒ solution to a steady state of the ODE in U

dϕ

dτ= −Q∇ϕJ (ϕ) on τ ∈ (0,∞) (pseudo–time),

ϕ = ϕ0 at τ = 0.

I Typically well–behaved (quadratic) cost functionalsI Typically ill–behaved constraints: the Navier–Stokes

systemI nonlinear, nonlocal, multiscale, evolutionary PDE,

I Dimensions:I state: 106 — 107 DoF × 102 — 103 time levelsI control: 104 — 105 DoF × 102 — 103 time levels

I No hope of using “matrix” formulation ...

I Formulation equivalent to Lagrange Multipliers





Differential of the Cost Functional

I The cost functional:

J (ϕ) =1

2

∫ T

0

[power related to

the drag force

]+

[power needed tocontrol the flow

]dt

=1

2

∫ T

0

∮Γ0

[p(ϕ)n− µn ·D(v(ϕ))] · [ϕ (ez × r) + v∞] dσdt,

I Expression for the Gateaux differential:

J ′(ϕ; h) =1

2

∫ T

0

∮Γ0

[p′(h)n− µn ·D (v′(h))] · [ϕ (ez × r) + v∞] +

[p(ϕ)n− µn ·D(v(ϕ))] · (ez × r) hdσ dt = B1

= (∇J (t), h)L2([0,T ])

The fields v′(h), p′(h) solve the linearized perturbation system.

I How to calculate the Gradient ∇J ?





Sensitivities and Adjoint States

I The linearized perturbation systemN[

v′

p′

]=

[∂v′

∂t+ (v ·∇)v′ + (v′ ·∇)v − µ∆v′ + ∇p′

−∇ · v′

]=

[00

]in Ω× (0,T ),

v′ = 0 at t = 0,

v′ = hτ on Γ× (0,T )

I Duality pairing defining the adjoint operator⟨N[

v′

p′

],

[v∗

p∗

]⟩L2(0,T ;L2(Ω))

=

⟨[v′

p′

],N ∗

[v∗

p∗

]⟩L2(0,T ;L2(Ω))

+ B1 + B2

I The adjoint system ( terminal value problem !! )N ∗

[v∗

p∗

]=

[− ∂v∗

∂t− v ·

[∇v∗ + (∇v∗)T

]− µ∆v∗ + ∇p∗

−∇ · v∗

]=

[00

]in Ω× (0,T ),

v∗ = 0 at t = T ,

v∗ = r × (ϕez ) + v∞ on Γ× (0,T )





Cost Functional Gradient

I The adjoint state and duality pairing can now be usedto re–express the cost functional differential as:

J ′(ϕ; h) =1

2

∫ T

0

∮ΓµRn ·D(v∗) · τ + µn ·D(v(ϕ)) · (ez × r) h dσ dt

I Identification of the cost functional gradient

J ′(ϕ; h) = (∇J (t), h)L2([0,T ]) =

∫ T

0

∇J (t) h dt

∇J (t) =1

2

∮Γ

µRn ·D(v∗) · τ + µn ·D(v(ϕ)) · (ez × r) dσ





Optimality (KKT) system

I Complete optimality system for ϕopt , [vopt , popt ], and [v∗, p∗]

1

2

∮ΓµRn ·D(v∗) · τ + µn ·D(v(ϕopt )) · (ez × r) dσ = 0[

∂v∂t

+ (v ·∇)v − µ∆v + ∇p∇ · v

]=

[00

]in Ω× (0,T ),

v = 0 at t = 0,

v = ϕoptτ on ΓN ∗

[v∗

p∗

]=

[− ∂v∗

∂t− v ·

[∇v∗ + (∇v∗)T

]− µ∆v∗ + ∇p∗

−∇ · v∗

]=

[00

]in Ω× (0,T ),

v∗ = 0 at t = T ,

v∗ = r × (ϕopt ez ) + v∞ on Γ

I A counterpart of the Euler–Lagrange equation

I Solved with an iterative Gradient Algorithm(e.g., Conjugate Gradients, quasi–Newton, etc.)





An Iterative Optimization Procedure

0. provide initial guess ϕ0

1. Solve for

v(ϕi ); p(ϕi )

on [0,T ]

2. Solve for

v∗(ϕi ); p∗(ϕi )

on [0,T ]

3. Use

v(ϕi ); p(ϕi )

and

v∗(ϕi ); p∗(ϕi )

to compute ∇J i (t) on [0,T ]

4. update control according to ϕi+1(t) = ϕi (t)− αiγi (∇J (t))

5. iterate 1. through 4. until convergence, i.e. until ∇J i (t) ' 0





Primal and Adjoint Simulationsfor Cylinder Rotation as Control





Results

I No Control

I Flow Pattern Modifications due to Control (T = 6)

I Optimal Control ϕopt , drag coefficient cD , transverse velocity v

50 60 70 80 90 100 110 120

t

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Ropt/|V|

50 60 70 80 90 100 110 120

t

0.85

0.9

0.95

1.0

1.05

1.1

1.15

1.2

1.25

1.3

cD

T=6.0

Basic flow

No control

50 60 70 80 90 100 110 120

t

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

vNo control




Euler System & Inverse FormulationSolution ApproachResults

PART II

Inverse Problem of Vortex

Reconstruction

joint work Ionut Danaila (Universite de Rouen)





I Ubiquitous Vortex Rings

Lim & Nickels, 1995

t = 20t = 2

z

r

Danaila & Heiles, 2008

I Models of Vortex Rings:I based on linearized equations (Kaplanski & Rudi, 1999, 2005)

I obtained with perturbation techniques (Fukumoto, 2010)

I inviscid models: Hill’s and Norbury-Fraenkel’s vortices

I Present Approach:Optimal Vortex Rings via Inverse Formulation





I Inviscid vortex ring in a moving frame of reference

zΩb

r(a) (b) r

Ωc

zψ=κ

ψ=

0Ωb

ω

r=

f (ψ) in Ωb,0 elsewhere,

f (ψ) — Vorticity Function(unspecified)

I 3D Axisymmetric Euler System

Lψ = −r f (ψ) in Ω,

ψ = 0 on γ.

where L := ∂∂z

(1r∂∂z

)+ ∂

∂r

(1r∂∂r

)= ∇ ·

(1r ∇)

and ∇ :=[∂∂z ,

∂∂r

]T.

I Special solutions:I f (ψ) = C in Ωb =⇒ Hill’s vortexI f (ψ) = C ∀ψ > k and f (ψ) = 0 ∀ψ ≤ k =⇒ Norbury-Fraenkel’s vortex





I Key Idea: determine vorticity function f (ψ) to match someobservation data =⇒ Inverse Problem

I Measurements of the tangential velocity component

+ n

r

γ b :

: ψ

= 0

Ωγ

z :: ψ = 0

z

B

C

O A

ψ = k

m := v · n⊥ =1

r

∂ψ

∂non boundary segments γz and γb

I Cost Functional

J (f ) :=αb

2

∫γb

(1

r

∂ψ

∂n

∣∣∣∣γb

−m

)2

dσ +αz

2

∫γz

(1

r

∂ψ

∂n

∣∣∣∣γz

−m

)2

dσ,

I Variational Minimization Problem: f := argminf ∈H1(I) J (f )

I nonnegativity constraint f (ψ) ≥ 0 ∀ψB. Protas Numerical Optimization of PDEs




I Inverse problem with unusual structure — reconstruction of a nonlinearsource term f (ψ)

I Assumptions

1. domain: f : I → R, I := [0, ψmax] — identifiability interval

2. smoothness: f ∈ H1(I) (square-integrable derivatives)

I Optimality condition: ∀f ′∈H1(I) J ′(f ; f ′) = 0

I Gradient iterations f = limk→∞ f (k)

f (k+1) = f (k) − τk∇J (f (k)), k = 1, 2, . . .

f (1) = f0,

f0 — initial guess, τk — step seize at k-th iteration

I Positivity enforcement via transformationf+ = (1/2)g2, Jg (g) := J ((1/2)g2)





I Gradient Expression — sensitivity of cost functional J (f ) with respectto perturbations of the vorticity function f (ψ)

∇L2J (s) = −∫γs

ψ∗ r

(∂ψ

∂n

)−1

dσ, s ∈ [0, ψmax].

γs := x ∈ Ω : ψ(x) = s — streamfunction level sets

0.05

0.1

0.15

0.20.25

0.3

00

z

r

1 0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

γs

γz

γbΩ

γs





I ψ∗ — solution of adjoint system

∇ ·(

1

r∇ψ∗

)+ rfψ(ψ)ψ∗ = 0 in Ω,

ψ∗ = αb

(1

r

∂ψ

∂n

∣∣∣∣γb

−m

)on γb,

ψ∗ = αz

(1

r

∂ψ

∂n

∣∣∣∣γz

−m

)on γz ,

I Smoothness ensured via Sobolev gradients:

J ′(f ; f ′) =⟨∇L2J (f ), f ′

⟩L2(I)

=⟨∇H1

J (f ), f ′⟩

H1(I)

=⇒

(I − `2 d2

ds2

)∇H1

J = ∇L2J in I,

∇H1

J = 0 at s = 0,

d

ds∇H1

J = 0 at s = ψmax,

I Algorithm easily implemented in FreeFEM++





Reconstruction of Hill’s Vortex

1e-05

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5 6

J(f

(k) )

/ J(f

(0) )

k

1e-05

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5 6

|Γ(f

(k) )

/ Γ

Hill-1

|, |I(

f(k) )

/ I H

ill-1

|, |E

(f(k

) ) / E

Hill-1

|

k

CirculationImpulseEnergy

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25 0.3

f(ψ

)

ψ

fHill

ψmax

f0-f





Reconstruction of Vortex Rings from DNS Data(Re = 17, 000)

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

f(ψ

)

ψ

DNS

fDNS-f

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.05 0.1 0.15 0.2

f(ψ

)

ψ

DNS

fDNS-f

· · · DNS data - - - empirical fit fDNS —– optimal reconstruction f





Reconstruction of Vortex Rings from DNS Data(Re = 17, 000)

z

r

0.5 0 0.50

0.5

1

1.5

(a) DNS

z

r

0.5 0 0.50

0.5

1

1.5

(b) present reconstruction

z

r

0.5 0 0.50

0.5

1

1.5

(c) NorburyFraenkel model

z

r

0.5 0 0.50

0.5

1

1.5

(d) KaplanskiRudi model

Vorticity distribution in space




Motivation & Mathematical ModelOptimization ProblemResults

PART III

Geometry Optimization

in Heat Transfer

joint work Xiaohui Peng and Katya Niakhai(former Master’s students at McMaster)





I Problem: Efficient cooling of a battery system

I Goal: determine optimal shape of cooling channels for aprescribed heat distribution

I Few mathematically precise results in literature=⇒ need to develop new tools





I 2D thermally isolated domain

I time–independent

I heat conduction only

I cooling channel — line heat sinkmodelled with Newton’s law of cooling

S = γ(u − u0)

u0 — temperature of the coolant fluidin the channel modelled by the coil C,

u0(s) = Ta +Tb − Ta

Ls, s ∈ [0, L],

I want to maintain prescribedtemperature u in the subdomain A(revised optimization objective)

Ω2

Ω1

δΩ

n

c

A

n

Ω1

Ω2

δΩ

A

nn

c

s





I Governing System

−k∆u1 = q in Ω1,

−k∆u2 = q in Ω2,

u1 = u2 on C∂u2

∂n− ∂u1

∂n= γ (u1 − u0) on C

∂u2

∂n= 0 on ∂Ω,

whereI Ω1 — the interior of the curve C,I Ω2 — the exterior of the curve C,I ui (x) is the temperature distribution u restricted in the domain

Ωi , for i = 1, 2,I k is the heat conductivity coefficient (a known material

property),I q is the distribution of heat sources (battery heating),I n are the unit outer normal on C and ∂ Ω





I Assuming:I a given distribution of heat sources q(x),I heat transfer described by governing equation,I a fixed length L =

∮C ds of the cooling channel C,

find the shape of the curve C which ensures that over thesubdomain A the actual temperature u(x , y) is as close aspossible to the prescribed temperature u

I Define

J (C) =

∫A

(u − u)2 dΩ

I Formal statement of optimization problem

maxCJ (C),

subject to: Governing System,∮Cds = L





I Optimal shape C characterized by the condition

J ′(C,Z) = 0 for all shape perturbations Z

I Gradient descent algorithm

x(n+1)C = x

(n)C − τn n∇J (C(n)), n = 1, 2, . . . ,

x(0)C = xC0 ,

where ∇J (C(n)) is the gradient of the cost functional

c

δΩ

Ω





I Problem of Shape Optimization (contour geometry),I Shape Calculus: parametrization of geometry

x(t,Z) = x + tZ for x ∈ ΓSL(0),

where Z : ΩSL → R2 is the perturbation “velocity” field.I Gateaux Shape Differential

J ′(ΓSL(0); Z) , limt→0

J (ΓSL(t,Z))− J (ΓSL(0))

t.

I Main Theorem [shape–differentiation of integrals w.r.t. theshape of the domain]: ∫

Ω(t,Z)

f dΩ +

∫∂Ω(t,Z)

g ds

′

=

∫Ω(0)

f ′ dΩ +

∫∂Ω(0)

g ′ ds+

+

∫∂Ω(0)

(f + κ g +

∂g

∂n

)Z · n ds,

I How to compute the gradient ∇J ?





I L2 Gradient ∇L2J (C(n)) computed as follows

∇L2J (C(n)) =γ

k(u1 − u0)

(∂u∗1∂n− κ u∗1

)− γ

k

∂u2

∂nu∗1 − λκ on C(n)

where u∗1 and u∗2 are solutions of the following adjointsystem

k∆u∗1 = (u − u)χA1 in Ω1,

k∆u∗2 = (u − u)χA2 in Ω2,

u∗1 − u∗2 = 0 on C(n),

k

(∂u∗2∂n− ∂u∗1

∂n

)= −γ u∗1 on C(n),

∂u∗2∂n

= 0 on ∂Ω2

I Optimal step size τn computed via line–minimization (usingBrent’s method)

τn = argminτ>0J (C(n) − τ∇J (C(n))B. Protas Numerical Optimization of PDEs




I Incorporation of the Length Constraint∮Cds = L0

I Modified (augmented) cost functional:

Jα(C) := J (C) +α

2

(∮Cds − L0

)2

,

where α ∈ R is a parameter

I After shape–differentiating the constraint, modified gradient

∇L2Jα(C) = ∇L2J (C) + α

(∮C(m)

ds − L0

)κ





I Gradients obtained using Riesz Representation Theorem

J ′(C; ζn) =⟨∇XJ , ζ

⟩X (C)

X — selected Hilbert space

I What is the required regularity of the gradients ∇J ?I xC(s) must be (at least) continuous

I L2 gradients ∇L2J (C) [X = L2(C)] may be discontinuous ...

I Need Sobolev Gradients [X = H1(C)]⟨∇H1

J , ζ⟩

H1(C)=

∫ L

0

∇H1

J ζ + `2 ∂∇H1

J∂s

∂ζ

∂sds, ∀ζ∈H1(C)

=⇒

(1− `2 ∂2

∂s2

)∇H1

J = ∇L2J on (0, L),

Periodic boundary conditions (P1),

∂

∂s∇H1

J∣∣∣

s=0,L= 0 (P2).





I Reformulation of the Governing System:

u = up + uh in Ω,

where ∀x∈Ω\C uh(x) = − 12π

∮C ln |x− xC |µ(xC) dσ.

I The new dependent variables up(x), x ∈ Ω;µ(x), x ∈ C satisfy

−k ∆ up = q in Ω,

µ(x) +γ

2π k

∮C

ln |x− xC |µ(xC) dσ =γ

k(up + uh − u0) on C,

∂up

∂n= −∂uh

∂non ∂Ω.

I Analogously for the Adjoint System with u∗p(x), x ∈ Ω;µ∗(x), x ∈ C





I Two coupled subproblems:

I Poisson equation for up (resp., u∗p )

I Singular Boundary Integral Equation for µ (resp., µ∗)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y





I Optimal discretization for each subproblem:

I spectral Chebyshev method for up (resp., u∗p ) in Ω

∆NU = f + q,

I spectral boundary-integral method with an analytic treatment of thesingular kernel for µ (resp., µ∗) on C(

I +γ

kK1 +

γ

kK2

)m +

γ

kPU =

γ

ku01,

I spectral interpolation P to couple up and µ (resp., u∗p and µ∗)[−∆N Bγk P I + γ

k K1 + γk K2

] [Um

]=

1

k

[q

γ u0 1

].





CASE I: α = 0

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

10

20

30

40

50

60

70

80

90

q(x , y)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

18.5

19

19.5

20

20.5

u0 10 20 30 40 50 60 70 80 90 100

100

101

102

J (C(m))

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

C(m)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

20

25

30

35

40

45

50

Initial Solution u

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

16

16.5

17

17.5

18

18.5

19

19.5

20

20.5

21

21.5

Final Solution u





CASE II: α = 0

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

38

40

42

44

46

48

50

52

54

56

58

60

q(x , y)

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

14.2

14.4

14.6

14.8

15

15.2

15.4

15.6

15.8

u0 5 10 15 20 25 30

10−1

100

101

102

103

104

J (C(m))

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

C(m)

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

25

30

35

40

45

50

Initial Solution u

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

13.8

14

14.2

14.4

14.6

14.8

15

15.2

15.4

15.6

15.8

Final Solution u





CASE III: α = 0, 1, 10, 102, 103; L0 = 2.3

x

y

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

−80

−60

−40

−20

0

20

40

60

80

100

120

140

q(x , y)x

y

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

30

40

50

60

70

80

90

100

110

120

130

140

u 1 2 3 4 5 6 7 810

0

101

102

103

104

n

J

J (C(m))

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

y

C1 2 3 4 5 6 7 8

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

n

L(C

(n) )

Length of C(m)x

y

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

27

28

29

30

31

32

33

34

35

36

37

Final Solution u (α = 103)





Conclusions

I Formulation of PDE control and estimation problems as constrainedoptimization

I PDE–constrained gradients via Adjoint EquationsI Vorticity form of the adjoint equationsI Optimization of free boundary problems via shape–differential calculus

I Inverse Problem of Vortex ReconstructionI Nonintuitive insights revealed by reconstruction from DNS dataI Big Question: what are the fundamental accuracy limits for

representation of real flows in terms of inviscid models?

I Shape-optimization approach for a model of 2D steady heat transferI Shape calculusI Spectrally-accurate solution of the governing and adjoint PDE systems





I References Part I — Open–Loop Control:I B. Protas, T. R. Bewley and G. Hagen, “A comprehensive framework for the

regularization of adjoint analysis in multiscale PDE systems”, Journal ofComputational Physics 195(1), 49-89, 2004.

I B. Protas, “On the “Vorticity” Formulation of the Adjoint Equations and itsSolution Using Vortex Method”, Journal of Turbulence 3, 048, 2002.

I B. Protas and A. Styczek, “Optimal Rotary Control of the Cylinder Wake in theLaminar Regime”, Physics of Fluids 14(7), 2073–2087, 2002.

I B. Protas, “Vortex Design Problem”, Journal of Computational and AppliedMathematics 236, 1926–1946, 2012.

I References Part II — Inverse Problem of Vortex Reconstruction:I V. Bukshtynov, O. Volkov, and B. Protas, “On Optimal Reconstruction of

Constitutive Relations”, Physica D 240, 1228–1244, 2011.I V. Bukshtynov and B. Protas, “Optimal Reconstruction of Material Properties in

Complex Multiphysics Phenomena”, Journal of Computational Physics 242,889–914, 2013.

I I. Danaila and B. Protas, “Optimal Reconstruction of Inviscid Vortices”,Proceedings of the Royal Society A (in press), 2015.

I References Part III — Shape Optimization:I X. Peng, K. Niakhai and B. Protas, “A Method for Geometry Optimization in a

Simple Model of Two-Dimensional Heat Transfer”, SIAM Journal on ScientificComputing 35, B1105–B1131, 2013.


numerical optimization of partial di erential equationsbprotas/rouen2018_ecole_part3_v2.pdf ·...

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