A Narrow-Stencil Finite Difference Method for Hamilton-Jacobi-Bellman Equations

Transcription

1 A Narrow-Stencil Finite Difference Method for Hamilton-Jacobi-Bellman Equations Xiaobing Feng Department of Mathematics The University of Tennessee, Knoxville, U.S.A. Linz, November 23, 2016

2 Collaborators Tom Lewis, North Carolina Stefan Schnake, Tennessee The work to be presented here has been partially supported by NSF

3 Outline Motivation and Background A Narrow-Stencil Finite Difference Method High Order Extensions Numerical Experiments Conclusion

4 Outline Motivation and Background A Narrow-Stencil Finite Difference Method High Order Extensions Numerical Experiments Conclusion

5 We consider second order fully nonlinear PDEs F(D 2 u, Du, u, x) = 0 Two best known classes of equations: Monge-Ampére equation: det(d 2 u) = f HJB equations: inf ν V (L ν u f ν ) = 0, where L ν u := A ν (x) : D 2 u + b ν (x) u + c ν (x)u, ν V. Both equations arise from many applications such as differential geometry, optimal mass transfer, stochastic optimal control, mathematical finance etc. Remark: c ν 0 in several applications (e.g., stochastic optimal control, Bellman reformulation of Monge-Ampère equation).

6 Example: (Stochastic Optimal Control) Suppose a stochastic process x(τ) is governed by the stochastic differential equation dx(τ) = f (τ, x(τ), u(τ)) dt + σ (τ, x(τ), u(τ)) dw (τ), τ (t, T ] x(t) = x Ω R n, W : Wiener process u : control vector and let [ ] T J (t, x, u) = E t x L (τ, x(τ), u(τ)) dτ + g (x(t )). t Stochastic optimal control problem involves minimizing J (t, x, u) over all u U for each (t, x) (0, T ] Ω.

7 Bellman Principle Suppose u U such that and define the value function u argmin J (t, x, u), u U v (t, x) = J (t, x, u ). Then, v is the minimal cost achieved starting from the initial value x(t) = x, and u is the optimal control that attains the minimum.

8 Bellman Principle (Continued) Let Ω R n, T > 0, and U R m. The Bellman Principle says v is the solution of for with v t = F(D 2 v, v, v, x, t) in (0, T ] Ω, (1) F(D 2 v, v, v, x, t) = inf (L uv h u ), u U n n n L u v = ai,j u (t, x)v x i x j + bi u (t, x)v xi + c u (t, x) v i=1 j=1 i=1 A u := 1 2 σσt b u := f (t, x, u) c u := 0 h u := L (t, x, u)

9 Ellipticity Definition: Let F[u] := F (D 2 u, u, u, x) and A, B SL(n). (a) F is said to be uniformly elliptic if Λ > λ > 0 such that λtr(a B) F(A, p, r, x) F(B, p, r, x) Λtr(A B) A B. (b) F is said to be proper elliptic if F(A, p, v, x) F(B, p, w, x) A B; v, w R d, v w. (c) F is said to be degenerate elliptic if F(A, p, r, x) F(B, p, r, x) A B.

10 Viscosity solutions Definitions: Assume F is elliptic in a function class A B(Ω) (set of bounded functions), (i) u A is called a viscosity subsolution of F[u] = 0 if ϕ C 2, when u ϕ has a local maximum at x 0 then F (D 2 ϕ(x 0 ), Dϕ(x 0 ), u (x 0 ), x 0 ) 0 (ii) u A is called a viscosity supersolution of F[u] = 0 if ϕ C 2, when u ϕ has a local minimum at x 0 then F (D 2 ϕ(x 0 ), Dϕ(x 0 ), u (x 0 ), x 0 ) 0 (iii) u A is called a viscosity solution of F[u] = 0 if u is both a sub- and supersolution of F[u] = 0 where u (x) := lim sup u(x ) and u (x) := lim inf u(x ) are the x x x x upper and lower semi-continuous envelops of u

11 Barles-Souganidis Framework I For approximating viscosity solutions, we first recall the Barles-Souganidis framework. Theorem (Barles-Souganidis ( 91)) Suppose that the elliptic problem F[u](x) = 0 in Ω satisfies the comparison principle. Assume that the (approximation) operator S : R + Ω R B(Ω) R is consistent, monotone and stable (as well as admissible), then the solution u ρ of problem: S(ρ, x, u ρ (x), u ρ ) = 0 in Ω, converges locally uniformly to the unique viscosity of u.

12 Barles-Souganidis Framework II (i) Admissibility and Stability. For all ρ > 0, there exists a solution u ρ B(Ω) to the following problem: S(ρ, x, u ρ (x), u ρ ) = 0 in Ω. Moreover, there exists a ρ-independent constant C > 0 such that u ρ L (Ω) C. (ii) Monotonicity. For all x Ω, t R and ρ > 0 S(ρ, x, t, u) S(ρ, x, t, v) u, v B(Ω), u v. (iii) Consistency. For all x Ω and φ C (Ω) there hold lim sup ρ 0 y x ξ 0 lim inf ρ 0 y x ξ 0 S(ρ, y, φ(y) + ξ, φ + ξ) ρ S(ρ, y, φ(y) + ξ, φ + ξ) ρ F (D 2 φ(x), φ(x), φ(x), x), F (D 2 φ(x), φ(x), φ(x), x).

13 Wasow-Motzkin Theorem Theorem (Wasow-Motzkin ( 53)) Any monotone and consistent method has to be a wide stencil scheme. Theorem (Bonnans and Zidani ( 03)) If A ν in the HJB equation is not diagonally dominant, then wide stencils are required to preserve monotonicity. Remark: The difficulty is the directional resolution. Wider stencils are used to increase the resolution.

14 Remarks on Monotone Schemes and Wide-stencils Why monotonicity? For the numerical scheme to identify the correct viscosity solution, it needs to respect ordering in some sense. The monotonicity provides such an ordering, which is the best known one so far. A drawback" of monotonicity is that one must use wide stencils according to Wasow-Motzkin ( 53), which could be very problematic for anisotropic problems because very fine directional resolution is required, besides the difficulty from handling boundary conditions.

15 Remarks on Monotone Schemes and Wide-stencils Why monotonicity? For the numerical scheme to identify the correct viscosity solution, it needs to respect ordering in some sense. The monotonicity provides such an ordering, which is the best known one so far. A drawback" of monotonicity is that one must use wide stencils according to Wasow-Motzkin ( 53), which could be very problematic for anisotropic problems because very fine directional resolution is required, besides the difficulty from handling boundary conditions. Consequently, in order to avoid using wide stencils, one must relax (or abandon) the concept of monotonicity (in the sense of Barles and Souganidis).

16 Outline Motivation and Background A Narrow-Stencil Finite Difference Method High Order Extensions Numerical Experiments Conclusion

17 Goals: To construct finite difference methods (FDMs) whose solutions converge to viscosity solutions of the underlying fully nonlinear 2nd order PDE problems, especially, to go beyond the domain of Barles-Souganidis framework and to be more suitable for FDMs and DG methods. Remark: A few existing narrow-stencil" methods are Glowinski et al. ( 04-12): Mixed FE for MA eqns (H 2 solns). Brenner et al. ( 09-13): DG for MA eqns (classical solns). Jensen-Smears ( 12): Linear FE for isotropic HJB eqns. Smears-Süli ( 14): DG-FE for (Cordes-) HJB (H 2 solns). F.-Neilan ( 07-11): FE and DG based on the vanishing moment approach.

18 Finite Difference Operators Let {e j } d j=1 denote the canonical basis of Rd. Define δ + x k,h k v(x) v(x + h ke k ) v(x) h k, δ x k,h k v(x) v(x) v(x h ke k ) h k δ µν x k,h k v(x) δ ν x k,h k δ µ x k,h k v(x), δ µν x k,h k ;x l,h l v(x) δ ν x l,h l δ µ x k,h k v(x) Discrete Gradients: Two natural "sided" choices [ ] ± h k δ± x k,h k Discrete Hessians: Four natural "sided" choices [ D µν h ]k,l δµν x k,h k ;x l,h l, µ, ν {, +} Remark: Low-regularity can be resolved by using sided" gradient and Hessian approximations.

19 Ideas Used for 1st Order Hamilton-Jacobi Equations (Crandall and Lions, 84) FD schemes with the form Ĥ( h U α, + h U α, U α, x α ) = 0 converge to the viscosity solution of a Hamilton-Jacobi equation assuming Consistency: Ĥ(q, q, u, x) = H(q, u, x), Monotonicity: Ĥ(,, u, x). Ĥ is called a numerical Hamiltonian. Ĥ is a function of both h U α and + h U α. The monotonicity requirement is compatible with a discrete first derivative test.

20 Vanishing Viscosity and Numerical Viscosity H( u, u, x) = 0 ɛ u ɛ + H( u ɛ, u ɛ, x) = 0 (E. Tadmor, 97) Every convergent monotone finite difference scheme for HJ equations implicitly approximates the differential equation βh u + H( u, x) = 0 for sufficiently large and possibly nonlinear β > 0, where βh u is called a numerical viscosity. Note: If h k h, then 1 ( d + h U α h U α) = h h U α h δx 2 k,hu α. Lax-Friedrichs numerical Hamiltonian k=1 ( q Ĥ(q, q + + q + ), u, x) H, u, x b (q + q ) 2 }{{} Numerical Viscosity

21 FD Approximations of F(D 2 u, u, u, x) = 0 Since u xi x j may be discontinuous at x α, it needs be approximated from multiple directions.

22 FD Approximations of F(D 2 u, u, u, x) = 0 Since u xi x j may be discontinuous at x α, it needs be approximated from multiple directions. There are 4 possible FD approximations of u xi,x j (x α ): and u xi,x j (x α ) δ µ x i δ ν x j u(x α ) D 2 u(x α ) D µν h u(x α) µ, ν {+, }. µ, ν {+, }.

23 FD Approximations of F (D 2 u, u, u, x) = 0 Inspired by the above observations, we propose the following form of FD methods for F(D 2 u, u, u, x) = 0: F (D ++ h U α, D + h U α, D + h U α, D h U α, h U α, + h U α, U α, x α ) = 0 F is called a numerical operator, which needs to satisfy

24 FD Approximations of F (D 2 u, u, u, x) = 0 Inspired by the above observations, we propose the following form of FD methods for F(D 2 u, u, u, x) = 0: F (D ++ h U α, D + h U α, D + h U α, D h U α, h U α, + h U α, U α, x α ) = 0 F is called a numerical operator, which needs to satisfy Consistency: F(P, P, P, P, q, q, u, x) = F(P, q, u, x), Generalized Monotonicity: F(,,,,,,, x), uses the natural (partial) orderings for symmetric matrices, vectors, and scalars Solvability/Admissibility and Stability: h 0 > 0 and C 0 > 0, which is independent h, such that F [U α, x α ] = 0 has a (unique) solution U and U l (T h ) < C 0 for h < h 0.

25 Remarks on Numerical Operator F F depends on all four "sided" Hessians and both "sided" gradients. The generalized monotonicity requirement is an extension of the Crandall and Lions framework that enforces the elliptic structure of the PDE with respect to the mixed discrete Hessians D ± h. If F is not continuous, F may not be continuous either. In this case the consistency definition should be replaced by lim inf F (P 1, P 2, P 3, P 4, r, s, y) F (P, q, t, x), P k P,r q s t,y x lim sup F (P 1, P 2, P 3, P 4, r, s, y) F (P, q, t, x). P k P,r qn s t,y x Above consistency and monotonicity are different from Barles-Souganidis definitions.

26 Examples of Numerical Operators F I Lax-Friedrichs-like schemes: 1-D (F-Lewis-Kao, 14) p1 + p 2 + p ) 3 F 1 (p 1, p 2, p 3, x) := F(, x + α(p 1 2p 2 + p 3 ) 3 F 2 (p 1, p 2, p 3, x) := F(p 2, x) + α(p 1 2p 2 + p 3 ) p1 + p ) 3 F 3 (p 1, p 2, p 3, x) := F(, x + α(p 1 2p 2 + p 3 ) 2

27 Examples of Numerical Operators F II Godunov-like schemes: 1-D (F-Lewis-Kao, 14): F 4 (p 1, p 2, p 3, x) := ext F(p, x) p I(p 1,p 2,p 3 ) where I(p 1, p 2, p 3 ) := [p 1 p 2 p 3, p 1 p 2 p 3 ] and ext := p I(p 1,p 2,p 3 ) min p I(p 1,p 2,p 3 ) max p I(p 1,p 2,p 3 ) if p 2 max{p 1, p 3 }, if p 2 min{p 1, p 3 }, min if p 1 < p 2 < p 3, p 1 p p 2 min if p 3 < p 2 < p 1. p 3 p p 2

28 Examples of Numerical Operators F III Godunov-like schemes: 1-D (F-Lewis-Kao, 14) (continued): F 5 (p 1, p 2, p 3, x) := extr F(p, x) p I(p 1,p 2,p 3 ) where I(p 1, p 2, p 3 ) := [p 1 p 2 p 3, p 1 p 2 p 3 ] and extr := p I(p 1,p 2,p 3 ) min p I(p 1,p 2,p 3 ) max p I(p 1,p 2,p 3 ) if p 2 max{p 1, p 3 }, if p 2 min{p 1, p 3 }, max if p 1 < p 2 < p 3, p 2 p p 3 max if p 3 < p 2 < p 1. p 2 p p 1

29 Higher Dimensional Lax-Friedrichs-like Schemes Central Discrete Hessian: D 2 hu α 1 4 Central Discrete Gradient: ( D ++ h + D + h + D + h h U α 1 2 ( + h + h ) Uα Lax-Friedrichs-like Numerical Operator: ) F [U α, x α ] F (D 2 hu α, 2 h U α, U α, x α + D ) h Uα + A(U α, x α ) : ( D ++ h U α D + h U α D + h b(u α, x α ) ( + h U α h U ) α U α + D h U ) α

30 Numerical Moment I The discrete operator A(U α, x α ) : ( D ++ h U α D + is called a numerical moment. Observation: h U α D + h U α + D h U ) α ( δ + x k,h k δ + x l,h l δ + x k,h k δ x l,h l δ x k,h k δ + x l,h l + δ x k,h k δ x l,h l ) Uα = h k h l δ 2 x k,h k δ 2 x l,h l U α = h k h l δ 2 x l,h l δ 2 x k,h k U α, an O ( h 2 k + h2 l ) approximation of uxk x k x l x l (x α ) scaled by h k h l. 1 d d : ( D ++ h U α D + h U α D + h U α + D h U α) h 2 2 u(x α ),

31 Numerical Moment II Remark: Here the numerical moment plays a similar role for the 2rd order fully nonlinear PDEs to what the numerical viscosity does for the 1st order fully nonlinear PDEs. The introduction of numerical moments is very much consistent with the vanishing moment method (at the PDE level) proposed and analyzed by F. and Neilan ( 07-12): ε 2 u ε + F(D 2 u ε, u ε, u ε, x) = 0, with the special choice of the parameter ε = αh 2. Is there an analogue of E. Tadmor s result for 2rd order PDEs?

32 Convergence I Assumption: F is uniformly elliptic and Lipschitz continuous in the Hessian argument, and A and b are uniformly bounded for the family of linear operators defining the HJB problem with Dirichlet boundary data. Assume c 0. Theorem (F-Lewis, 16) The Lax-Friedrichs-like scheme is admissible, consistent, stable, and generalized-monotone. Main ingredients of proof: Consistency is trivial. Generalized monotonicity is also easy" by choosing the numerical moment large enough (related to the Lipschitz constant of F).

33 Convergence II Admissibility and stability are the most difficult to verify. The idea is to show that the mapping M ρ : U α Ûα defined by h Û α = h U α ρ F[U α ] is monotone and a contraction for small enough ρ > 0. h is an enhanced discrete Laplacian. Stability follows from applying Crandall-Tartar lemma to M ρ which commutes with additive constants (this is the reason to assume c 0).

34 Convergence III Theorem (F-Lewis, 16) In addition, suppose F[u] = 0 satisfies the comparison principle. Let U be the solution to the Lax-Friedrichs-like scheme and u h be its piecewise constant extension. Then u h converges to u locally uniformly as h 0 +. Main ingredients of proof Follow and modify Barles-Souganidis proof to show that u(x) := lim inf u h (x) and u(x) := lim sup u h (x) are respectively viscosity super- and sub-solution. First work with quadratic test functions, then with general test functions. Use the numerical moment to control the off-diagonal entries in the discrete Hessians (this is the most difficult and technical step). Use the consistency and comparison principle to get u = u.

35 The Local Stencil (2D) The extra nodes in the Cartesian directions smooth the approximation through the diagonal components of the vanishing moment. Directional resolution is no longer necessary avoiding the need for a wide-stencil.

36 Overcoming a Lack of Monotonicity v v x 0 x k x 0 x k δ 2 x k,h k v(x 0 ± h k e k ) < δ 2 x k,h k v(x 0 ) < 0 δ 2 x k,h k v(x 0 ) δ 2 x k,h k v(x 0 ) < 0 δ 2 x k,h k v(x 0 ± h k e k ) > 0 and δ 2 x k,h k v(x 0 ) < 0 δ 2 x k,h k v(x 0 ) δ 2 x k,h k v(x 0 ) > 0

37 Outline Motivation and Background A Narrow-Stencil Finite Difference Method High Order Extensions Numerical Experiments Conclusion

38 Extensions Goals: To develop high order methods and to use unstructured meshes Bad" news: It is not possible to construct higher than 2nd order monotone FD schemes (we have yet given up since g-monotonicity is a weaker requirement)

39 Extensions Goals: To develop high order methods and to use unstructured meshes Bad" news: It is not possible to construct higher than 2nd order monotone FD schemes (we have yet given up since g-monotonicity is a weaker requirement) Good" news: Inspired by a work of Yan-Osher (JCP, 11) for fully nonlinear 1st order Hamilton-Jacobi equations, we are able to construct high order MDG (mixed DG) and LDG (local DG) methods. Ideas of MDG: write F(u xx, x) = 0 as F ( p R, p M, p R, x) = 0 p L = u xx p M = u a xx p R = u + xx and use different numerical fluxes on the linear equations.

40 Extensions (continued) Ideas of LDG: write F(u xx, u x, u, x) = 0 as F( p 1, p 2, p 3, p 4, q 1, q 2, u, x) = 0, q 1 = u x (x ), q 2 = u x (x + ), p 1 = q 1x (x ), p 2 = q 1x (x + ), p 3 = q 2x (x ), p 4 = q 2x (x + ), where u x (x ) (resp. q jx (x )) and u x (x + ) (resp. q jx (x + )) denote the left and right limits of u x (resp. q jx ) at x. They dictate how numerical fluxes should be chosen in the discretization. and stands for monotone increasing and decreasing, respectively. r = 0 is allowed!

41 Outline Motivation and Background A Narrow-Stencil Finite Difference Method High Order Extensions Numerical Experiments Conclusion

42 1-D simulations Test 1: Consider the problem uxx = 0, 0 < x < 1 u(0) = 0, u(1) = 0.5 This problem has the pointwise solutions u + (x) = 0.5x 2 (convex), u (x) = 0.5x 2 + x (concave) Computed using Lax-Friedrichs-like scheme F 1 with α = 1 (left) and α = 1 (right)

43 Test 2: Consider the problem min A θu xx S(x) = 0, 1 < x < 1 θ {1,2} u( 1) = 1, u(1) = 1 { 12x 2, if x < 0 A 1 = 1, A 2 = 2, S(x) = 24x 2, if x 0 This problem has the exact solution u(x) = x x 3

44 Test 3: Consider the problem u 3 xx + 8 sign(x) = 0, 1 < x < 1 u( 1) = 1, u(1) = 1. This problem has the exact solution u(x) = x x C 1 ([ 1, 1]) which is not classical

45 2-D simulations Test 4: Let Ω = (0, 1) 2. Consider Monge-Ampére equation det(d 2 u) = 1 u = 0 in Ω on Ω Remark: No explicit solution formula and solution is not classical.

46 ( α < 0: concave solution) ( α > 0: convex solution)

47 Test 5: Let Ω = ( 1, 1) 2. Consider Monge-Ampére equation det(d 2 u(x, y)) = 0 u = g in Ω on Ω Choose g such that the exact solution is u(x, y) = x.

48 h x L norm order L 2 norm order 2.50E E E E E E E E E Top: α = I, r = 1, and h y = 1/3 fixed. Bottom: α = I, r = 1, and odd number of intervals in the x-direction.

49 3-D Simulations Test 6: Let Ω = (0, 1) 3. Consider Monge-Ampére equation det(d 2 u) = f u = g in Ω on Ω f (x, y, z) = (1 + x 2 + y 2 + z 2 )e 3(x2 +y 2 +z 2 ) 2 g(x, y, z) = e y2 +z 2 2 if x = 0 e x2 +z 2 2 if y = 0 e x2 +y 2 2 if z = 0 e y2 +z if x = 1 e x2 +z if y = 1 e x2 +y if z = 1 The exact solution is u 0 (x, y, z) = e x2 +y 2 +z 2 2

50 (computed solution) (error function)

51 Test 7: Let Ω = (0, 1) 3. Consider Monge-Ampére equation det(d 2 u) = 1 u = 0 in Ω on Ω Remark: No explicit solution formula and solution is not classical.

52 computed solution (α > 0)

53 Outline Motivation and Background A Narrow-Stencil Finite Difference Method High Order Extensions Numerical Experiments Conclusion

54 Concluding Remarks The generalized monotone structure allows us to deal with low regularity functions by considering multiple gradient and Hessian approximations. The key new concept is that of a numerical moment which can be used to design generalized monotone methods such as the Lax-Friedrichs-like scheme. By using narrow stencil schemes that can be expressed using only forward and backward difference quotients, the methods can be easily extended to higher-order and non-cartesian domains/grids using the DG techniques (or DG Finite Element Calculus). Formulation easily extends to Monge-Ampère type equations either directly or based on its HJB reformulation, although the analysis is not yet. Many open problems/issues: such as extension to degenerate PDES, rate of convergence in C 0 -norm, boundary layers, parabolic PDEs, nonlinear solvers, etc.

55 References XF, R. Glowinski and M. Neilan, Recent Developments in Numerical Methods for Fully Nonlinear 2nd Order PDEs", SIAM Review, 55:1-64, 2013 M. Neilan, A. Salgado and W. Zhang, Numerical Analysis of Strongly Nonlinear PDEs", arxiv.org/abs/ , to appear in Acta Numerica. Check arxiv.org for more (and new) references.

56 Thanks for Your Attention!