arXiv subjects

Olivier Bokanowski

Publications and source records attributed to Olivier Bokanowski.

12 recordsLinked to original sources

arXiv2026

Neural feedback approximation for stochastic control with degenerate diffusions: error estimates and numerical analysis

We study finite-horizon stochastic optimal control problems and approximate the resulting time-discrete formulation by a direct policy-learning problem over neural-network feedback maps. We prove a quantitative convergence estimate, in an averaged sense, for the error between the time-discrete value and the value induced by an approximately optimized neural policy. The bound separates the approximation of near-optimal feedback policies, the localization of stochastic trajectories on compact sets, and the optimization tolerance in training. The analysis does not require transition-density assumptions and covers possibly degenerate diffusions and deterministic controlled dynamics in a unified framework. Numerical experiments are provided for a degenerate stochastic radial target problem, a Hamilton--Jacobi--Bellman benchmark, and a gas storage problem, illustrating the approach and separating the main error sources: time discretization, restriction to piecewise-constant policies, neural-network approximation, and Monte Carlo evaluation.

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arXiv2026

Solving Hamilton-Jacobi equations by residual minimization of monotone finite-difference discretizations

We introduce a method for solving Hamilton--Jacobi equations, both inviscid and viscous, by minimizing the squared residuals of monotone finite-difference discretizations on grids of varying resolution. The method is designed to leverage neural networks and modern GPUs to solve these equations in higher dimensions; consequently, the setting for our analysis is the minimization of the residual functionals via gradient-based optimization. We establish a well-posedness theory for this approach: any critical point of the finite-difference loss solves the monotone scheme together with the prescribed Dirichlet boundary conditions, and the error of an approximation is controlled by its residual. We then derive the rate of convergence of the gradient flow that minimizes the residual in several settings, noting that the rate may depend on the grid resolution and the domain dimension. Building on this foundation, we propose a multi-level training algorithm that exploits the faster convergence available on the coarser grids of the discretization. Combined with the convergence theorem of Barles and Souganidis for monotone and consistent schemes, our results guarantee convergence to the unique viscosity solution as the grid is refined. We illustrate the approach on eikonal equations, level-set problems, and a Hamilton--Jacobi--Isaacs equation arising from a stochastic differential game, in dimensions up to eight.

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arXiv2026

Numerical Approximation for Path-Dependent McKean-Vlasov Control with Non-Asymptotic Error Estimates

Path-dependent McKean--Vlasov (MKV) control models large interacting populations with history-dependent dynamics and costs. This paper develops a unified approximation-and-learning framework for continuous time path-dependent MKV problem under open-loop controls. First, an Euler discretization scheme with piecewise-constant controls is shown to achieve a non-asymptotic error of $O(h^{1/4})$. Second, we establish a discrete dynamic programming principle and prove value equivalence between open-loop and history-dependent feedback controls, enabling optimization on a reduced filtration. Third, an interacting particle system is introduced to approximate the continuous-time value, yielding an overall error bound of $O(h^{1/4}) + O(M^{-γ})$ for $M$ particles and an explicitly given $γ> 0$. Finally, we propose a fully implementable neural-network policy-gradient method using pathwise features. Numerical experiments, including a path-dependent linear-quadratic benchmark, demonstrate the effectiveness of the algorithm.

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arXiv2024

Representation results and error estimates for differential games with applications using neural networks

We study deterministic optimal control problems for differential games with finite horizon. We propose new approximations of the strategies in feedback form, and show error estimates and a convergence result of the value in some weak sense for one of the formulations. This result applies in particular to neural networks approximations. This work follows some ideas introduced in Bokanowski, Prost and Warin (PDEA, 2023) for deterministic optimal control problems, yet with a simplified approach for the error estimates, which allows to consider a global optimization scheme instead of a time-marching scheme. We also give a new approximation result between the continuous and the semi-discrete optimal control value in the game setting, improving the classical convergence order under some assumptions on the dynamical system. Numerical examples are performed on elementary academic problems related to backward reachability, with exact analytic solutions given, as well as a two-player game in presence of state constraints. We use stochastic gradient type algorithms in order to deal with the min-max problem.

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arXiv2022

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 Huré 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.

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arXiv2020

Backward Differentiation Formula finite difference schemes for diffusion equations with an obstacle term

Finite difference schemes, using Backward Differentiation Formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term, of the form $$\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v- φ(t,x))= f(t,x).$$ For the scheme building on the second order BDF formula (BDF2), we discuss unconditional stability, prove an $L^2$-error estimate and show numerically second order convergence, in both space and time, unconditionally on the ratio of the mesh steps. In the analysis, an equivalence of the obstacle equation with a Hamilton-Jacobi-Bellman equation is mentioned, and a Crank-Nicolson scheme is tested in this context. Two academic problems for parabolic equations with an obstacle term with explicit solutions and the American option problem in mathematical finance are used for numerical tests.

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arXiv2019

Computation of avoidance regions for driver assistance systems by using a Hamilton-Jacobi approach

We consider the problem of computing safety regions, modeled as nonconvex backward reachable sets, for a nonlinear car collision avoidance model with time-dependent obstacles. The Hamilton-Jacobi-Bellman framework is used. A new formulation of level set functions for obstacle avoidance is given and sufficient conditions for granting the obstacle avoidance on the whole time interval are obtained, even though the conditions are checked only at discrete times. Different scenarios including various road configurations, different geometry of vehicle and obstacles, as well as fixed or moving obstacles, are then studied and computed. Computations involve solving nonlinear partial differential equations of up to five space dimensions plus time with nonsmooth obstacle representations, and an efficient solver is used to this end. A comparison with a direct optimal control approach is also done for one of the examples.

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arXiv2018

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.

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arXiv2017

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.

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arXiv2016

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-Ampère 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.

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arXiv2015

Convergence of discontinuous Galerkin schemes for front propagation with obstacles

We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jacobi equation of the form $\min(u_t + c u_x, u - g(x))=0$, in one space dimension. New convergence results and error bounds are obtained for Lipschitz regular data. These "low regularity" assumptions are the natural ones for the solutions of the studied equations.

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arXiv2015

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 $ε$-monotone property. Convergence results and precise error estimates are given, of the order of $\sqrt{Δx}$ where $Δx$ 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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