A Narrow-Stencil Finite Difference Method for - PowerPoint PPT Presentation

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

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

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

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

We consider second order fully nonlinear PDEs F ( D 2 u , Du , u , x ) = 0 Two best known classes of equations: det ( D 2 u ) = f • Monge-Ampére equation: • 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).

Example: (Stochastic Optimal Control) Suppose a stochastic process x ( τ ) is governed by the stochastic differential equation d x ( 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 )) d τ + 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 .

Bellman Principle Suppose u ∗ ∈ U such that u ∗ ∈ argmin J ( t , x , u ) , u ∈ U and define the value function 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.

Bellman Principle (Continued) Let Ω ⊂ R n , T > 0, and U ⊂ R m . The Bellman Principle says v is the solution of v t = F ( D 2 v , ∇ v , v , x , t ) in ( 0 , T ] × Ω , (1) for F ( D 2 v , ∇ v , v , x , t ) = inf u ∈ U ( L u v − h u ) , n n n � � � i ( t , x ) v x i + c u ( t , x ) v a u b u L u v = i , j ( t , x ) v x i x j + i = 1 j = 1 i = 1 with A u := 1 b u := f ( t , x , u ) 2 σσ T c u := 0 h u := L ( t , x , u )

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 λ tr ( A − B ) ≤ F ( A , p , r , x ) − F ( B , p , r , x ) ≤ Λ tr ( A − B ) ∀ A ≥ B . (b) F is said to be proper elliptic if ∀ A ≥ B ; v , w ∈ R d , v ≤ w . F ( A , p , v , x ) ≤ F ( B , p , w , x ) (c) F is said to be degenerate elliptic if F ( A , p , r , x ) ≤ F ( B , p , r , x ) ∀ A ≥ B .

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 ) , D ϕ ( 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 ) , D ϕ ( 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 x ′ → x u ( x ′ ) are the x ′ → x upper and lower semi-continuous envelops of u

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.

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 S ( ρ, y , φ ( y ) + ξ, φ + ξ ) ≤ F ( D 2 φ ( x ) , ∇ φ ( x ) , φ ( x ) , x ) , lim sup ρ ρ → 0 y → x ξ → 0 S ( ρ, y , φ ( y ) + ξ, φ + ξ ) ≥ F ( D 2 φ ( x ) , ∇ φ ( x ) , φ ( x ) , x ) . lim inf ρ ρ → 0 y → x ξ → 0

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.

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.

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).

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

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. ◮ · · ·

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

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

Vanishing Viscosity and Numerical Viscosity − ǫ ∆ u ǫ + H ( ∇ u ǫ , u ǫ , x ) = 0 H ( ∇ u , u , x ) = 0 �− → ( E. Tadmor, ’97 ) Every convergent monotone finite difference scheme for HJ equations implicitly approximates the differential equation − β h “∆ u ” + H ( ∇ u , x ) = 0 for sufficiently large and possibly nonlinear β > 0, where − β h “∆ u ” is called a numerical viscosity . d � � � ∇ + h U α − ∇ − δ 2 Note : If h k ≡ h , then 1 · = h ∆ h U α ≡ h x k , h U α . h U α k = 1 Lax-Friedrichs numerical Hamiltonian � q − + q + � − b · ( q + − q − ) � H ( q − , q + , u , x ) ≡ H , u , x � �� � 2 Numerical Viscosity