Worst-case shape optimization

for the Dirichlet energy

Giuseppe Buttazzo

Dipartimento di Matematica

Università di Pisa

buttazzo@dm.unipi.it

http://cvgmt.sns.it

“A Mathematical Tribute to Ennio De Giorgi”Pisa, September 19–23, 2016

Joint work

Worst-case shape optimization for the Dirich-let energy

José Carlos Bellido (Castilla La Mancha, Spain)Giuseppe Buttazzo (Pisa, Italy)Bozhidar Velichkov (Grenoble, France)

To appear on Nonlinear Analysis TMA

available at:http://cvgmt.sns.it

http://arxiv.org

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Optimization problems:

min{F (u) : u ∈ X

}.

Problems in the calculus of variations are ofthis type, u is the state variable varying in aspace of functions and F is usually a func-tional expressed in an integral form. In thisformulation we only observe the system with-out intervening.

Example:

min{ ∫

Ωj(x, u,∇u) dx : u ∈ H10(Ω)

}.

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Optimal control problems:

min{F (u, v) : u ∈ X, v ∈ Y, u ∈ A(v)

}.

The class A(v) is usually given through a dif-ferential equation (elliptic PDE in our case),

called state equation, v is the control vari-

able and represents the way we can operate

in the system.

Example:

minu∈H10(Ω), v∈L2(Ω)

{ ∫Ω|u−u0|2+αv2 dx : −∆u = f+v

}.

6

Let us emphasize the presence of given datain the optimal control problem, that we de-note by f . Then the problem is written as

min{F (u, v) : u ∈ A(v, f)

}.

For instance, the example above describesthe vertical displacement of a membrane fixedon ∂Ω under the action of the exterior loadf and of an extra load v that we may add.

Consider the case when the data f are onlyknown only up to some degree of uncer-tainty; nevertheless, we still want to find anoptimal solution in some sense.

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A possibility (we do not deal with) is to as-sume that f is known with a probability P ;in this case the average cost functional canbe optimized and we are in the interestingframework of stochastic optimization.

We want on the contrary consider the worstcase for f ; more precisely, we assume thatthe data can be perturbed as f + g with‖g‖Lp ≤ δ and we optimize the worst casecost

Fwc(u, v) = sup‖g‖Lp≤δ

{F (u, v) : u ∈ A(v, f+g)

}.

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We are interested in the case when the con-trol is the domain Ω; we are then in theframework of shape optimization problems.Other cases of worst case optimization prob-lems are considered in Allaire-Dapogny (M3AS2014). We want to show the existence of anoptimal domain for

min{F(Ω) : Ω ⊂ D, |Ω| ≤ m

}where D is a prescribed bounded subset ofRd and F is a worst-case functional given by

F(Ω) = sup{F (Ω, f + g) : ‖g‖Lp(D) ≤ δ

},

being F a given shape functional.

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We start by considering as F the energy func-tional

E(Ω, f) = infu∈H10(Ω)

∫Ω

(1

2|∇u|2 − fu

)dx.

The worst-case functional F is:F(Ω) = sup

‖g‖Lp(D)≤δE(Ω, f + g)

= infu∈H10(Ω)

∫D

(1

2|∇u|2 − fu

)dx+ δ‖u‖

Lp′(D)

and the worst-case shape optimization prob-lem becomes

min{F(Ω) : Ω ⊂ D, |Ω| ≤ m

}.

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Before stating the existence of an optimaldomain for the problem above let us recall avery general existence theorem based on theDal Maso-Mosco (AMO 1987) characteriza-tion of relaxed Dirichlet problems.

Theorem [Buttazzo-Dal Maso (ARMA 1993)]Let F (Ω) be such that:• F is γ-lower semicontinuous;• F is decreasing for set inclusion.Then the shape optimization problem

min{F (Ω) : |Ω| ≤ m

}admits a solution.

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The assumptions above are verified in the

worst-case shape optimization problem, and

so we have that for every δ and m there exists

an optimal domain Ωδ,m solving

min{Fδ(Ω) : Ω ⊂ D, |Ω| ≤ m

}where

Fδ(Ω) = infu∈H10(Ω)

∫D

(1

2|∇u|2−fu

)dx+δ‖u‖

Lp′(D)

.

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Radial case

We consider the case of a right-hand side

f of radial type; more precisely, we assume

f = f(|x|) with f(r) decreasing.

Theorem If D is large enough (to contain

a ball of measure m) the optimal domain

Ωδ,m is a ball of measure m (centered at the

origin).

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Uncertainty only in the state equation

We consider the case of a shape optimal con-trol problem

max|Ω|≤m

∫Ωh(x)uΩ dx

where h ≥ 0 and uΩ is the solution of

−∆u = f in Ω, u ∈ H10(Ω).We assume that h is perfectly known, whilef is uncertain.

Example: best shape for the average tem-perature under partially known heat sources.

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Writing f + g instead of f and taking the

worst-case situation, denoting by R the re-

solvent operator of −∆, we have the worstcase functional

Fδ(Ω) = sup‖g‖p≤δ

−∫DhR(f + g) dx

= sup‖g‖p≤δ

−∫D

(fR(h) + gR(h)

)dx

=∫D−f(x)wΩ dx+ δ‖wΩ‖Lp′(D)

where

−∆wΩ = h in Ω, wΩ ∈ H10(Ω).

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Notice that Fδ is still γ-lower semicontinu-ous but it is not monotone decreasing. Then

the Buttazzo-Dal Maso theorem for the ex-

istence of an optimal shape cannot be used.

Nevertheless, the following result holds.

Theorem Assume:

• h ≥ 0 and h ∈ Ld(D);• f ∈ Lp(D) with p ≥ 2d/(d+ 2);• f ≥ c > 0 on D.Then, there exists δ̄ > 0 such that for every

0 < δ ≤ δ̄, there exists a solution Ωδ to theworst-case shape optimal control problem.

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A numerical example

D = [0,1]× [0,1], p = 2, δ = 0.25

f =

1 on [0, 12]× [0,1]2 on [12,1]× [0,1]It is numerically convenient to simulate a do-

main Ω by a potential V (x) taking the value

0 in Ω and +∞ outside. The measure |Ω| isthen simulated through the quantity∫

De−αV (x) dx with α small.

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More precisely this approximation has to be

stated in terms on Γ-convergence, proved in

[BGRV, JEP 2014].

The simulation has been made by J.C. Bel-

lido using:

• FreeFEM++• the Method of Moving Asymptotes (a kindof gradient method widely used for Topology

and Structural Optimization problems)

• a mesh of 50× 50 elements.

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0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0

200

400

600

800

Optimal potential for the unperturbed case

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0

200

400

600

800

Results for the perturbed case with δ = 0.25

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0

2

4

6

·10−2

Optimal state for the unperturbed case

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0

1

2

3

4

·10−2

Optimal state for the perturbed case with δ = 0.25

0

0.20.4

0.60.8

1 00.2

0.40.6

0.8

10

500

1,000

0

200

400

600

800

Optimal potential (3D view) for the unperturbed case

0

0.20.4

0.60.8

1 00.2

0.40.6

0.8

10

500

0

200

400

600

800

Optimal potential (3D view) for the case with δ = 0.25

In progress: It would be very interesting to

make an asymptotic analysis (often called Γ

development) of the sets Ωδ for δ small.

The expected result is that Ωδ is (asymptot-

ically) equal to Ω with a boundary layer Σδof local thickness δh(σ)

Σδ ={x = tν(σ), σ ∈ ∂Ω, −δh−(σ) < t < δh+(σ)

}for a suitable function h to be characterized,

with∫∂Ω h dσ = 0.

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