Numerical schemes for Hamilton-Jacobi equations, control problems and games

Transcription

1 Numerical schemes for Hamilton-Jacobi equations, control problems and games M. Falcone H. Zidani SADCO Spring School Applied and Numerical Optimal Control April 23-27, 2012, Paris Lecture 2/3 M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 1 / 58

2 Outline 1 Introduction 2 Fast Marching Methods 3 Classical Domain decomposition 4 Patchy vectorfields 5 The Patchy Domain Decomposition Method M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 2 / 58

3 OUTLINE OF THE COURSE: Lecture 1: Approximation of optimal control problems via DP Lecture 2: Efficient methods and perspectives Lecture 3: Approximation of reachable sets and state constrained problems M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 3 / 58

4 Outline 1 Introduction 2 Fast Marching Methods 3 Classical Domain decomposition 4 Patchy vectorfields 5 The Patchy Domain Decomposition Method M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 4 / 58

5 The numerical solution of optimal control problems via the Dynamic Programming approach is mainly motivated by the search for feedback controls for generic nonlinear Lipschitz continuous vectorfields and costs. The solution of the corresponding Bellman equation in high dimension is a computationally intensive task and this bottleneck has limited the applications of this theory to industrial cases. We want to overcome this technical problem developing new efficient algorithms with limited (and controlled) memory allocations and reasonable CPU times. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 5 / 58

6 DP s advantages and disadvantages PROS 1. The characterization of the value function is valid for all classical problems in any dimension. 2. The approximation is based on a-priori error estimates in L, is valid in any dimension and does not require structured grids. 3. The computation of feedback controls is almost built in and there are nice results in low dimension. CONS The curse of dimensionality makes the problem difficult to solve in high dimension due to 1. computational cost 2. huge memory allocations. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 6 / 58

7 DP s advantages and disadvantages PROS 1. The characterization of the value function is valid for all classical problems in any dimension. 2. The approximation is based on a-priori error estimates in L, is valid in any dimension and does not require structured grids. 3. The computation of feedback controls is almost built in and there are nice results in low dimension. CONS The curse of dimensionality makes the problem difficult to solve in high dimension due to 1. computational cost 2. huge memory allocations. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 6 / 58

8 Tecnical difficulties The bottleneck is the approximation of the value function v, however this remains the main goal since v allows to get back to feedback controls in a rather simple way. For control problems a argmin[f (x, a) v(x) + l(x, a)] For games (a e, a p) argminmax[f (x, a e, a p ) v(x) + l(x, a e, a p )] M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 7 / 58

9 The Tag-chase game in a courtyard, V p > V e M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 8 / 58

10 Efficient methods In order to overcome these technical difficulties and reduce the memory allocation requirements we have modified the standard approximation scheme based on a fixed point iteration of a discrete Hamilton-Jacobi equation derived by DP. Fast Marching Methods/Low memory algorithms Parallel Fast Marching Method Domain decomposition methods M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 9 / 58

11 Efficient methods In order to overcome these technical difficulties and reduce the memory allocation requirements we have modified the standard approximation scheme based on a fixed point iteration of a discrete Hamilton-Jacobi equation derived by DP. Fast Marching Methods/Low memory algorithms Parallel Fast Marching Method Domain decomposition methods M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 9 / 58

12 Efficient methods In order to overcome these technical difficulties and reduce the memory allocation requirements we have modified the standard approximation scheme based on a fixed point iteration of a discrete Hamilton-Jacobi equation derived by DP. Fast Marching Methods/Low memory algorithms Parallel Fast Marching Method Domain decomposition methods M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 9 / 58

13 Efficient methods In order to overcome these technical difficulties and reduce the memory allocation requirements we have modified the standard approximation scheme based on a fixed point iteration of a discrete Hamilton-Jacobi equation derived by DP. Fast Marching Methods/Low memory algorithms Parallel Fast Marching Method Domain decomposition methods M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 9 / 58

14 Outline 1 Introduction 2 Fast Marching Methods 3 Classical Domain decomposition 4 Patchy vectorfields 5 The Patchy Domain Decomposition Method M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 10 / 58

15 Classical fixed point algorithm The classical algorithm is based on fixed-point iterations u k+1 i,j = S(u k i,j, u k i±1,j, u k i,j±1) where all the grid nodes are visited in some fixed order at each iteration. S represents the fully discrete operator (FD, semi-lagrangian,...). Drawbacks This algorithm can be slow, especially for d 3, due to the huge number of nodes. WARNING At each iteration a large part of grid nodes are NOT really updated so these computations are completely useless. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 11 / 58

16 Classical fixed point algorithm The classical algorithm is based on fixed-point iterations u k+1 i,j = S(u k i,j, u k i±1,j, u k i,j±1) where all the grid nodes are visited in some fixed order at each iteration. S represents the fully discrete operator (FD, semi-lagrangian,...). Drawbacks This algorithm can be slow, especially for d 3, due to the huge number of nodes. WARNING At each iteration a large part of grid nodes are NOT really updated so these computations are completely useless. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 11 / 58

17 Classical fixed point algorithm The classical algorithm is based on fixed-point iterations u k+1 i,j = S(u k i,j, u k i±1,j, u k i,j±1) where all the grid nodes are visited in some fixed order at each iteration. S represents the fully discrete operator (FD, semi-lagrangian,...). Drawbacks This algorithm can be slow, especially for d 3, due to the huge number of nodes. WARNING At each iteration a large part of grid nodes are NOT really updated so these computations are completely useless. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 11 / 58

18 The Fast Marching Method The FMM was introduced by Tsitsiklis in 1995 and then J. A. Sethian in It is an acceleration method for the classical iterative FD scheme for the eikonal equation. Main Idea Processing the nodes in a special ordering one can compute the solution in just 1 iteration. Remark This special ordering corresponds to the increasing values of u. The main feature of these methods is that they have an O(N ln(n)) complexity. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 12 / 58

19 The Fast Marching Method The FMM was introduced by Tsitsiklis in 1995 and then J. A. Sethian in It is an acceleration method for the classical iterative FD scheme for the eikonal equation. Main Idea Processing the nodes in a special ordering one can compute the solution in just 1 iteration. Remark This special ordering corresponds to the increasing values of u. The main feature of these methods is that they have an O(N ln(n)) complexity. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 12 / 58

20 The Fast Marching Method The FMM was introduced by Tsitsiklis in 1995 and then J. A. Sethian in It is an acceleration method for the classical iterative FD scheme for the eikonal equation. Main Idea Processing the nodes in a special ordering one can compute the solution in just 1 iteration. Remark This special ordering corresponds to the increasing values of u. The main feature of these methods is that they have an O(N ln(n)) complexity. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 12 / 58

21 The Fast Marching Method, contn d The nodes are divided in three sets: ACCEPTED, NARROW BAND and FAR. At each step, the NB node with the minimal value is labelled as ACCEPTED and the NB zone is updated. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 13 / 58

22 The semi-lagrangian FMM for the eikonal equation { max a A x Rn \Ω T (x) = 0 x Ω Let us examine the FM method based on the semi-lagrangian scheme. Via the Kružkov transform v(x) = 1 e T (x) the SL-scheme can be written as (E) { vh (x i ) = min a B(0,1) {e h v h (x i hc(x i )a)} + 1 e h v h (x i ) = 0 x i R n \ Ω x i Ω where the nodes x i belong to a structured grid G. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 14 / 58

23 Convergence of FM-SL method Theorem (Cristiani, F. 2007) Let the following assumptions hold true: c( ) Lip(R n ) L (R n ) c(x) 0 Then, SL-FM method computes an approximate solution of (E). Moreover, this solution is exactly the same solution of SL classical iterative scheme. Remark NO coupling conditions on c and x are required! M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 15 / 58

24 TEST 1. Numerical comparison Distance from the origin. Ω = {(0, 0)}, c(x, y) 1 Exact solution: T (x, y) = (x 2 + y 2 ) M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 16 / 58

25 TEST 1. Numerical comparison Distance from the origin. Ω = {(0, 0)}, c(x, y) 1 Exact solution: T (x, y) = (x 2 + y 2 ) method x L error L 1 error CPU time (sec) SL (46 its.) FM-FD FM-SL M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 16 / 58

26 TEST 1. Numerical comparison Distance from the origin. Ω = {(0, 0)}, c(x, y) 1 Exact solution: T (x, y) = (x 2 + y 2 ) method x L error L 1 error CPU time (sec) SL (46 its.) FM-FD FM-SL M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 16 / 58

27 TEST 1. Numerical comparison Distance from the origin. Ω = {(0, 0)}, c(x, y) 1 Exact solution: T (x, y) = (x 2 + y 2 ) method x L error L 1 error CPU time (sec) SL (46 its.) FM-FD FM-SL M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 16 / 58

28 TEST 2. Minimum time problem with obstacles M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 17 / 58

29 TEST 2. Optimal trajectory M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 18 / 58

30 TEST 3. Minimum time problem Eikonal equation with velocity c(x, y) = x + y. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 19 / 58

31 Patchy decomposition Main Idea Since the patches are invariant with respect to the patchy vector fields, we can split the computation of the solution in D sub-problems, each corresponding to a patchy sub-domain and use a parallel algorithm to compute the value function in the whole domain. This patchy domain decomposition method has shown to be more efficient with respect to standard (static) domain decomposition techniques as we will show by some numerical tests. The algorithm has been proposed in [Cacace, Cristiani, F. and Picarelli, SIAM J. Sci. Com, in print]. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 20 / 58

32 Outline 1 Introduction 2 Fast Marching Methods 3 Classical Domain decomposition 4 Patchy vectorfields 5 The Patchy Domain Decomposition Method M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 21 / 58

33 The model problem Let us consider, for example, the infinite horizon optimal control problem which leads to the Hamilton-Jacobi-Bellman equation λv(x) + max { f (x, a) v(x) l(x, a)} = 0, x a A Ω where A is a compact subset of R m and f, l are given functions, λ > 0. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 22 / 58

34 The infinite horizon problem Dynamics { ẏ(t) = f (y(t), α(t)) t > 0 Admissible controls y(0) = x α( ) A {α : [0, + [ A, measurable} Cost Value function J(x, α( )) = v(x) = 0 l(y(t), α(t))e λt dt inf J(x, α( )) α( ) A M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 23 / 58

35 The infinite horizon problem For numerical purposes we have to deal the problem in a bounded domain Ω λv(x) + max { f (x, u) v(x) l(x, u)} = 0, x Ω u U Domain splitting Let us consider a splitting of Ω into D subdomains Ω d, d = 1,..., D Ω = d Ω d and a grid G with a number of nodes N Ω N Ω N N D M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 24 / 58

36 Sketch of Classical DD MAIN CICLE REPEAT STEP 1 Compute one iteration of the numerical operator S restricted to every domain Ω d, d = 1,..., D STEP 2 Couple the information on the overlapping zones (Transmission Conditions) UNTIL a stopping rule is satisfied M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 25 / 58

37 Sketch of Classical DD MAIN CICLE REPEAT STEP 1 Compute one iteration of the numerical operator S restricted to every domain Ω d, d = 1,..., D STEP 2 Couple the information on the overlapping zones (Transmission Conditions) UNTIL a stopping rule is satisfied M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 25 / 58

38 Transmission Conditions The correct transmission condition is the min operator min{s 1, S 2,..., S D }. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 26 / 58

39 Outline 1 Introduction 2 Fast Marching Methods 3 Classical Domain decomposition 4 Patchy vectorfields 5 The Patchy Domain Decomposition Method M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 27 / 58

40 Main goal We want to construct a domain decomposition which is based on the patches defined by Ancona and Bressan. PROS patches are invariant with respect to the optimal dynamics CONS we need a dynamic construction of the patches. Let be Ω R n an open domain with smooth boundary Ω and g a smooth vector field defined on a neighborhood of Ω. Definition We say that the pair (Ω, g) is a patch if Ω is a positive-invariant region for g, i.e. at every boundary point x Ω the inner product of g with the outer normal n satisfies g(x), n(x) < 0. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 28 / 58

41 Patchy vectorfields A patchy vector field on a domain Ω R n is a superposition of patches, as reported in the following Definition We say that g : Ω R n is a patchy vector field if there exists a family of patches {(Ω α, g α ) : α I} such that - I is a totally ordered index set, - the open sets Ω α form a locally finite covering of Ω, - the vector field g can be written in the form g(x) = g α (x) if x Ω α \ Ω β. β>α M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 29 / 58

42 Outline 1 Introduction 2 Fast Marching Methods 3 Classical Domain decomposition 4 Patchy vectorfields 5 The Patchy Domain Decomposition Method M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 30 / 58

43 Goal To build a domain decomposition such that the solution in each patch does not depend on the solution in other patches; there is no transmission condition through the boundaries of the patches. In this way the computation can be fully parallelized. The final solution is obtained by merging all the patches at the end. To this end we need an a-priori knowledge of characteristics which is not available PRE-COMPUTATIONS M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 31 / 58

44 The Patchy Algorithm tep1. (Computation on the coarse grid). We solve the equation on a coarse grid G coarse by means of the classical domain decomposition technique. This leads to the function u coarse. tep2. (Interpolation on a fine grid). We compute a first approximation u (0) of the solution on a fine grid G fine by means of a simple bilinear interpolation of the values u coarse. We also compute the optimal control a coarse(x i ) = arg max a { f (x i, a) u (0) (x i )}, x i G fine Note that a coarse is defined on G fine. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 32 / 58

45 The Patchy Algorithm tep3. (Partition of target). On G fine, we divide the target in N p parts denoted by Ω j 0, j = 1,..., N p. tep4. (Main cycle) For any j = 1,..., N p, Step4.1. (Creation of j-th patch). We use the (coarse) optimal control acoarse to find the nodes of the grid G fine which have Ω j 0 in their numerical domain of dependence. This procedure defines the j-th patch. (NEXT SLIDE) Step4.2. (Computation in patches). We solve iteratively the equation in the j-th patch until convergence is reached, imposing state constraints boundary conditions. tep5. (Merge). All the solutions computed in the N p patches are assembled in the grid G fine. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 33 / 58

46 The Patchy Algorithm: creation of a patch For j = 1,..., N p 1 (Initialization) Set { φ i = 1, x i Ω j 0 0, x i G fine \Ω j 0, i = 1,..., N. 2 (Iteration) Solve iteratively the following ad hoc numerical scheme, until convergence is reached. φ i = φ(x i + h i f (x i, a coarse(x i ))), i = 1,..., N. Note that the solution φ takes values in [0, 1]. 3 (Projection) Project the color j into a binary value φ i = { 1, φ(xi ) 1 2 0, φ(x i ) < 1 2, i = 1,..., N. The sub-domain P j = {x i : φ(x i ) = 1} is the j-th patch. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 34 / 58

47 The Patchy Algorithm: creation of a patch Initialization Iteration Projection M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 35 / 58

48 The Patchy Algorithm: invariant domain decomposition Patchy domain decomposition (4 patches) M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 36 / 58

49 Numerical Tests Numerical Tests in dimension 2 M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 37 / 58

50 Test 1: Eikonal f (x 1, x 2, a) = a, A = B(0, 1), Ω 0 = B(0, 0.5). Patchy domain decomposition (8 patches) M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 38 / 58

51 Test 1: Eikonal Patchy Error We compute the difference between the patchy solution and the DD solution. Since the scheme is the same, this error is due to the fact that patches are not perfectly independent. Error table in norm 1 ( ) depending on the space steps k coarse and k fine. k f =0.08 k f =0.04 k f =0.02 k f =0.01 k f =0.005 k c = (0.960) (1.856) (0.048) (0.034) (0.026) k c = (0.046) (0.023) (0.042) (0.008) k c = (0.029) (0.013) (0.008) k c = (0.016) (0.010) k c = (0.008) A = B(0, 1) is discretized with 32 points and the number of patches is 16. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 39 / 58

52 Test 1: Eikonal Patchy solution The subsolutions merge quite well! M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 40 / 58

53 Test 1: Eikonal Patchy error The error is localized on the boundaries of the patches! M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 41 / 58

54 Test 1: Eikonal Patchy method vs classical Domain Decomposition CPU times (in seconds) depending on the number of processors and the number of patches Controls: 16. Grid: domains 4 domains 8 domains 16 domains Best DD 1 proc procs procs Controls: 32. Grid: domains 4 domains 8 domains 16 domains Best DD 1 proc procs procs M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 42 / 58

55 Test 2: Fan f (x 1, x 2, a) = x 1 + x a, A = B(0, 1), Ω 0 = {x 1 = 0}. Patchy domain decomposition (8 patches) M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 43 / 58

56 Test 2: Fan Patchy error Error table in norm 1 ( ) depending on the space steps k coarse and k fine. k f = 0.08 k f = 0.04 k f = 0.02 k f = 0.01 k f = k c = (3.023) (1.507) (0.315) (0.263) (0.263) k c = (1.502) (0.149) (0.095) (0.095) k c = (0.111) (0.061) (0.037) k c = (0.079) (0.037) k c = (0.037) A = B(0, 1) is discretized with 32 points and the number of patches is 16. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 44 / 58

57 Test 2: Fan Patchy Error The error is localized on the boundaries of the patches! M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 45 / 58

58 Test 2: Fan Patchy method vs classical Domain Decomposition CPU times (in seconds) depending on the number of processors and the number of patches Controls: 32. Grid: domains 4 domains 8 domains 16 domains Best DD 1 proc procs procs M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 46 / 58

59 Test 3: Zermelo f (x 1, x 2, a) = 2.1a + (2, 0), A = B(0, 1), Ω 0 = B(0, 0.5). Patchy domain decomposition (8 patches) M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 47 / 58

60 Test 3: Zermelo Patchy Error Error table in norm 1 ( ) depending on the space steps k coarse and k fine. k f = 0.08 k f = 0.04 k f = 0.02 k f = 0.01 k f = k c = (0.293) (0.059) (0.057) (0.027) (0.016) k c = (0.063) (0.041) (0.023) (0.016) k c = (0.039) (0.023) (0.016) k c = (0.020) (0.015) k c = (0.016) A = B(0, 1) is discretized with 32 points and the number of patches is 16. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 48 / 58

61 Test 3: Zermelo Patchy Error The error is localized on the boundaries of the patches! M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 49 / 58

62 Test 3: Zermelo Patchy method vs classical Domain Decomposition Cpu times (in seconds) depending on the number of processors and the number of patches Controls: 32. Grid: domains 4 domains 8 domains 16 domains Best DD 1 proc procs procs M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 50 / 58

63 Numerical Tests Numerical Tests in dimension 3 Here several add-on s are enabled! (ordering of the nodes (FMM-like), reduced controls,...) M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 51 / 58

64 Test 1: Eikonal f (x 1, x 2, x 3, a) = a, A = B(0, 1), Ω 0 = B(0, 0.5). dynamics grid size CPU time Error L 1 Error L Eikonal 3D Eikonal 3D A = B(0, 1) is discretized with 189 points (then reduced when working on the fine grid) and the number of patches is 8. Processors are 4. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 52 / 58

65 Test 1: Eikonal One level set of the patchy solution The error is localized on the boundaries of the patches! M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 53 / 58

66 Test 2: Fan f (x 1, x 2, x 3, a) = x 1 + x 2 + x a, A = B(0, 1), Ω 0 = {x 1 = 0}. dynamics grid size CPU time Error L 1 Error L Fan 3D Fan 3D A = B(0, 1) is discretized with 189 points (then reduced when working on the fine grid) and the number of patches is 8. Processors are 4. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 54 / 58

67 Test 2: Fan Patchy domain decomposition (8 patches) and level sets of the patchy solution M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 55 / 58

68 Test 2: Fan Some optimal trajectories to the target M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 56 / 58

69 Conclusions and Future directions We developed an approximation method for the solution of Hamilton-Jacobi equations which combines a patchy decomposition of the domain and a dynamic programming scheme. The method can handle: - more general control problems (minimum time, finite horizon,...) - state constraints - pursuit evasion games The numerical tests show a very small and localized error (on the patches boundaries). Future directions We want to analyze the method (convergence, error estimates) and combine this technique with efficient fast marching techniques. M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 57 / 58

70 References E. Carlini, M. Falcone, N. Forcadel, R. Monneau,Convergence of a Generalized Fast Marching Method for an eikonal equation with a velocity changing sign, SIAM J. Numer. Anal. 46 (2008), M. Falcone, R. Ferretti, Semi-Lagrangian approximation schemes for linear and Hamilton-Jacobi equations, SIAM, book in preparation S. Cacace, E. Cristiani, and M. Falcone, A local ordered upwind method for Hamilton- Jacobi and Isaacs equations, Proceedings of 18th IFAC World Congress S. Cacace, E. Cristiani, M. Falcone, A. Picarelli, A patchy domain decomposition method for a class of Hamilton-Jacobi-Bellman equations, preprint 2011, submitted to SIAM J. Sci. Comp. available at M. Falcone, P. Lanucara, and A. Seghini, A splitting algorithm for Hamilton-Jacobi-Bellman equations, Applied Numerical Mathematics, 15 (1994), pp M. Falcone & H. Zidani ( ) Approximation of Control and Game Pbs 58 / 58