Haut PDF On the numerical approximation of first order Hamilton Jacobi equations

On the numerical approximation of first order Hamilton Jacobi equations

On the numerical approximation of first order Hamilton Jacobi equations

Unité de recherche INRIA Futurs Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex France Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy[r]

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Error Estimates for Second Order Hamilton-Jacobi-Bellman Equations. Approximation of Probabilistic Reachable Sets

Error Estimates for Second Order Hamilton-Jacobi-Bellman Equations. Approximation of Probabilistic Reachable Sets

ϑ  (T, x) = Φ  (x) in R d where H(t, x, p, Q) := sup a∈U −b(t, x, a) · p − Tr([σσ T ](t, x, a)Q). In the case when the the drift b and the diffusion σ are bounded and when the value function ϑ  is itself bounded, error estimates of monotone schemes have been obtained first by Krylov [ 27 ] for a case where σ is a constant function. These results were developed further in [ 28 , 29 , 6 , 7 , 8 ] by introducing new tools that allow to consider the case where σ can depend on time, space and also on the control variable. Several other extensions of the theory have been analysed in the literature, let us mention some of these extensions for stopping-game problems [ 13 ], for impulsive control systems [ 14 ], for integro-partial differential HJB equations [ 18 , 9 , 10 ], and for a general class of coupled HJB systems [ 16 ]. Note also that the case of fully uniformly elliptic operators have been also studied in [ 17 ] using a different approach than the one introduced by Krylov. Here, we extend the theory of error estimates to an unbounded Lipschitz setting. The proof is still based on “Krylov regularization” and on some refined consistency estimates. To the best of our knowledge, this is the first result in the case where b, σ and the solution to the HJB equation itself are unbounded with respect to the space variable (with linear growth).
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Nonlinear Perron-Frobenius theory and max-plus numerical methods for Hamilton-Jacobi equations

Nonlinear Perron-Frobenius theory and max-plus numerical methods for Hamilton-Jacobi equations

We show that if ˜ V 6 V is not equal to V , then the set of differentiability points x 0 such that H(x 0 , ∇ ˜ V ) > 0 is of positive measure (Proposition 8.11). To find such points, we randomly generate a number of points and select those at which the value of Hamiltonian is positive. This also takes a linear computation time (with respect to the current number of basis functions) for each sampled point. The advantages of this method are as follows. First adding a basis function only requires a number of arithmetical operations that is linear to the number of basis functions. Thus, such an ele- mentary step can be done quickly. Secondly, the time discretization step is automatically adapted by the method, and each added basis function is guaranteed to be useful in some region of the state space, at least at the step when we add it. Numerical experiments (Section 8.4) show that to reach a back- substitution of the same order, the new algorithm takes much less time than the SDP based pruning algorithm (103s vs >10h for the instance used in [MDG08]). We might make a connection between this randomized method for infinite horizon optimal control problems and the so-called "point-base value iteration" [CLZ97, ZZ01] for Partially Observable Markov Decision Process (POMDP). Al- though developed in very different settings, the two methods share the idea that improving quickly the approximate value function at randomly generated witness points leads to a better performance than improving uniformly but slowly the approximated value function.
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Value iteration convergence of "-monotone schemes for stationary Hamilton-Jacobi equations

Value iteration convergence of "-monotone schemes for stationary Hamilton-Jacobi equations

Example 5.6. We consider again the problem of Example 5.5 and use the filtering scheme in which Π A k was chosen as the cubic spline interpolation already presented in Example 5.5, cf. Figure 5.2 (right). The numerical parameters were chosen as space and time step k = h = 0.06, resulting in 51 nodes, and the minimum in (4.2) was computed over a discrete set of controls U discretizing the interval [0, 0.4] with 51 equidistant values. Figure 5.3 shows the evolution of the value iteration plotting the difference kw j+1 − w j k depending on j for different filtering parameter ε. One clearly observes that the iteration converges to increasingly smaller sets for shrinking filtering parameter, i.e., for increasing weight on the first order monotone scheme. Obviously, the filtering significantly improves the convergence behavior of the value iteration.
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Homogenization of Hamilton-Jacobi equations and applications to traffic flow modelling

Homogenization of Hamilton-Jacobi equations and applications to traffic flow modelling

The second chapter is dedicated to the homogenization of a Hamilton-Jacobi equation for traffic lights. We consider an infinite road where lights are equally spaced and with a constant phase shift between two lights. This model takes the form of a first order Hamilton-Jacobi equation with an Hamiltonian that is discontinuous in the space variable and the notion of viscosity solution is the one introduced in [ 52 ]. Each light is modelled as a time-periodic flux limiter and the traffic flow between two lights corresponds to the classical LWR model. The global Hamiltonian will be time-periodic but not periodic in space for a general phase shift. We first show that the rescaled solution converges toward the solution of the expected macroscopic model where the effective Hamiltonian depends on the phase shift. In a second time, numerical simulations are used to analyse the effect of the phase shift on the effective Hamiltonian and to reveal some properties of the effective Hamiltonian from the numerical observations. In the third chapter, we are interested in some homogenization problems of Hamilton-Jacobi equations within the almost periodic setting which generalizes the usual periodic one. The first problem is the evolutionary version of the work [ 54 ], with the same stationary Hamiltonian. The second problem has already been solved in the second chapter but we use here almost periodic arguments for the time periodic and space almost periodic Hamiltonian. We only study the ergodicity of the associated cell problems. We finally discuss open problems, the first one concerning a space and time almost periodic Hamiltonian and the second one being a microscopic model for traffic flow modelling where the Hamiltonian is almost periodic in space.
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An adaptive sparse grid semi-lagrangian scheme for first order Hamilton-Jacobi Bellman equations

An adaptive sparse grid semi-lagrangian scheme for first order Hamilton-Jacobi Bellman equations

ORDER HAMILTON-JACOBI BELLMAN EQUATIONS OLIVIER BOKANOWSKI, JOCHEN GARCKE, MICHAEL GRIEBEL, AND IRENE KLOMPMAKER A BSTRACT . We propose a semi-Lagrangian scheme using a spatially adaptive sparse grid to deal with non-linear time-dependent Hamilton-Jacobi Bellman equations. We focus in particular on front propagation models in higher dimensions which are related to control problems. We test the numerical efficiency of the method on several benchmark problems up to space dimension d = 8, and give evidence of convergence towards the exact viscosity solution. In addition, we study how the complexity and precision scale with the dimension of the problem.
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Large time behavior of unbounded solutions of first-order Hamilton-Jacobi equations in $\mathbb{R}^N$

Large time behavior of unbounded solutions of first-order Hamilton-Jacobi equations in $\mathbb{R}^N$

For Lemma 4.2, we refer the reader to [1, Theorem 9.2, p.90]. References [1] Yves Achdou, Guy Barles, Hitoshi Ishii, and Grigory L. Litvinov. Hamilton-Jacobi equa- tions: approximations, numerical analysis and applications, volume 2074 of Lecture Notes in Mathematics. Springer, Heidelberg; Fondazione C.I.M.E., Florence, 2013. Lecture Notes from the CIME Summer School held in Cetraro, August 29–September 3, 2011, Edited by Paola Loreti and Nicoletta Anna Tchou, Fondazione CIME/CIME Foundation Subseries. [2] M. Bardi and I. Capuzzo Dolcetta. Optimal control and viscosity solutions of Hamilton-
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Large time behavior of solutions of viscous Hamilton-Jacobi Equations with superquadratic Hamiltonian

Large time behavior of solutions of viscous Hamilton-Jacobi Equations with superquadratic Hamiltonian

guaranted when f is bounded even for simple ordinary (y ′ (t) = −1) or partial (u t − u xx + |u x | = −1) equations. It is then be hopeless for fully nonlinear PDE as we can note from (1.6) where u(x, t) could go to −∞ for some f < 0. Taking this into account, we find that the study of the long time behavior of the solution u of (1.1)-(1.2)-(1.3) first lead us to the study of a stationary ergodic problem. More precisely, like in the work of lasry & Lions [37], we are interested in findind an appropriate constant c such that the function u(·, t)+ct remains bounded and c is the unique constant for which the stationnary equation with state constraint boundary condition
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AN EFFICIENT FILTERED SCHEME FOR SOME FIRST ORDER HAMILTON-JACOBI-BELLMAN EQUATIONS

AN EFFICIENT FILTERED SCHEME FOR SOME FIRST ORDER HAMILTON-JACOBI-BELLMAN EQUATIONS

Abstract. We introduce a new class of “filtered” schemes for some first order non-linear Hamilton-Jacobi- Bellman equations. The work follows recent ideas of Froese and Oberman (SIAM J. Numer. Anal., Vol 51, pp.423- 444, 2013). The proposed schemes are not monotone but still satisfy some -monotone property. Convergence results and precise error estimates are given, of the order of √ ∆x where ∆x is the mesh size. The framework allows to construct finite difference discretizations that are easy to implement, high–order in the domains where the solution is smooth, and provably convergent, together with error estimates. Numerical tests on several examples are given to validate the approach, also showing how the filtered technique can be applied to stabilize an otherwise unstable high–order scheme.
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Stability and convergence of second order backward differentiation schemes for parabolic Hamilton-Jacobi-Bellman equations

Stability and convergence of second order backward differentiation schemes for parabolic Hamilton-Jacobi-Bellman equations

HAMILTON-JACOBI-BELLMAN EQUATIONS OLIVIER BOKANOWSKI ∗ , ATHENA PICARELLI † , AND CHRISTOPH REISINGER † Abstract. We study a second order BDF (Backward Dierentiation Formula) scheme for the numerical approximation of parabolic HJB (Hamilton-Jacobi-Bellman) equations. The scheme under consideration is implicit, non-monotone, and second order accurate in time and space. The lack of monotonicity prevents the use of well-known convergence results for solutions in the viscosity sense. In this work, we establish rigorous stability results in a general nonlinear setting as well as convergence results for some particular cases with additional regularity assumptions. While most results are presented for one-dimensional, linear parabolic and non-linear HJB equations, some results are also extended to multiple dimensions and to Isaacs equations. Numerical tests are included to validate the method.
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Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations

Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations

The paper is organized as follows: in Section 2 we expose the basic tools of our analysis with particular reference to invariance results for multivalued vector fields, Filippov Approximation Theorem ad differential properties of the interface. Section 3 makes precise the setting with assumptions, definition of the involved dynamics, and statements of main results. More important, it is written down the HamiltonJacobi–Bellman equation we propose for our model, see (HJB). In Section 4 we study the continuous character of the value function, while Sections 5, 6 represent the core of the paper with results showing the relationship between being sub/supersolution to (HJB) and enjoying sub and superoptimality properties. In Section 7 we provide the demonstrations of main theorems. Finally, the first appendix is about augmented dynamics with the proof of its Lipschitz character on the interface, and the second is centered on the notion of ε partitions for a given curve, minimal ε–partitions and the introduction of an index related to it, see Definition B.4, on which induction arguments of Sections 5, 6 are based.
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On the        regularizing effect for unbounded solutions of first-order        Hamilton-Jacobi equations

On the regularizing effect for unbounded solutions of first-order Hamilton-Jacobi equations

where A may be degenerate. In both cases, we provide regularizing effects for bounded from below solutions. The main improvement in the assumptions is easy to describe in the coercive case since we just require that A, f are continuous in x (no uniform continuity assumptions) and, in particular, f may have some growth at infinity. In the non-coercive case, analogous results hold except that we have to impose far more restrictive assumptions on the t-dependence of the equation.

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APPROXIMATION OF CONTROLS FOR LINEAR WAVE EQUATIONS: A FIRST ORDER MIXED FORMULATION

APPROXIMATION OF CONTROLS FOR LINEAR WAVE EQUATIONS: A FIRST ORDER MIXED FORMULATION

tions of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [Cindea & M¨ unch, A mixed formulation for the direct ap- proximation of the control of minimal L 2 -norm for linear type wave equations], we have introduced a space-time variational approach ensuring strong convergent approximations with respect to the discretization parameter. The method, which relies on generalized ob- servability inequality, requires H 2 -finite element approximation both in time and space.
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On the existence of correctors for the stochastic homogenization of viscous hamilton-jacobi equations

On the existence of correctors for the stochastic homogenization of viscous hamilton-jacobi equations

If H is convex in p and A is independent of p, our proof implies that, for any p ∈ R d , the limit lim δ→9 δv δ,p (0, ·) exists in probability; see Proposition 3.10. This result and its the proof are very much in the flavor of [14]. Finally we note that our arguments also yield the existence of a corrector in some directions and, thus, homogenization, for nonconvex Hamiltonians and p dependent A. More precisely, for any direction p, there exists a constant c such that p belongs to the convex hull of directions p ′ for which a corrector exists with associated homogenized constant equal to c;
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Hamilton–Jacobi equations for optimal control on junctions and networks

Hamilton–Jacobi equations for optimal control on junctions and networks

Both [1] and [14] contain the first comparison and uniqueness results: in [1], suitably modified geodesic distances are used in the doubling variables method for proving comparison theorems under rather strong continuity as- sumptions. In [14], Imbert, Monneau and Zidani used a completely different argument based on the explicit solution of a related optimal control problem, which could be obtained because it was assumed that the Hamil- tonians associated with each edge did not depend on the state variable.

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Hamilton Jacobi equations on metric spaces and transport-entropy inequalities

Hamilton Jacobi equations on metric spaces and transport-entropy inequalities

1.1. General framework. In this section we give the general setting of this article. 1.1.1. Assumptions on the space. In all the paper, (X, d) will be a complete and separable metric space in which closed balls are compact. This latter assumption could be removed at the expense of additional standard technicalities. We will sometimes assume that (X, d) is a geodesic space, meaning that for every two points x, y ∈ X there is at least one curve (γ t ) t∈[0,1] with γ 0 = x, γ 1 = y and such that d(γ s , γ t ) = |t − s|d(x, y) for all s, t ∈ [0, 1]. Such
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Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions

Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions

The assumption of finite range dependence is well-motivated physically and is analogous to the standard assumption of i.i.d. in discrete probability models. It is not an assumption made for simplicity: we do not know how to relax it even to allow very quick decaying correlations of the coefficients. However, by stability arguments, we can obtain homogenization results for coefficients fields which are uniform limits of finite-range fields. This covers many typical exam- ples, including for instance coefficients fields built by convolutions of smooth (but not compactly supported) functions against Poisson point clouds.
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Dirichlet Problems for some Hamilton-Jacobi Equations With  Inequality Constraints

Dirichlet Problems for some Hamilton-Jacobi Equations With Inequality Constraints

Very few results applicable to highway traffic are available for control of first order hyperbolic conservation laws. Differential flatness [47] has been successfully applied to Burgers equation (and therefore to the Lighthill-Whitham-Richards equation) in [72] order to avoid the formation of such shockwaves. This analysis does not so far extend to the presence of shocks. Lyapunov based techniques have also been applied to the Burgers equation [59]. Adjoint based methods have been successfully applied to networks of Lighthill- Whitham-Richards equations in [54]; these results seem so far the most promising, but they do not have guarantees to provide an optimal control policy. Questions of interest in controlling first-order partial differential equations, and in particular, Lighthill-Whitham-Richards equations, are still open and difficult to solve due to the presence of shocks occurring in the solutions of these partial differential equations.
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On the Large Time Behavior of Solutions of the Dirichlet problem for Subquadratic Viscous Hamilton-Jacobi Equations

On the Large Time Behavior of Solutions of the Dirichlet problem for Subquadratic Viscous Hamilton-Jacobi Equations

 Now, we turn to the proof of the interior H¨older estimates on v with respect to the x-variable uniformly in t ∈ (0, +∞) which is based on an idea introduced by Ishii and Lions [12]. This idea has been already used for instance in Barles [1] and Barles and Souganidis [7] to show gradient estimates of viscosity solutions to quasilinear elliptic ans parabolic PDE with Lipschitz initial conditions, by Barles and Da Lio [5] to prove local H¨older estimates up to the boundary of bounded solutions to fully non linear elliptic PDE with Neumann boundary conditions and by Da Lio [10] to obtain C 0,ν -
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On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions

On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions

HAMILTON-JACOBI EQUATIONS ASSOCIATED WITH NONLINEAR BOUNDARY CONDITIONS GUY BARLES, HITOSHI ISHII AND HIROYOSHI MITAKE Abstract. In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy-Neumann problems by using two fairly different methods : the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the “weak KAM approach” which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry-Mather sets.
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