Shape optimization under uncertainty Rahel Br ugger, Roberto Croce, - PowerPoint PPT Presentation

Shape optimization under uncertainty Rahel Br¨ ugger, Roberto Croce, Marc Dambrine, Charles Dapogny, Helmut Harbrecht, Michael Multerer, and Benedicte Puig Helmut Harbrecht Department of Mathematics and Computer Science University of Basel (Switzerland)

Overview I Shape optimization in case of geometric uncertainty I Shape optimization in case of random diffusion I Shape optimization in case of random right-hand sides Helmut Harbrecht

Free boundary problems Problem. Seek the free boundary Γ such that u satisfies � ∆ u = f in D u = g on Σ Σ Γ D u = 0 , � ∂ u ∂ n = h on Γ I Growth of anodes. f ⌘ 0 , g ⌘ 1 , h ⌘ const Bernoulli’s free boundary problem I Electromagnetic shaping. Exterior boundary value problem, uniqueness ensured by volume constraint. Different formulations as shape optimization problem. 9 Z k ∇ v k 2 � 2 fv + h 2 � J 1 ( D ) = d x ! inf > > > D > > 8 Z � ∆ v = f � ∆ w = f > in D D k ∇ ( v � w ) k 2 d x ! inf > J 2 ( D ) = > > > > on Σ = < v = g w = g where ◆ 2 ✓ ∂ v � ∂ w Z J 3 ( D ) = ∂ n + h d x ! inf on Γ v = 0 ∂ n = h > > > > > : Γ > > > Z Γ w 2 d x ! inf > J 4 ( D ) = > > ; Helmut Harbrecht

Free boundary problems Problem. Seek the free boundary Γ such that u satisfies � ∆ u = f in D u = g on Σ Σ Γ D u = 0 , � ∂ u ∂ n = h on Γ I Growth of anodes. f ⌘ 0 , g ⌘ 1 , h ⌘ const Bernoulli’s free boundary problem I Electromagnetic shaping. Exterior boundary value problem, uniqueness ensured by volume constraint. Different formulations as shape optimization problem. 9 Z k ∇ v k 2 � 2 fv + h 2 � J 1 ( D ) = d x ! inf > > > D > > 8 Z � ∆ v = f � ∆ w = f > in D D k ∇ ( v � w ) k 2 d x ! inf > J 2 ( D ) = > > > > on Σ = < v = g w = g where ◆ 2 ✓ ∂ v � ∂ w Z J 3 ( D ) = ∂ n + h d x ! inf on Γ v = 0 ∂ n = h > > > > > : Γ > > > Z Γ w 2 d x ! inf > J 4 ( D ) = > > ; Helmut Harbrecht

Free boundary problem with geometric uncertainty Problem. Seek the free boundary Γ ( ω ) such that u ( ω ) satisfies ∆ u ( ω ) = 0 in D ( ω ) u ( ω ) = 1 on Σ ( ω ) Σ Γ D u ( ω ) = 0 , � ∂ u ∂ n ( ω ) = h on Γ ( ω ) for all ω 2 Ω . The questions to be addressed in the following are I How to model the random domain D ( ω ) ? Is the problem well-posed in the sense of D ( ω ) being almost surely well-defined? I Since it is a free boundary problem, we are looking for a free boundary. I Indeed, we are looking for the statistics of the domain itself. But how to define the expectation of a random domain? I How to compute the solution to the random free boundary problem numerically? Helmut Harbrecht

Statistical quantities I Expectation or mean. Z E [ v ]( x ) : = Ω v ( x , ω ) d P ( ω ) I Correlation. Z Ω v ( x , ω ) v ( y , ω ) d P ( ω ) = E [ v ( x ) v ( y )] Cor [ v ]( x , y ) : = I Covariance. Z � �� � v ( x , ω ) � E [ v ]( x ) v ( y , ω ) � E [ v ]( y ) d P ( ω ) Cov [ v ]( x , y ) : = Ω = Cor [ v ]( x , y ) � E [ v ]( x ) E [ v ]( y ) I Variance. Z � 2 d P ( ω ) � V [ v ]( x ) : = v ( x , ω ) � E [ v ]( x ) Ω � x = y � E [ v ] 2 ( x ) = Cov [ v ]( x , y ) � � = Cor [ v ]( x , y ) � x = y I k -th moment. Z M [ v ]( x 1 , x 2 ,..., x k ) : = Ω v ( x 1 , ω ) v ( x 2 , ω ) ··· v ( x k , ω ) d P ( ω ) Helmut Harbrecht

Existence and uniqueness of solutions Remarks. I The solution Γ to the free boundary problem exists if h > 0 is sufficiently large. I If the interior boundary Σ is convex, then the solution is unique. I If the interior boundary Σ is not convex, multiple solutions might exist. I In case of a starshaped boundary Σ , the solution is unique and also starshaped. Parametrization. Assume that Σ ( ω ) is P -almost surely starlike. Then, we can parametrize x = σ ( φ , ω ) 2 R 2 : σ ( φ , ω ) = q ( φ , ω ) e r ( φ ) , φ 2 [ 0 , 2 π ] � Σ ( ω ) = , x = γ ( φ , ω ) 2 R 2 : γ ( φ , ω ) = r ( φ , ω ) e r ( φ ) , φ 2 [ 0 , 2 π ] � Γ ( ω ) = . Theorem (H/Peters [2015]). Assume that q ( φ , ω ) satisfies 0 < r  q ( φ , ω )  R for all φ 2 [ 0 , 2 π ] and P -almost every ω 2 Ω . Then, there exists a unique free boundary Γ ( ω ) , for almost every ω 2 Ω . Espe- cially, with some constant R > R , the radial function r ( φ , ω ) of the associated free boundary satisfies q ( φ , ω ) < r ( φ , ω )  R for all φ 2 [ 0 , 2 π ] and P -almost every ω 2 Ω . Helmut Harbrecht

Expectation and variance Definition (Parametrization based expectation). The parametrization based ex- pectation E P [ D ] of the boundaries Σ ( ω ) and Γ ( ω ) is given by x 2 R 2 : x = E [ q ( φ , · )] e r ( φ ) , φ 2 [ 0 , 2 π ] � E P [ Σ ] = , x 2 R 2 : x = E [ r ( φ , · )] e r ( φ ) , φ 2 [ 0 , 2 π ] � E P [ Γ ] = . Remark. The expected domain E P [ D ] is thus given by x = ( ρ , φ ) 2 R 2 : E [ q ( φ , · )]  ρ  E [ r ( φ , · )] � E P [ D ] = . This is also called the radius-vector expectation . Theorem (H/Peters [2015]). The variance of the domain D ( ω ) in the radial direction is given via the variances of its boundaries parameterizations in accordance with x 2 R 2 : x = V [ q ( φ , · )] e r ( φ ) , φ 2 [ 0 , 2 π ] � V P [ Σ ( ω )] = , x 2 R 2 : x = V [ r ( φ , · )] e r ( φ ) , φ 2 [ 0 , 2 π ] � V P [ Γ ( ω )] = . The parametrization based expectation depends on the particular parametrization! Helmut Harbrecht

Stochastic quadrature method I Random parametrization of the interior boundary. N for y = [ y 1 ,..., y N ] | 2 ⇤ : = [ � 1 / 2 , 1 / 2 ] N . ∑ q ( φ , y ) = E [ q ]( φ )+ q k ( φ ) y k k = 1 It then holds Z Z E [ q ]( φ ) = Ω q ( φ , ω ) d P ( ω ) = ⇤ q ( φ , y ) ρ ( y ) d y , Z � 2 = Z � 2 d P ( ω ) � � 2 ρ ( y ) d y � � 2 . � � � � V [ q ]( φ ) = q ( φ , ω ) E [ q ]( φ ) q ( φ , y ) E [ q ]( φ ) Ω ⇤ I Solution map. Let F : L ∞ � ! L ∞ � � � Ω ; C per ( 0 , 2 π ) Ω ; C per ( 0 , 2 π ) , q ( φ , ω ) 7! r ( φ , ω ) denote the solution map. Then, the expectation and the variance of r ( φ , ω ) are given by E [ r ]( φ ) = E [ F ( q )]( φ ) V [ r ]( φ ) = V [ F ( q )]( φ ) . and I (Quasi-) Monte Carlo quadrature. The high-dimensional integrals are approximated by means of a sampling method. Helmut Harbrecht

Numerical example p 10 2 ∑ � q ( φ , ω ) = q ( φ , ω )+ sin ( k φ ) Y 2 k � 1 ( ω )+ cos ( k φ ) Y 2 k ( ω ) k k = 1 0.4 0.35 0.3 0.25 Radius 0.2 E [ r ] 0.15 F ( E [ q ]) 0.1 E [ q ] std[ r ] 0.05 0 0 1 2 3 4 5 6 Polar angle Helmut Harbrecht

Vorob’ev expectation I Leading idea. Identify the random set D ( ω ) with its characteristic function ( 1 , if x 2 D ( ω ) , 1 D ( ω ) ( x ) = 0 , otherwise . This embeds the problem into the linear space L ∞ ( R 2 ) . I Coverage function. The average of characteristic func- tions is not a characteristic function anymore but belongs to the cone { q 2 L ∞ ( R 2 ) : 0  q  1 } . The limit object is the so-called coverage function � � p ( x ) = P x 2 D ( ω ) . Definition (Vorob’ev expectation). The Vorob’ev expectation E V [ D ] of D ( ω ) is defined as the set { x 2 R 2 : p ( x ) � µ } for µ 2 [ 0 , 1 ] which is determined from the condition Z L ( { x 2 R 2 : p ( x ) � λ } )  R 2 p ( x ) d x  L ( { x 2 R 2 : p ( x ) � µ } ) for all λ > µ . Helmut Harbrecht

Numerical example Helmut Harbrecht

Free boundary problem with random diffusion Problem. Seek the free boundary Γ ( ω ) such that u ( ω ) satisfies � � α ( ω ) ∇ u ( ω ) = 0 in D ( ω ) div u ( ω ) = 1 on Σ Σ Γ D u ( ω ) = 0 , � α ( ω ) ∂ u ∂ n ( ω ) = h on Γ ( ω ) for all ω 2 Ω , where 0 < α  α ( ω )  α < ∞ . � � ugger/Croce/H [2018]). For ω 2 Ω , the solution u ( ω ) , Γ ( ω ) Theorem (Br¨ is given by the shape optimization problem α ( ω ) k ∇ u ( ω ) k 2 + h 2 ⇢ � Z J ( D , ω ) = d x ! inf α ( ω ) D subject to � � α ( ω ) ∇ u ( ω ) = 0 in D div u ( ω ) = 1 on Σ u ( ω ) = 0 on Γ Helmut Harbrecht

Free boundary problem with random diffusion I We shall minimize α ( ω ) k ∇ u ( ω ) k 2 + h 2 ⇢ � Z Z ⇥ ⇤ J ( D , ω ) = d P ( ω ) d x ! min . E α ( ω ) Ω D I A minimizer exists since we have an energy type shape functional. I The shape gradient reads α ( ω ) k ∇ u ( ω ) k 2 + h 2 ⇢ � Z Z ⇥ ⇤ δ E J ( D , ω ) [ V ] = Γ h V , n i d P ( ω ) d σ . α ( ω ) Ω I Compute the Karhunen-Lo` eve expansion of the diffusion coefficient M ∑ α ( x , ω ) = E [ α ]( x )+ α k ( x ) Y k ( ω ) , k = 1 where the coefficient functions { α k ( x ) } k are elements of C 1 ( D ) and the random vari- ables { Y k ( ω ) } k are independently and uniformly distributed in [ � 1 / 2 , 1 / 2 ] yields a parametric problem on ⇤ = [ � 1 / 2 , 1 / 2 ] M I Use a quasi Monte-Carlo method to approximate the integral over Ω by an integral over over ⇤ . Helmut Harbrecht