Finite element approximation of the elliptic Isaacs equation Abner - PowerPoint PPT Presentation

Finite element approximation of the elliptic Isaacs equation Abner J. Salgado Department of Mathematics University of Tennessee RICAM November 24, 2016 Joint work with Wujun Zhang (Rutgers U.) Supported by NSF grant DMS-1418784

Outline Motivation Building blocks Approximation by an integro-differential operator The scheme Rate of convergence The solution scheme Conclusions

Where do elliptic equations come from? Diffusion and elliptic equations: • Particles diffuse in a medium Ω . • When reaching the boundary ∂ Ω , say at point y , the particle releases a certain amount of energy (payoff) ϕ ( y ) . • What is the average energy released when starting at point x ? • If the medium is isotropic, then this is given by � ∆ u = 0 in Ω u = ϕ on ∂ Ω .

Where do elliptic equations come from? Diffusion and elliptic equations: • If the medium is not isotropic, the diffusion process can be described by a matrix A : � A ( x ) : D 2 u ( x ) = 0 in Ω u = ϕ on ∂ Ω . • If λI � A � Λ I the equation is uniformly elliptic.

The HJB equation • Let β ∈ B be a set of design parameters. • We can choose the material to maximize payoff.  [ A β ( x ) : D 2 u ( x )] = 0 sup in Ω  β ∈B u = ϕ on ∂ Ω .  • Hamilton Jacobi Bellman equation.

The HJB equation  F [ x, D 2 u ] := sup [ A β ( x ) : D 2 u ( x ) − f β ( x )] = 0 in Ω  β ∈B u = ϕ on ∂ Ω .  • The operator F is convex. • Solutions are u ∈ C 2 ,s (Ω) , for some s ∈ (0 , 1) (Evans 1982; Krylov 1982). For this the convexity of F is essential.

The Isaacs equation • Let α ∈ A and β ∈ B be a set of design parameters. • Two players compete, the first one wants to maximize payoff, while the second minimize it.  [ A α,β ( x ) : D 2 u ( x )] = 0 α ∈A sup inf in Ω  β ∈B u = ϕ on ∂ Ω .  • Isaacs equation.

Differential games • Two players try to control the process d X s = ℓ ( X s , s, α, β ) d s + σ ( X s , s, α, β ) d W s , X 0 = x and get a certain payoff �� T � J ( x, t, α, β ) = E x,t f ( s, X s , α, β ) d s + g ( X T ) . t • The first wants to maximize it, while the second wants to minimize it α ∈A sup inf J ( x, t, α, β ) . β ∈B

Differential games • Define the value function J ( x, t, α, β ) = J ( x, t, α ⋆ , β ⋆ ) . u ( x, t ) = inf α ∈A sup β ∈B • Bellman/Isaacs optimality principle ( ⊕ some stochastic calculus) yields that � ∂ t u ( x, t ) + A α,β : D 2 u ( x, t ) + b α,β · ∇ u ( x, t ) − f α,β � α ∈A sup inf = 0 β ∈B with A α,β ( x, t ) = 1 2 σ ( x, t, α, β ) σ ( x, t, α, β ) ⊺ , b α,β ( x, t ) = ℓ ( x, t, α, β ) , f α,β ( x, t ) = f ( x, t, α, β ) . • Parabolic Isaacs equation!

Every equation is Isaacs Let F : Ω × S d → R be nondecreasing with respect to its second argument (elliptic). As shown by (Evans 1980), we have the following representation: �� 1 ∂F � F [ x, M ] = inf α ∈ S d sup ∂M ( x, (1 − t ) α + tβ ) : ( M − β ) d t + F ( x, α ) β ∈ S d 0 Set � 1 ∂F A α,β ( x ) = ∂M ( x, (1 − t ) α + tβ ) d t, 0 f α,β ( x ) = A α,β : α − F ( x, α ) , A = B = S d . Notice that A α,β ( x ) � 0 and we have � � A α,β ( x ) : M − f α,β ( x ) F [ x, M ] = inf α ∈A sup . β ∈B

The Isaacs equation � �  F [ x, D 2 u ] := inf A α,β ( x ) : D 2 u ( x ) − f α,β ( x ) α ∈A sup = 0 in Ω  β ∈B u = ϕ on ∂ Ω .  • The operator F is not convex. • What can we say about the regularity of the solution? • (Nirenberg 1953) showed that, if d = 2 , the solution to every elliptic equation is locally C 2 ,s for some s ∈ (0 , 1) .

The counterexample of Nadirashvili and friends • Let d = 5 . • Define 1 + 3 P 5 ( x ) = x 3 x 2 3 + x 2 4 − 2 x 2 5 − 2 x 2 � � 2 x 1 , 2 1 w ( x ) = | x | 1+ δ P 5 ( x ) , δ ∈ [0 , 1) • We have that w ∈ C 1 , 1 − δ ( ¯ B 1 ) \ C 2 ( B 1 ) . • There exists a uniformly elliptic Isaacs operator that depends only on M , it is Lipschitz and such that F [ D 2 w ] = 0 , in B 1 w = P 5 on ∂B 1 . • Solutions of the Isaacs equation (for d ≥ 5 ) are not C 2 . • Open question: What can we say for d = 3 , 4 ?

Outline Motivation Building blocks Approximation by an integro-differential operator The scheme Rate of convergence The solution scheme Conclusions

Construction of numerical schemes I Let F h [ u h ] = 0 be an approximation scheme for the elliptic problem F [ u ] = 0 . As shown in (Barles, Souganidis 1991), if • The scheme is monotone, that is: If u h − v h has a nonnegative maximum at z , then F h [ u h ]( z ) ≤ F h [ v h ]( z ) . • The scheme is consistent, i.e., if I h is the interpolation onto the mesh F h [ I h w ] ≈ F [ w ] whenever w is smooth. Then the scheme is convergent: u h → u as h → 0 .

Construction of numerical schemes II Question: How do we construct schemes with these properties? Finite Differences: • Kuo and Trudinger in a series of papers (1990-1992) devised a program to construct consistent and monotone finite difference schemes. Question: How do we obtain rates of convergence? From (Kuo, Trudinger 1992): So far, such estimates have eluded us and to the best of our knowledge they remain a challenging and important open problem.

Analysis of numerical schemes I Let F be elliptic and u ε be a smooth subsolution such that � u − u ε � L ∞ (Ω) ≈ ε κ 1 . • By consistency F h [ u ε ] ≈ h κ 2 , • By monotonicity � u ε − u h � L ∞ (Ω) ≈ h κ 2 , • Triangle inequality � u − u h � L ∞ (Ω) ≈ ε κ 1 + h κ 2 , and choosing ε we get a rate of convergence.

Analysis of numerical schemes II Question: How do we construct u ε ? If the operator F is convex • (Krylov 1997; 2000) “shake” the coefficients. • (Barles, Jakobsen 2002) “shake” the coefficients of the scheme. • (Barles, Jakobsen 2005) relate to a QVI. • ... What if F is not convex? • (Caffarelli, Souganidis 2008) Use the “sup” convolution � � u ( y ) − 1 u + 2 ε | x − y | 2 ε ( x ) = sup y ∈ ¯ Ω and very subtle properties of viscosity solutions to get a rate � u − u h � L ∞ (Ω) � h σ . The rate σ cannot be determined.

Outline Motivation Building blocks Approximation by an integro-differential operator The scheme Rate of convergence The solution scheme Conclusions

Approximation: construction • Introduced in (Caffarelli, Silvestre 2010). For Ω ⊂ R d consider � �  A α,β ( x ) : D 2 u ( x ) F [ u ]( x ) := inf α ∈A sup = f ( x ) in Ω  β ∈B u = 0 on ∂ Ω ,  with λI ≤ A α,β ≤ Λ I ∀ α ∈ A , ∀ β ∈ B . 2 I ≤ A α,β − λ Then λ 2 I and we can write �� � � F [ u ]( x ) = λ A α,β ( x ) − λ : D 2 u ( x ) 2 ∆ u + inf α ∈A sup 2 I . β ∈B

Approximation: construction Let ϕ be a radially symmetric, smooth function, supported in the R d | y | 2 ϕ ( y ) d y = d and � unit ball, such that d w ( x, y ) = w ( x + y ) − 2 w ( x ) + w ( x − y ) . For ε > 0 define 1 � � 1 � I α,β εM α,β ( x ) − 1 y [ w ] ( x ) = R d d w ( x, y ) ϕ d y. ε ε d +2 det M α,β ( x ) � 1 / 2 . A α,β ( x ) − λ with M α,β ( x ) = � 2 I We approximate F [ w ]( x ) by F ε [ w ]( x ) := λ � � I α,β 2 ∆ w + inf α ∈A sup [ w ] ( x ) . ε β ∈B

Approximation: construction And consider, instead, � F ε [ u ε ]( x ) = f ( x ) in Ω u ε = 0 in Ω ε \ Ω y ∈ R d : dist ( y, Ω) < Qε � � � where Ω ε = with Q = 2 /λ . This is a good idea because Lemma If w ∈ C 2 (¯ Ω ε ) then, as ε → 0 , we have � A α,β ( x ) − λ � I α,β : D 2 w ( x ) [ w ] ( x ) → 2 I ε

Approximation: properties Proposition (CS ‘10) Let A α,β ( x ) = A α,β , Ω be bounded and convex, f ∈ C 0 , 1 (¯ Ω) . Then there exists a unique function u ε ∈ C 2 ,s (Ω) that solves the approximate equation. Moreover, there is s ∈ (0 , 1] , such that � u ε � C 1 ,s (Ω) + � u ε � C 0 , 1 (¯ Ω) � � u ε � L ∞ (Ω) + � f � L ∞ (Ω) . In addition, there is a σ > 0 such that � u − u ε � L ∞ (Ω) � ε σ � f � C 0 , 1 (¯ Ω) . Corollary There is a s > 0 such that u ∈ C 1 ,s (Ω) ∩ C 0 , 1 (¯ Ω) . We will discretize the integro-differential equation!

Outline Motivation Building blocks Approximation by an integro-differential operator The scheme Rate of convergence The solution scheme Conclusions

Notation • We follow (Nochetto, Zhang 2014). • T h is a quasiuniform triangulation of Ω of size h . • N h are the interior and N ∂ h boundary nodes, respectively. • V h is a P 1 continuous finite element space, V 0 h = V h ∩ H 1 0 (Ω) . • { φ z } z ∈N h is the Lagrange nodal basis of V 0 h . • For w h ∈ V h and z ∈ N h the discrete Laplacian ∆ h is � − 1 � �� ∆ h w h ( z ) = − φ z ( x ) d x ∇ w h ( x ) ∇ φ z ( x ) d x. Ω Ω

Description of the scheme Define, for z ∈ N h , h [ w h ]( z ) := λ � � F ε I α,β 2 ∆ h w h ( z ) + inf α ∈A sup [ w h ] ( z ) . ε β ∈B We seek for u ε h ∈ V 0 h such that F ε h [ u ε h ]( z ) = f z ∀ z ∈ N h , where � − 1 � �� f z = φ z ( x ) d x f ( x ) φ z ( x ) d x. Ω Ω