Wavelet Methods for PDE-Constrained Control Problems ...naconf/07/kunoth_dundee07.pdfWavelet Methods for PDE-Constrained Control Problems: Optimal Preconditioners, Fast Iterative Solvers

Wavelet Methods for PDE-Constrained Control Problems:

Optimal Preconditioners, Fast Iterative Solvers and Adaptivity

Angela Kunoth

Institut fur Angewandte Mathematik & Institut fur Numerische Simulation

Universitat Bonn, Germany

Central goal: Development of efficient solution algorithms with optimal linear complexity

Central issues: Iterative solvers, multilevel preconditioning and adaptivity

Problem classes:

(I) Optimal control problems constrained by linear elliptic PDEs with distributed or Neumann

boundary control ; single operator equation as constraint

(II) . . . . . . with Dirichlet boundary control ; saddle point problem as constraint

(III) Problem (I) with additional inequality constraints on the control

Supported by SFB 611 (Deutsche Forschungsgemeinschaft) 1

(I) PDE-Constrained Optimal Control Problem with Distributed Control

given y∗, f

ω > 0

minimize J(y, u) = 12‖y − y∗‖2H1−s(Ω)

+ ω2‖u‖2

(H1−t(Ω))′

subject to −∆y + y = f + u in Ω ⊂ Rd

∂y∂n

= 0 on ∂Ω

0 ≤ s ≤ 1 smoothness parameter for state y

0 ≤ t smoothness parameter for control u

A : H1(Ω)→ (H1(Ω))′ 〈Av,w〉 :=R

Ω(∇v · ∇w + vw)dx

; weak formulation nontrivial solution for y∗ 6≡ A−1f


+ ω2‖u‖2

(H1−t(Ω))′

subject to Ay = f + u

Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 2

(II) PDE-Constrained Optimal Control Problem with Dirichlet Boundary Control

given y∗, f

ω > 0

ΩΓyΓ

minimize J(y, u) = 12‖Ty − y∗‖2

Hs− 1

2 (Γy)+ ω

2‖u‖2

Ht(Γ)

subject to −∆y + y = f in Ω ⊂ Rd

y = u on Γ control boundary

∂y∂n

= 0 on ∂Ω \ Γ

s, t ∈ [ 12, 32] smoothness parameters for state and control

〈Av,w〉 :=R

Ω(∇v · ∇w + vw)dx 〈Bv, q〉 :=R

Γ v q dΓ bilinear form on H1(Ω)× (H1/2(Γ))′

; weak formulation (appending essential Dirichlet b.c. by Lagrange multipliers)

minimize J(y, u) = 12‖y − y∗‖2Hs(Γy)

+ ω2‖u‖2

Ht(Γ)

subject to L` y

p

´

:=“

A BT

B 0

” “

yp

”

=“

fu

”

. . . allows also combination with fictitious domain method and changing boundary Γ . . .


(III) Distributed Control Problem and Control Inequality Constraints

Problem (I) with additional inequality constraints on control


+ ω2‖u‖2

(H1−t(Ω))′

subject to Ay = f + u

and u ≤ u ≤ u


PDE-constrained optimal control problems ; requires repeated solution of PDE constraint

Ay = f + u or L`yp

´

=`fu

´

; requires fast solver as core ingredient

Numerical Solution of a Single Elliptic PDE

Elliptic PDE Ay = f s.th. ‖Av‖Y ′ ∼ ‖v‖Y ; find y ∈ Y : 〈v,Ay〉 = 〈v, f〉 for all v ∈ Y

Conventional discretization on a uniform grid: Yh ⊂ Y dimYh <∞ ; Ah yh = fh

Obstructions:

Large linear systems of equations ; iterative solver

High desired accuracy ; small h ; larger problem ; worse condition cond2(Ah) ∼ h−2

LBB condition for saddle point problems

Resolution of singularities in data and/or geometry ; small h

Ingredients for Efficient Numerical Solution:

(i) Multilevel preconditioner Ch

multigrid methods, BPX preconditioner, wavelet discretization ; cond2(ChAh) ∼ 1

(ii) Nested iteration

(iii) Additionally: Adaptive refinement

a–posteriori error estimation ; local grid refinement ; convergence/convergence rates?

First goal: Realize discretization error accuracy ε with minimal amount of work O(N(ε))


A-priori Estimates for Finite Elements

Quality measure: Approximation in norm ‖y − yh‖L2(Ω) ≤ ε

A–priori error estimates: Ω ⊂ Rd dimYh = N ∼ h−d uniform grid

‖y − yh‖L2(Ω) <∼ hr ‖y‖Hr(Ω) yh ∈ Yh 0 ≤ r ≤ rmax

⇐⇒ ‖y − yN‖L2(Ω) <∼ N−r/d ‖y‖Hr(Ω)

‖y − yN‖H1(Ω)

<∼ N−(r−1)/d ‖y‖Hr(Ω)

N degrees of freedom ←→ accuracy O(N−r/d)

Approximation rate determined by

(i) approximation order rmax of Yh

(ii) space dimension d

(iii) amount of smoothness of y in L2

Target: Realize discretization error accuracy ε ∼ h2 ∼ 2−2J for grid with spacing h ∼ 2−J

Problem complexity: For h ∼ 2−J a total of N ∼ 2Jd unknowns

Optimal complexity for iterative solver: Minimal amount of work is O(N)


Ingredients for Efficient Numerical Solution: (i) Multilevel Preconditioner

Asymptotically optimal preconditioner: Ch such that cond2(ChAh) ∼ 1

and setup and application of Ch in optimal linear complexity O(N)

Schwarz iterative schemes based on subspace corrections ; multilevel schemes:

• multiplicative schemes ; multigrid methods Brandt, Braess, Bramble, Hackbusch . . .

• additive schemes ; BPX preconditioner; wavelet discretizationBramble, Pasciak, Xu, Yserentant, Oswald, Dahmen, Kunoth . . .

Relevant idea from Approximation Theory: Multilevel characterization of function spaces

and norm equivalences


Multilevel Characterization of Function Spaces

Multiresolution Yj0 ⊂ Yj0+1 ⊂ . . . ⊂ Yj ⊂ Yj+1 ⊂ . . . Y, closY

“

S∞j=j0

Yj

”

= Y

Linear (orthogonal) projectors Qj : Y → Yj s.th. QjQℓ = Qj for j ≤ ℓ ; Qj −Qj−1 projector

Theorem: [Dahmen, Kunoth ’92], [Oswald ’92]

(S) Φj uniformly stable basis for Yj : ‖c‖ℓ2 ∼ ‖cT Φj‖L2

(J) Jackson estimate infvj∈Yj

‖v − vj‖L2<∼ 2−sj‖v‖Hs v ∈ Hs 0 < s ≤ d

(B) Bernstein inequality ‖vj‖Hs <∼ 2sj‖vj‖L2

vj ∈ Yj s < t

=⇒ Norm equivalence

‖v‖2Hs ∼

∞X

j=j0

22sj‖(Qj −Qj−1)v‖2L2s ∈ (−σ, σ)

0 < σ := mind, t, 0 < σ := mind, t,

Proof: (J) and discrete Hardy inequality ; upper estimate for ‖ · ‖Hs .

(B), ‖Qj‖L2<∼ 1 and Whitney estimate ; lower estimate


Norm Equivalence for Optimal Preconditioning

Theorem: [Jaffard ’92], [Dahmen, Kunoth ’92], [Oswald ’92]

Y = Hs C−1J := Aj0Qj0 +

JX

j=j0

22sj(Qj −Qj−1)

is optimal preconditioner for AJ : YJ → YJ : cond2(C1/2J AJC

1/2J ) ∼ 1 as J →∞

Proof: Isomorphism ‖Av‖Y ′ ∼ ‖v‖Y on YJ combined with norm equivalence for Y = Hs

Realization of C−1J :

• Any s ∈ (−σ, σ): Explicit representation of (Qj −Qj−1)v ; wavelet basis together with

diagonal Ds := (2sj)j=j0...J ; Fast Wavelet Preconditioner (FWT) realizes preconditioning in

optimal linear complexity [Jaffard ’92], [Dahmen, Kunoth ’92]

• s > 0 : Replace CJ = Aj0Qj0 +J

X

j=j0

2−2sj(Qj −Qj−1) by spectrally equivalent preconditioner

C−1J := Aj0Qj0 +

JX

j=j0

2−2sjQj

; BPX preconditioner is also optimal preconditioner

developed by [Bramble, Pasciak, Xu ’90], optimality proved by [Dahmen, Kunoth ’92], [Oswald ’92]

Hierarchical basis preconditioner by [Yserentant ’89] not optimal


Building Blocks: (Biorthogonal Spline–) Wavelets

H Hilbert space with ‖ · ‖H H′ dual space for H with 〈·, ·〉

Ψ := ψλ : λ ∈ II ⊂ H Wavelets II (infinite) index set

(NE) Ψ Riesz basis for H

v ∈ H: v = vT Ψ :=X

λ∈II

vλ ψλ such that ‖v‖H ∼ ‖v‖ℓ2(II)

(L) Locality diam (suppψλ) ∼ 2−|λ| |λ| resolution

ψλ centered around 2−|λ|k

(CP) Vanishing moments

0 1

ψ2,2

ψ2,1

[Dahmen, Kunoth, Urban ’99] [Dahmen, Schneider ’99; Kunoth, Sahner ’05] [Harbrecht, Schneider ’00]


Numerical Results with Fast Wavelet Transform: Spectral Condition Numbers

Elliptic partial differential operator on Ω = (0, 1)d with FWT preconditioning

−∆ + 1 (−∆ + 1)CK

j 0 1 0 1

3 229 22.3 256 27.1

4 244 23.9 263 27.9

5 255 25.0 289 30.6

6 262 25.7 301 31.9

8 271 26.6 319 33.9

10 276 27.1 330 35.0

12 278 27.3 337 35.8

space dimension d = 1

−∆ + 1 (−∆ + 1)CK

j 0 1 4 5 0 1 3 4

3 519 78.2 76.0 49.5 256 27.8 17.3 9.64

4 627 129 128 124 308 33.4 20.9 11.8

5 646 149 149 147 372 40.4 25.3 14.3

6 664 165 165 165 416 45.1 28.2 16.0

8 681 179 179 179 480 52.1 32.6 18.4


−∆ + 1 (−∆ + 1)CK

j 0 9 0 1 4

3 1103 269 256 28.5 18.3

4 1917 1913 520 57.8 37.1

5 2228 2222 557 62.0 39.8

6 2459 2443 572 63.6 40.9


Uniformly bounded and absolutely small spectral condition numbers cond2(AJ ) [Burstedde ’05]


Ingredients for Efficient Numerical Solution: (ii) Nested Iteration

Recall goal: realize discretization error accuracy εJ ∼ h2 ∼ 2−2J for grid with spacing h ∼ 2−J

with minimal amount of work O(N) N ∼ 2Jd unknowns

Naive strategy:

• Iterate only on highest level J and iterate until discretization error accuracy

needs O(J) = O(− log εJ ) iterations to achieve prescribed discretization error accuracy

εJ ∼ 2−2J

• Each application of optimally conditioned AJ requires O(NJ ) arithmetic operations

; a total of O(J NJ ) arithmetic operations iterating on finest level only

Theorem:

Starting with coarsest level j0, solve Ajyj = fj on each level j up to discretization error

accuracy εj and prolongate result from level j to next level j + 1 as initial guess

; Optimal preconditioner + nested iteration yields method of optimal complexity O(NJ )

to reach discretization error accuracy on finest level J

Proof: For mupliplicative Schwarz schemes: known and known as full multigrid

For additive preconditioners: optimal condition of Aj ; fixed amount of iterations on each level to reach discretization error accuracy on that level;

spaces nested and Nj ∼ 2dj and geometric series argument [Dahmen, Kunoth, Schneider ’99]


Application to (I) PDE-Constrained Optimal Control Problem with Distributed Control

Control problem in wavelet coordinates

minimize J(y, u) = 12‖D−s (y − y∗)‖2 + ω

2‖Dt u‖2 0 ≤ s ≤ 1, 0 ≤ t

subject to Ay = f + u A : ℓ2 → ℓ2 automorphism ‖ · ‖ := ‖ · ‖ℓ2

Necessary and Sufficient Conditions

Lagr(y,u,p) := J(y,u) +˙

p, Ay − (f + D−t u)¸

δLagr = 0 ;

Ay = f + D−t u

AT p = −D−2s (y − y∗)

ωu = D−t p

⇐⇒ Qu = g

Q : ℓ2 → ℓ2 automorphism

whereQ := D−tA−T D−2sA−1D−t + ωI symmetric positive definite

g := D−tA−T D−2s(y∗ −A−1f)


Nested-Iteration-Inexact-Conjugate-Gradient Algorithm NIICG

Essential idea: Conjugate gradient (cg) iteration on Qu = g (outer loop) and cg iteration for primal

and dual systems (inner iterations); combined with nested iteration [Burstedde, Kunoth ’05]

Numerical Examples for Distributed Control Problem

d = 2: y∗ =q

|x− ( 13, 13)T | ω = 1 f ≡ 1 s = t = 0 runtime 177 sec

j ‖rj‖ #O #E #A #R ǫR(y) ǫP (y) ǫR(u) ǫP (u)

3 1.60e-04 3.73e-04 1.29e-05 3.51e-05

4 7.87e-06 6 4 1 18 1.41e-04 2.17e-04 5.11e-06 5.90e-06

5 3.89e-06 5 4 1 19 1.67e-05 8.34e-05 1.77e-06 1.79e-06

6 2.02e-06 4 4 2 17 1.43e-05 4.30e-05 7.68e-07 7.69e-07

7 1.13e-06 2 7 3 16 1.30e-05 2.39e-05 7.19e-07 7.19e-07

8 4.76e-07 5 3 1 15 1.42e-06 9.90e-06 1.85e-07 1.85e-07

9 2.25e-07 3 5 2 16 1.12e-06 4.51e-06 1.33e-07 1.33e-07

10 1.32e-07 3 5 2 16 9.48e-07 9.48e-07 1.10e-07 1.10e-07

d = 3: y∗ =q

|x− ( 13, 13, 13)T | ω = 1 f ≡ 1 s = t = 0 runtime 3502 sec

j ‖rj‖ #O #E #A #R ǫR(y) ǫP (y) ǫR(u) ǫP (u)

3 1.41e-04 2.92e-04 1.13e-05 2.36e-05

4 6.09e-06 10 9 1 49 1.27e-04 1.78e-04 3.46e-06 3.79e-06

5 3.25e-06 10 7 1 58 1.11e-05 6.14e-05 9.47e-07 9.53e-07

6 1.71e-06 7 6 1 57 1.00e-05 2.86e-05 5.03e-07 5.03e-07

7 8.80e-07 6 6 1 53 9.19e-06 9.19e-06 3.72e-07 3.72e-07

3.2GHz Pentium IV computer (family 15, model 4, stepping 1, with 1MB L2 Cache) [Burstedde ’05]


Application to (II) PDE-Constrained Optimal Control Problem with Dirichlet Boundary Control

Optimality conditions ; system of saddle point problems ; additional inner iterations

[Kunoth ’01]

L“y

p

”

=“f

u

”

ωRHt(Γ)u = µ

LT“z

µ

”

=“−TT R

Hs− 1

2 (Γy)(Ty − yΓy )

0

”

⇐⇒ NU = F :⇐⇒

0

@

L E

E LT

1

A

0

B

B

B

B

B

@

y

p

z

u

1

C

C

C

C

C

A

:=

0

B

B

B

B

B

@

A BT 0 0

B 0 0 −ω−1R−1Ht(Γ)

TT RH

s− 12 (Γ)

T 0 AT BT

0 0 B 0

1

C

C

C

C

C

A

0

B

B

B

B

B

@

y

p

z

u

1

C

C

C

C

C

A

=

0

B

B

B

B

B

@

f

0

−TT yΓy

0

1

C

C

C

C

C

A


Control Problem with Dirichlet Boundary Control: Solution Graphics

[Pabel ’05]

00.25

0.50.75

1

0

0.250.5

0.751

0

0.25

0.5

0.75

1

00.25

0.50.75

0

0.250.5

0.75

Ω = 100

Ω = 10

Ω = 5

Ω = 2

Ω = 1

Ω = 0.5

Ω = 0.4

Ω = 0.3

Ω = 0.2

Ω = 0.1

Ω = 0.01

00.25

0.50.75

1

0

0.250.5

0.751

0

0.25

0.5

0.75

1

00.25

0.50.75

0

0.250.5

0.75

s = 0.0

s = 0.1

s = 0.2

s = 0.3

s = 0.4

s = 0.5

s = 0.7

s = 0.9

s = 1.0

s = 1.5


Numerical Results for Control Problem with Dirichlet Boundary Control

All–In–One Solver: cg method & nested iteration on NT NU = NT F

Best results for inexact gradient iteration on u and Uzawa with conjugate directions for each of saddle

point problems & nested iteration

d = 2: yΓy ≡ 1 f ≡ 1 ω = 1 s = 12

L2(Γy) t = 12

H1/2(Γ) [Pabel ’05]

P with D−11 P O with D−1

a

J ‖r(kJ )J

‖ ‖y−y(kJ )J

‖ kJ#Int-It

kJ

#Int-ItkJ

‖r(kJ )J

‖ ‖y−y(kJ )J

‖ kJ#Int-It

kJ

#Int-ItkJ

4 1.6105e-02 7.7490e-00 0 – – 3.9807e-02 8.5283e-02 4 1.5 2.5

5 1.6105e-02 7.7506e-00 0 – – 1.3970e-02 2.5233e-02 2 1 1.5

6 6.3219e-03 1.7544e-02 2 1 1 1.4421e-02 1.2301e-02 1 1 1

7 5.8100e-03 3.3873e-02 0 – – 6.1354e-03 4.6339e-03 2 1 1.5

8 1.6378e-03 3.4958e-03 2 1 1 2.2413e-03 1.7083e-03 2 1.5 1.5

9 1.8247e-03 7.4741e-03 0 – – 9.2961e-04 8.8526e-04 2 1.5 1

10 4.3880e-04 9.2663e-04 2 1 1.5 9.2954e-04 8.7262e-03 0 – –

11 4.6181e-04 1.8486e-03 0 – – 4.4790e-04 3.8321e-04 2 1.5 1.5


Distributed Control Problem with Inequality Constraints

Treatment of control inequality constraints:

projected gradient method — interior point (IP) method — primal-dual active set (PDAS) method

u(x) ≡ −0.5 y∗(x) = x1(1− x1) · x2(1− x2)− 32[x1(1− x1) + x2(1− x2)]

f(x) :=

8

<

:

0.5 + 2[x1(1− x1) + x2(1− x2)] for x ∈ K•,

16x1(1− x1)x2(1− x2) + 2[x1(1− x1) + x2(1− x2)] otherwise

; exact solution y(x) = x1(1− x1) · x2(1− x2)

j J(y(k), u(k)) ε(u(k)j

) ε(y(k)j

) k ∅ (#CG) run time

4 6.963009e + 01 1.099181e − 02 1.125383e − 02 2 9 31.1186

5 6.577754e + 01 1.910503e − 03 5.154380e − 03 3 45 40.7679

6 6.391144e + 01 7.092608e − 04 2.361333e − 03 3 92 181.723

7 6.305831e + 01 2.237459e − 04 1.115190e − 03 3 148 1484.53

8 6.274058e + 01 8.386212e − 05 5.381711e − 04 3 186 11545.4

9 6.268009e + 01 2.803506e − 05 2.625252e − 04 3 230 100938

<∼ 2−3/2 j <

∼ 2−j ∼ const

PDAS [Hoffmann & K ’07]


So far: uniform grids . . . . . . further reduction of complexity by adaptive refinement

Recall: A-priori Estimates for Finite Elements

Quality measure: Approximation in norm ‖y − yh‖L2(Ω) ≤ ε

A–priori error estimates: Ω ⊂ Rd dimYh = N ∼ h−d uniform grid

‖y − yh‖L2(Ω) <∼ hr ‖y‖Hr(Ω) yh ∈ Yh 0 ≤ r ≤ rmax

⇐⇒ ‖y − yN‖L2(Ω) <∼ N−r/d ‖y‖Hr(Ω)

‖y − yN‖H1(Ω)

<∼ N−(r−1)/d ‖y‖Hr(Ω)

N degrees of freedom ←→ accuracy O(N−r/d)

Approximation rate determined by

(i) approximation order rmax of Yh

(ii) space dimension d

(iii) amount of smoothness of y in L2

‘Wish list’ for adaptive method

• uses no a–priori knowledge

• realizes rate under less smoothness assumptions

• universally usable: automatic realization of order


Paradigm for One Stationary PDE [Cohen, Dahmen, DeVore ’99–’01]

(i) Well–posed variational problem: given f ∈ y′, find y ∈ Y : 〈v,Ay〉 = 〈v, f〉 ∀v ∈ Y Ay = f

(MP) ‖Av‖Y ′ ∼ ‖v‖Y for all v ∈ Y

(ii) Wavelet basis for Y : Ψ = ψλ : λ ∈ II ⊂ Y

(NE) c ‖v‖ℓ2 ≤ ‖P

λ∈II vλψλ‖Y ≤ C ‖v‖ℓ2 for all v = (vλ) ∈ ℓ2

Ay := (〈ψλ, Ay〉)λ∈II f = (〈ψλ, f〉)λ∈II

;

Theorem Ay = f ⇐⇒ Ay = f A : ℓ2 → ℓ2 and Ay = f well-posed in ℓ2

(MP) + (NE) =⇒ (c2 cA)−1 ‖w‖ℓ2 ≤ ‖Av‖ℓ2 ≤ C2CA ‖w‖ℓ2 w ∈ ℓ2

(iii) (Idealized) iteration

yn+1 = yn + Cn(f −Ayn) n = 0, 1, 2, . . . ‖yn+1 − y‖ℓ2 ≤ ρ ‖yn − y‖ℓ2 ρ < 1

(iv) Approximate realization through adaptive evaluation of Ayn (CP)

Solve [ε,C,A, f ]→ y(ε)

(i) Initialize ‖y − v‖ℓ2 ≤ δ η = δ

(ii) perturbed update v + rη := v + Res [η,C,A, f ,v]→ v η/2→ η

until ‖rη‖ℓ2 “small enough”

(iii) Coarse [cδ,v]→ v δ → δ/2 η = δ go to (ii) until δ ≤ ε


Complexity Analysis

Based on benchmark:

decay rate s for (wavelet-)best N term approximation As := v ∈ ℓ2 : ‖v − vN‖ℓ2 <∼ N−s

Work/accuracy balance of best N term approximation:

Target accuracy ε (∼ N−s) ←→ Work ε−1/s (∼ N)

Convergence and Complexity

(Idealized) iteration yn+1 = yn + Cn(f −Ayn) update via Res [η,C,A, f ,v]→ rη

Theorem [CDD] Output rη of Res [η,C,F, f ,v] satisfies ‖rη‖As <∼ ‖v‖As + ‖u‖As

and # supp rη , #flops <∼ η−1/s (‖v‖

1/sAs + ‖u‖

1/sAs + 1)

=⇒ for every variational problem satisfying (MP) perturbed scheme Solve has properties:

(I) For every target accuracy ε > 0 Solve produces after finitely many steps

approximate solution uε such that ‖u− uε‖ℓ2 ≤ ε

(II) Exact solution u ∈ As =⇒ suppuε, # flops ∼ ε−1/s ∼ N


; construct Res [η,C,F, f ,v] such that output rη satisfies ‖rη‖As <∼ ‖v‖As + ‖u‖As

and #supp rη , #flops <∼ η−1/s (‖v‖

1/sAs + ‖u‖

1/sAs + 1)

F(u) = Au ; Properties of Res derived from s– compressibility of A

Core Ingredient of Solve : Compressible Operators

A s∗–compressible: for every 0 < s < s∗ there exists Aj

with ≤ αj2j nonzero entries per row and column such that

‖A−Aj‖ ≤ αj2−sj j ∈ N0

X

j∈N0

αj <∞

supp v <∞ ; v[j] := v2j best 2j approximations wj := Ajv[0] + Aj−1v[1] + · · ·+ A0v[j]


Generalization to (I) PDE-Constrained Optimal Control Problem with Distributed Control

Control problem in wavelet coordinates

minimize J(y, u) = 12‖D−s (y − y∗)‖2 + ω

2‖Dt u‖2 0 ≤ s ≤ 1, 0 ≤ t

subject to Ay = f + u A : ℓ2 → ℓ2 automorphism ‖ · ‖ := ‖ · ‖ℓ2

Necessary and Sufficient Conditions

Lagr(y,u,p) := J(y,u) +˙

p, Ay − (f + D−t u)¸

δLagr = 0 ;

Ay = f + D−t u

AT p = −D−2s (y − y∗)

ωu = D−t p

⇐⇒ Qu = g

Q : ℓ2 → ℓ2 automorphism

whereQ := D−tA−T D−2sA−1D−t + ωI symmetric positive definite

g := D−tA−T D−2s(y∗ −A−1f)


Convergence and Complexity Analysis

Essential idea: Res for Solve [. . . ,Q, . . .] reduced to Res for Solve [. . . ,A, . . .]

and system of Euler equations ←→ condensed system

Theorem [Dahmen, Kunoth ’05]

For any target accuracy ε > 0 Solve [ε,Q,g]→ uε converges in finitely many steps

‖u− uε‖ ≤ ε ‖y − yε‖ <∼ ε ‖p− pε‖ <

∼ ε uε,yε,pε finitely supported

u,y,p ∈ As =⇒

(# suppuε) + (# suppyε) + (# supppε) <∼ ε−1/s

“

‖u‖1/sAs + ‖y‖

1/sAs + ‖p‖

1/sAs

”

‖uε‖As + ‖yε‖As + ‖pε‖As <∼ ‖u‖As + ‖y‖As + ‖p‖As

#flops ∼ ε−1/s

Extension to PDE-constrained control problems with Dirichlet boundary control

; System of saddle point problems ; additional inner iterations [Kunoth ’05]


Numerical Example for Distributed Control Problem (1D)

min J(y, u) J(y, u) = 12‖y − y∗‖2H1(Ω)

+ 12‖u‖2

L2(Ω)(s = 0 t = 1)

under constraints

8

<

:

−y′′ + y = f + u in Ω := (0, 1)dydn

= 0 at 0, 1

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1

right hand side f

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

target state y*

right hand side f target state y∗

Wavelet coefficients: State y adjoint state p control u [Burstedde, Kunoth ’05]


Numerical Example for Distributed Control Problem (2D)

min J(y, u) J(y, u) = 12‖y − y∗‖2

H1/2(Ω)+ 1

2‖u‖2

L2(Ω)y∗ = h2 ⊗ h2 (s = 1/2 t = 0)

under constraints

8

<

:

−∆y + y = D3/2(h1 ⊗ h1) + u in Ω := (0, 1)2

dydn

= 0 on ∂Ω

Optimal rate in energy norm (r = 2 d = 2) is r−1d

= 12

j ‖rj‖ #O #E #A #R S Nad ǫP (y) ǫP (u)

3 1.31e-02 2.19e-04

4 3.09e-04 1 6 1 11 54.0% 156 1.02e-02 2.19e-04

5 3.55e-04 1 6 2 11 49.0% 534 5.08e-03 2.19e-04

6 1.80e-04 4 4 1 20 51.6% 2182 2.55e-03 2.19e-04

7 1.22e-04 6 6 1 21 43.1% 7169 1.31e-03 2.19e-04

8 5.61e-05 8 8 1 23 36.0% 23745 6.73e-04 2.19e-04

9 2.22e-05 10 9 1 23 30.6% 80525 3.33e-04 1.55e-04

10 1.15e-05 12 9 2 24 27.6% 289790 1.25e-04 1.07e-04

σ ≈ 0.55 [Burstedde ’05]


Numerical Example for Distributed Control Problem (2D) Target y∗

0 0.2 0.4 0.6 0.8 x1 00.20.40.60.8x2

0.40.60.8

1

[Burstedde ’05]

type e = (1, 0) type e = (0, 1)

4

5

6

7

1.69e-03

0.00e+00 4

5

6

7

1.69e-03

0.00e+00

4

5

6

7

2.08e-01

0.00e+00 4

5

6

7

2.08e-01

0.00e+00

4

5

6

7

4.34e-03

0.00e+00 4

5

6

7

4.34e-03

0.00e+00


Summary

• Fast iterative solution based on multiscale ideas of first order optimality conditions for

PDE-constrained control problems

• Uniformly bounded condition numbers of system matrices + nested iteration

=⇒ solver produces discretization error accuracy in optimal linear complexity O(NJ )

• Application to control problems constrained by linear elliptic PDEs with distributed or Dirichlet

boundary control

– Modelling of objective functional [Burstedde, Kunoth ’05]

– Uniformly bounded condition numbers of system matrices

– A–posteriori error estimators for coupled system of operator equations

– automated adaptive refinement for each of state y, control u and adjoint state p

– convergence proof and algorithmic efficiency: first optimal complexity estimates

method has optimal work/accuracy rate

Extensions, Generalizations and Outlook

• Goal-oriented error estimation [Dahmen, Kunoth, Vorloeper ’05]

• Attractive for control problems with linear parabolic PDE constraints

; PDEs coupled globally in time – space-time approach in weak form

• Control problem with nonlinear elliptic PDE as constraints: SQP methods . . .Angela Kunoth — Wavelet Methods for PDE Control: Optimal Preconditioners, Fast Iterative Solvers and Adaptivity 28

Numerical Results: BPX Preconditioner and Nested Iteration

Laplacian on sphere ∆S y = f f(x) = 2x exact solution y(x) = x

stopping criterion ‖rj‖ℓ2 ≤ 2−2j (∼ h2) piecewise linear finite elements

its: iterations on each level to reach stopping criterion [Maes, Kunoth, Bultheel ’05]

j cond ‖rj‖ℓ2its

1 3.1 2.4897e-05 12

2 3.7 1.6766e-05 9

3 4.6 4.7350e-06 11

4 5.5 4.5474e-06 11

5 6.2 1.6705e-06 12

6 6.7 1.0193e-06 12

7 7.0 6.2720e-07 12

8 7.4 1.6451e-07 13

Biharmonic equation on sphere ∆2S y = f C1 cubic conforming finite elements

j cond ‖rj‖ℓ2its

1 52.0 2.2290e-03 0

2 66.7 5.1424e-04 2

3 78.4 4.2928e-04 1

4 87.7 3.1846e-04 3

5 95.2 1.6570e-04 3

6 100.6 7.9261e-05 5

7 105.5 4.0583e-05 4

Optimality of BPX preconditioner [Kunoth ’94; Oswald ’94]

Numerical results [Maes, Kunoth, Bultheel ’05]


Documents

Wavelet Methods for PDE-Constrained Control Problems ...naconf/07/kunoth_dundee07.pdfWavelet Methods for PDE-Constrained Control Problems: Optimal Preconditioners, Fast Iterative Solvers