Semi-Lagrangian Methods for Monge-Amp` ere Equations Max Jensen University of Sussex joint work with Xiaobing Feng (University of Tennessee) http://arxiv.org/abs/1602.04758 Slides available at https://goo.gl/v42Zv8 1
Part 1 Monge-Amp` ere as Bellman problem Slides available at https://goo.gl/v42Zv8 2
excarvation pile of soil One motivation: optimal transport ◮ Consider soil ◮ f + ( x ) = height of pile at x , x ◮ f − ( x ) = depth of hole at x , y � � f + = f − < ∞ . with ◮ We want to move the soil into holes: transport plan x �→ s ( x ). ◮ Then f + = f − ( s ) det( Ds ) ( D gradient) ◮ Cost of transport related quadratically to distance: ◮ optimal transport plan s ∗ has a potential u ∗ s ∗ = ∂ u ∗ ◮ For smooth u we have the Monge-Amp` ere equation f + = f − ( Du ) det( D 2 u ) ( D 2 Hessian) ◮ Other motivation: prescribed Gaussian curvature. Slides available at https://goo.gl/v42Zv8 3
excarvation pile of soil One motivation: optimal transport ◮ Consider soil ◮ f + ( x ) = height of pile at x , x ◮ f − ( x ) = depth of hole at x , y � � f + = f − < ∞ . with ◮ We want to move the soil into holes: transport plan x �→ s ( x ). ◮ Then f + = f − ( s ) det( Ds ) ( D gradient) ◮ Cost of transport related quadratically to distance: ◮ optimal transport plan s ∗ has a potential u ∗ s ∗ = ∂ u ∗ ◮ For smooth u we have the Monge-Amp` ere equation f + = f − ( Du ) det( D 2 u ) ( D 2 Hessian) ◮ Other motivation: prescribed Gaussian curvature. Slides available at https://goo.gl/v42Zv8 3
Simple Monge-Amp` ere equation ◮ The simple Monge-Amp` ere equation is one of the model problems for the development of numerical methods. ◮ It combines the following properties: ◮ fully nonlinear ◮ degenerate elliptic ◮ anisotropic The simple Monge-Amp` ere equation Let Ω ⊂ R n be strictly convex. Let u | ∂ Ω = g and, with f ( x ) ≥ 0, � f � n − det D 2 u = 0 . n Slides available at https://goo.gl/v42Zv8 4
Ellipticity and convexity Definition (elliptic) The operator F in a differential equation F ( x , u ( x ) , Du ( x ) , D 2 u ( x )) = 0 is called (degenerate) elliptic if, x ∈ Ω , r ∈ R , p ∈ R n , F ( x , r , p , X ) ≤ F ( x , r , p , Y ) whenever Y − X is pos. semidefinite where X , Y are symmetric matrices. ◮ Set � f ( x ) � n F ( x , r , p , X ) = − det X . n Then F satisfies the ellipticity condition only for positive semi-definite X , Y . ◮ So a theory based on ellipticity is restricted to convex functions. Slides available at https://goo.gl/v42Zv8 5
Ellipticity and convexity Definition (elliptic) The operator F in a differential equation F ( x , u ( x ) , Du ( x ) , D 2 u ( x )) = 0 is called (degenerate) elliptic if, x ∈ Ω , r ∈ R , p ∈ R n , F ( x , r , p , X ) ≤ F ( x , r , p , Y ) whenever Y − X is pos. semidefinite where X , Y are symmetric matrices. ◮ Set � f ( x ) � n F ( x , r , p , X ) = − det X . n Then F satisfies the ellipticity condition only for positive semi-definite X , Y . ◮ So a theory based on ellipticity is restricted to convex functions. Slides available at https://goo.gl/v42Zv8 5
Viscosity solution of the Monge-Amp` ere equation Definition ◮ Let u be convex and f ∈ C (Ω), f ≥ 0. ◮ u is viscosity subsolution (supersolution) of � n − det D 2 u = 0 � f n if for all convex ψ ∈ C 2 (Ω) and x 0 ∈ Ω such that u − ψ has a local maximum (minimum) at x 0 then � n − det D 2 ψ ≤ 0 � n − det D 2 ψ ≥ 0 � f �� f � . n n ◮ If u is viscosity sub- and supersolution it is a viscosity solution. ◮ If Ω ⊂ R n is open bounded and strictly convex and f ≥ 0 continuous, then there exists a unique viscosity solution of the simple Monge-Amp` ere equation. Slides available at https://goo.gl/v42Zv8 6
Problem ◮ Restrict the approximation space to convex functions? Problem ◮ Convex finite element functions not dense in set of convex functions. ◮ To obtain convergence one needs to admit non-convex functions into the approximation space. ◮ What about ◮ ellipticity, ◮ uniqueness? ◮ Other methods, such as finite I difference schemes, have the same nterpolant of ( x 1 + x 2 ) 2 density issue. (Aguilera, Morin; 2009) Slides available at https://goo.gl/v42Zv8 7
Introduction to Hamilton-Jacobi-Bellman ◮ Generally (elliptic) Bellman equations of the type Hu = 0 where ( − A α : D 2 w + b α · ∇ w + c α w − r α ) . Hw := sup � �� � α ∈ A linear, 2nd order non-divergence form ◮ Hamilton-Jacobi-Bellman equations describe the optimal expected cost in a controlled stochastic process. ◮ Well-established field with own range of numerical tools. Slides available at https://goo.gl/v42Zv8 8
Classical equivalence ◮ We define the Hamiltonian √ � � n H : S → R , A �→ sup − B : A + det Bf , B ∈ S 1 where ◮ S is the set of self-adjoint linear mappings R n → R n , ◮ S 1 := { A ∈ S : trace A = 1 and A ≥ 0 } . ◮ We define the Monge-Amp` ere mapping � f � n M : S → R , A �→ − det( A ) . n Krylov (1987) H A = 0 ⇔ M A = 0 and A ≥ 0 . Slides available at https://goo.gl/v42Zv8 9
Uniqueness — on the whole space! ◮ Contour lines for diagonal A ∈ R 2 × 2 and f = 1. 2 2 � 1 � 1 0 � 2 2 � 1 1 1 1 1 0 0 0 0 b b 2 0 1 1 � 1 � 1 2 � 2 2 � 1 � 2 � 2 � 2 � 1 0 1 2 � 2 � 1 0 1 2 a a �� a 0 �� �� a 0 �� M H 0 b 0 b ◮ From the numerical point now uniqueness does not need to be enforced by a constraint on the approximation space, because the HJB operator only vanishes in first quadrant. Slides available at https://goo.gl/v42Zv8 10
Viscosity solutions of HJB Viscosity Solution of HJB ◮ v is a subsolution of HJB if for all smooth ψ H ( D 2 ψ ( x )) ≤ 0 at every x 0 which maximises v − ψ with v ( x ) = ψ ( x ). ◮ Supersolution similar. Subsolution + supersolution =: solution. ◮ Definition similar to that for Monge-Amp` ere, but for HJB there is no convexity condition as ellipticity holds on the whole space. ◮ We complement the differential equation with Dirichlet conditions in the pointwise sense: u ( x ) = g ( x ) , x ∈ ∂ Ω . Slides available at https://goo.gl/v42Zv8 11
Equivalence in viscosity sense ◮ From the general theory we have existence of a viscosity solution of the Monge-Amp` ere problem. Theorem (Feng, J (2016)) ◮ Let Ω be strictly convex and f ≥ 0 . Then { u ∈ C (Ω) convex and solves Mu = 0 in viscosity sense } = { u ∈ C (Ω) solves Hu = 0 in viscosity sense } Observe, regarding the proof: ◮ Monge-Amp` ere has a smaller set of test functions (convexity). ◮ Monge-Amp` ere only allows non-negative forcing term, otherwise difference u − ψ could be moved into forcing term and classical equivalence can be used. Slides available at https://goo.gl/v42Zv8 12
Comparison principle ◮ A comparison principle gives uniqueness. Theorem (Feng, J (2016)) ◮ Let u be a subsolution and v be a supersolution of the Bellman problem. ◮ Then u ≤ v on Ω if u ≤ v on ∂ Ω . ◮ Notice that comparison is based on pointwise and not on viscosity boundary conditions. Slides available at https://goo.gl/v42Zv8 13
Summary of Part 1 ◮ Simple Monge-Amp` ere equation � f � n − det D 2 u = 0 n Conditional uniqueness: comparison principle/well-posedness only for convex functions. ◮ Numerical difficulties: e.g. set of convex FE functions not dense in set of convex functions. ◮ We propose to use for numerics an HJB formulation √ � � − B : D 2 u + n sup det Bf = 0 . B ∈ S 1 ◮ The HJB formulation has the same viscosity solution as the Monge-Amp` ere problem, but unconditional comparison principle and well-posedness. Slides available at https://goo.gl/v42Zv8 14
Part 2 Semi-Lagrangian discretisation Slides available at https://goo.gl/v42Zv8 15
Theorem (Barles-Souganidis (1991)) If a numerical method is ◮ monotone ◮ pointwise consistent on Ω in the viscosity sense ◮ stable in L ∞ and it is ensured that numerical solutions exist, then the numerical solutions converge in L ∞ loc to the viscosity solution of the (elliptic) PDE. Theorem (Wasow-Motzkin (1952); Kocan (1995)) ◮ Consider a second-order elliptic PDE with ‘irrational angle of anisotropy‘ (even constant coefficient, linear PDE). ◮ Consider essentially a finite difference grid. ◮ If a numerical scheme is ◮ monotone, ◮ pointwise consistent, then its stencil grows unboundedly as h → 0 . Slides available at https://goo.gl/v42Zv8 16
Theorem (Barles-Souganidis (1991)) If a numerical method is ◮ monotone ◮ pointwise consistent on Ω in the viscosity sense ◮ stable in L ∞ and it is ensured that numerical solutions exist, then the numerical solutions converge in L ∞ loc to the viscosity solution of the (elliptic) PDE. Theorem (Wasow-Motzkin (1952); Kocan (1995)) ◮ Consider a second-order elliptic PDE with ‘irrational angle of anisotropy‘ (even constant coefficient, linear PDE). ◮ Consider essentially a finite difference grid. ◮ If a numerical scheme is ◮ monotone, ◮ pointwise consistent, then its stencil grows unboundedly as h → 0 . Slides available at https://goo.gl/v42Zv8 16
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