Galerkin methods for the Monge–Ampère problem with transport boundary conditions Omar Lakkis Computer Science and Technology — Free University of Bozen-Bolzano IT and Mathematics — University of Sussex — Brighton, England GB with the precious contribution of Tristan Pryer University of Reading, GB and Ellya Kawecki University of Oxford, GB talk given 24 November 2016 at Numerical methods for Hamilton–Jacobi equations in optimal control and related fields Special Semester on Computational Methods in Science and Engineering – Workshop 3 Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 1 / 53
Outline From Monge–Kantorovitch to Monge–Ampère 1 PDE Background: fully nonlinear elliptic PDE’s 2 History and competing approaches 3 Finite differences Finite elements Iterative nonlinear solvers 4 Fixed point Newton Hessian recovery A non-variational FEM (NVFEM) solver 5 Convergence 6 Experiments 7 Transport (second) boundary conditions 8 Conclusions 9 10 References Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 2 / 53
Monge’s mass transportation problem ∫ ∫ Consider ρ , σ > 0 , σ , find t field such that ρ = for any Lebesgue-measurable subset A of � d d ∈ � � � (3.1) ∀ A (Borel) ⊆ Ω . ρ ( x ) d x = σ ( y ) d y ρ ( x ) d x ρ A t ( A ) � � t (3.2) σ ( y ) d y = σ ( t ( x )) | detD t ( x ) | d x . t ( A ) A σ This works for any positive measure densities ρ , σ . If ρ , σ are functions, then by du Bois-Reymond’s lemma σ ( y ) d y ∀ x ∈ � d . (3.3) σ ( t ( x )) | detD t ( x ) | = ρ ( x ) Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 3 / 53
Monge’s mass transportation problem Optimise the cost Find t such that (4.1) ( detD t ) σ ◦ t = ρ which minimises the transportation cost functional � | t ( x ) − x | 2 ρ ( x ) d x . (4.2) Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 4 / 53
From Monge to Monge–Ampère Following Brenier, (1991) Under convexity and regularity assumptions, the Monge-Ampere equation detD 2 u ( x ) = f ( x , ∇ u ( x )) coupled to the second boundary condition second boundary condition (5.1) ∇ u ( Ω ) = Υ , and the right-hand side ρ ( x ) (5.2) f ( x , ∇ u ( x )) = g ( ∇ u ( x )) provides a solution to the Monge problem (5.3) t = ∇ u . Full analysis with viscosity solutions was performed by Urbas, (1997). For regularity and related issues see L. A. Caffarelli, (1990), L. Caffarelli, Nirenberg and Spruck, (1985) and Luis A. Caffarelli, (1990) . Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 5 / 53
Fully nonlinear elliptic PDE’s Definition and notation Given a real-valued nonlinear function F of matrices F : Sym ( � d ) → � . (FNFun) Consider the equation � [ u ] : = F ( D 2 u ) − f = 0 and u | ∂ Ω = 0 (FNE) Conditional ellipticity condition, i.e., � � � � � N ξ � ≤ F ( M + N ) − F ( M ) ≤ Λ sup � N ξ � λ ( M ) sup | ξ | = 1 | ξ | = 1 (NL-Ellip) ∀ M ∈ � ⊆ Sym ( � d ) , N ∈ Sym ( � d ) . for some ellipticity domain � and “constants” λ ( · ) , Λ > 0 . Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 6 / 53
Fully nonlinear elliptic PDE’s The ellipticity fauna � [ u ] : = F ( D 2 u ) − f = 0 Conditionally elliptic ∃ � ⊆ Sym ( � d ) , λ ( · ) , Λ > 0 : � � � � � N ξ � ≤ F ( M + N ) − F ( M ) ≤ Λ sup � N ξ � λ ( M ) sup | ξ | = 1 | ξ | = 1 ∀ M ∈ � ⊆ Sym ( � d ) , N ∈ Sym ( � d ) . Unconditionally elliptic if � = Sym ( � d ) . Uniformly elliptic inf λ > 0 . Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 7 / 53
Characterisation of the ellipticity condition in the smooth case Ellipticity condition, i.e., � � � � � ≤ F ( M + N ) − F ( M ) ≤ Λ sup � N ξ � N ξ � λ ( M ) sup | ξ | = 1 | ξ | = 1 (NL-Ellip) ∀ M ∈ � ⊆ Sym ( � d ) , N ∈ Sym ( � d ) . for some ellipticity “constants” λ ( · ) , Λ > 0 . If F is differentiable then (NL-Ellip) is satisfied if and only if for each M ∈ � there exists µ > 0 such that � � � 2 ξ ⊺ F ′ ( M ) ξ ≥ µ � ξ ∀ ξ ∈ � d . (8.1) Furthermore � = Sym ( � d ) and µ is independent of M if and only if F is uniformly elliptic. Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 8 / 53
The Monge–Ampère–Dirichlet problem A classical fully nonlinear elliptic PDE Boundary value problem detD 2 u = f in Ω (MAD) on ∂ Ω u = 0 admits a unique solution in the cone of convex functions when f > 0 . [Luis A. Caffarelli and Cabré, 1995] Derivative of nonlinear function F ( X ) = det X yields F ′ ( X ) = Cof X . Problem elliptic if and only if � � ξ ⊺ CofD 2 u ξ ≥ λ � 2 � ξ ∀ ξ ∈ � d for some λ > 0 . Conotonic constraint Restriction on unknown functions u : they must be globally either convex or concave ( conotonic ). Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 9 / 53
A simple fully nonlinear elliptic PDE Consider problem � [ u ] : = sin ( ∆ u ) + 2 ∆ u − f = 0 in Ω , u = 0 on ∂Ω . Differentiating, we see that D � [ v ] w = ( cos ( ∆ v ) + 2 ) I :D 2 w = ( cos ( ∆ v ) + 2 ) ∆ w . Hence problem uniformly elliptic. Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 10 / 53
A Krylov-type cubic elliptic PDE The problem is for d = 2 � [ u ] : = ( ∂ 11 u ) 3 + ( ∂ 22 u ) 3 + ∂ 11 u + ∂ 22 u − f = 0 in Ω (Krylov) on ∂ Ω . u = 0 Problem is uniformly elliptic since its differentiation gives: � � 3 x 2 22 + 1 0 F ′ ( X ) = . 3 x 2 0 11 + 1 Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 11 / 53
Pucci’s equation Consider F : Sym ( � d ) → � to be the extremal function � d α i λ i ( N ) where λ i ( N ) eigenvalues of N F ( N ) = (Pucci) i = 1 for some given α 1 ,..., α d ∈ � . Special case when d = 2 , α 1 = α ≥ 1 and α 2 = 1 yields equation � � 1 / 2 . ( � 2 Pucci) ( ∆ u ) 2 − 4detD 2 u 0 = ( α + 1 ) ∆ u + ( α − 1 ) The problem is unconditionally elliptic. Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 12 / 53
Classes of fully nonlinear equations A rough guide See Luis A. Caffarelli and Cabré, 1995 for a more systematic classification. Isaacs form: inf β sup α L αβ u = 0 . Bellman type: Isaacs with only one β ( ⇔ no inf). Related to Hamilton–Jacobi–Bellman, stochastic control and differential game theory. Isaacs form is very general: “non-algebraic” and harder to treat numerically. We don’t, yet. [M. Jensen and Iain Smears, 2012; Lio and Ley, 2010, e.g.] . Hessian invariants (algebraic): Monge–Ampère, Pucci, Laplace (!). Subdivided into unconeditionally elliptic (Pucci, Laplace) and coneditionally elliptic (Monge–Ampère). Other algebraic FNE’s (Krylov, algebraic nonlinearities, etc.) Aronson equations and infinite-harmonic functions, nicely reviewed in Barron, L. C. Evans and R. Jensen, 2008. (These aren’t proper FNE’s, as they are quasilinear, nevertheless, Hessian recovery applies well.) Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 13 / 53
Finite difference approaches 1 Earliest known provided approximations of the Monge–Ampère (and other equations) by Oliker and Prussner, 1988. 2 Kuo and Trudinger, 1992 gave mostly theoretical work introduced the concept of wide stencils and proving convergence for wide enough stencils. 3 Benamou and Brenier, 2000 proposed an approach based on the Brenier-solution concept related to fluid-dynamics and mass-trasportation. 4 Oberman, 2008 introduced more practically effective work working out the details, proiding a bound on the wide stencil’s width. See also Froese, 2011 and Benamou, Froese and Oberman, 2012 for second boundary conditions. Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 14 / 53
Galerkin (mainly finite element) methods Dean and Glowinski, 2006 (and earlier work) introduced a FE least square method to solve Monge–Ampère equation. Böhmer, 2010 (and earlier papers) advocates (mostly theoretically, proving convergence) the use of C 1 /spline finite elements to directly compute the Hessian. Practical schemes have been constructed recently by Davydov and Saeed, 2013. Feng and M. Neilan, 2009 introduce the vanishing moment method a fourth order semilinear approximation: ε ∆ 2 u + F [ u ] = 0 and take ε → 0 . (Similar to early vanishing viscosity methods for first order PDE’s.) Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 15 / 53
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