Semi-Lagrangian discontinuous Galerkin schemes for some first and second-order partial differential equations

Explicit, unconditionally stable, high-order schemes for the approximation of some first- andsecond-order linear, time-dependent partial differential equations (PDEs) are proposed.The schemes are based on a weak formulation of a semi-Lagrangian scheme using discontinuous Galerkin (DG) elements.It follows the ideas of the recent works of Crouseilles, Mehrenberger and Vecil (2010), Rossmanith and Seal (2011),for first-order equations, based on exact integration, quadrature rules, and splitting techniques for the treatment of two-dimensionalPDEs. For second-order PDEs the idea of the schemeis a blending between weak Taylor approximations and projection on a DG basis.New and sharp error estimates are obtained for the fully discrete schemes and for variable coefficients.In particular we obtain high-order schemes, unconditionally stable and convergent,in the case of linear first-order PDEs, or linear second-order PDEs with constant coefficients.In the case of non-constant coefficients, we construct, in some particular cases,"almost" unconditionally stable second-order schemes and give precise convergence results.The schemes are tested on several academic examples

High-order filtered schemes for time-dependent second order HJB equations

In this paper, we present and analyse a class of "filtered" numerical schemes for second order Hamilton-Jacobi-Bellman equations. Our approach follows the ideas introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Amp\`ere partial differential equation, SIAM J. Numer. Anal., 51(1):423--444, 2013, and more recently applied by other authors to stationary or time-dependent first order Hamilton-Jacobi equations. For high order approximation schemes (where "high" stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by "filtering" them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward difference formulae (BDF) for constructing the high order schemes.Comment: 27 pages, 16 figures, 4 table

An efficient filtered scheme for some first order Hamilton-Jacobi-Bellman equations

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 $\epsilon$-monotone property. Convergence results and precise error estimates are given, of the order of $\sqrt{\Delta x}$ where $\Delta x$ 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.Comment: 20 pages (including references), 26 figure

Minimal Time Problems with Moving Targets and Obstacles

International audienceWe consider minimal time problems governed by nonlinear systems under general time dependant state constraints and in the two-player games setting. In general, it is known that the characterization of the minimal time function, as well as the study of its regularity properties, is a difficult task in particular when no controlability assumption is made. In addition to these difficulties, we are interested here to the case when the target, the state constraints and the dynamics are allowed to be time-dependent. We introduce a particular "reachability" control problem, which has a supremum cost function but is free of state constraints. This auxiliary control problem allows to characterize easily the backward reachable sets, and then, the minimal time function, without assuming any controllability assumption. These techniques are linked to the well known level-set approachs. Partial results of the study have been published recently by the authors in SICON. Here, we generalize the method to more complex problems of moving target and obstacle problems. Our results can be used to deal with motion planning problems with obstacle avoidance

Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinuous data

International audienceOn étudie un schéma non monotone pour l'équation Hamilton Jacobi Bellman du premier ordre, en dimension 1. Le schéma considèré est lié au schéma anti-diffusif, appellé UltraBee, proposé par Bokanowski-Zidani (publié en 2007 dans J. Sci. Compt.). Ici, on prouve la convergence, en norme $L^1$, à l'ordre 1, pour une condition initiale discontinue. Le caractère anti-diffusif du schéma est illustré par quelques exemples numériques

Some convergence results for Howard's algorithm

International audienceThis paper deals with convergence results of Howard's algorithm for the resolution of \min_{a\in \cA} (B^a x - b^a)=0 where $B^a$ is a matrix, $b^a$ is a vector (possibly of infinite dimension), and \cA is a compact set. We show a global super-linear convergence result, under a monotonicity assumption on the matrices $B^a$. In the particular case of an obstacle problem of the form $\min(A x - b,\, x-g)=0$ where $A$ is an $N\times N$ matrix satisfying a monotonicity assumption, we show the convergence of Howard's algorithm in no more than $N$ iterations, instead of the usual $2^N$ bound. Still in the case of obstacle problem, we establish the equivalence between Howard's algorithm and a primal-dual active set algorithm (M. Hintermüller et al., {\em SIAM J. Optim.}, Vol 13, 2002, pp. 865-888). We also propose an Howard-type algorithm for a "double-obstacle" problem of the form $\max(\min(Ax-b,x-g),x-h)=0$. We finally illustrate the algorithms on the discretization of nonlinear PDE's arising in the context of mathematical finance (American option, and Merton's portfolio problem), and for the double-obstacle problem

An efficient data structure to solve front propagation problems

International audienceIn this paper we develop a general efficient sparse storage technique suitable to coding front evolutions in d>= 2 space dimensions. This technique is mainly applied here to deal with deterministic target problems with constraints, and solve the associated minimal time problems. To this end we consider an Hamilton-Jacobi-Bellman equation and use an adapted anti-diffusive Ultra-Bee scheme. We obtain a general method which is faster than a full storage technique. We show that we can compute problems that are out of reach by full storage techniques (because of memory). Numerical experiments are provided in dimension d=2,3,4

Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

International audienceThis work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton- Jacobi-Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach

Neural networks for first order HJB equations and application to front propagation with obstacle terms

We consider a deterministic optimal control problem with a maximum running cost functional, in a finite horizon context, and propose deep neural network approximations for Bellman's dynamic programming principle, corresponding also to some first-order Hamilton-Jacobi-Bellman equations. This work follows the lines of Hur\'e et al. (SIAM J. Numer. Anal., vol. 59 (1), 2021, pp. 525-557) where algorithms are proposed in a stochastic context. However, we need to develop a completely new approach in order to deal with the propagation of errors in the deterministic setting, where no diffusion is present in the dynamics. Our analysis gives precise error estimates in an average norm. The study is then illustrated on several academic numerical examples related to front propagations models in the presence of obstacle constraints, showing the relevance of the approach for average dimensions (e.g. from $2$ to $8$), even for non-smooth value functions

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

We study a second order BDF (Backward Differentiation 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