Olivier Bokanowski

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This paper deals with convergence results of Howard’s algorithm for the resolution of mina∈A(B x − b) = 0 where B is a matrix, b is a vector (possibly of infinite dimension), and A is a compact set. We show a global super-linear convergence result, under a monotonicity assumption on the matrices B. In the particular case of an obstacle problem of the form(More)
We consider a target problem for a nonlinear system under state constraints. We give a new continuous level-set approach for characterizing the optimal times and the backward-reachability sets. This approach leads to a characterization via a Hamilton-Jacobi equation, without assuming any controllability assumption. We also treat the case of time-dependent(More)
We propose two new antidiffusive schemes for advection (or linear transport), one of them being a mixture of Roe’s Super-Bee scheme and of the “Ultra-Bee” scheme. We show how to apply these schemes to treat time-dependent first order Hamilton-Jacobi-Bellman equations with discontinuous initial data, possibly infinitely-valued. Numerical tests are proposed,(More)
This paper is concerned with the numerical approximation of viability kernels. The method described here provides an alternative approach to the usual viability algorithm. We first consider a characterization of the viability kernel as the value function of a related optimal control problem, and then use a specially relevant numerical scheme for its(More)
We consider minimal time problems governed by nonlinear systems under general time dependent state constraints and in the twoplayer 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 controllability assumption is made. In(More)
We study a superreplication problem of European options with gamma constraints, in mathematical finance. The initially unbounded control problem is set back to a problem involving a viscosity PDE solution with a set of bounded controls. Then a numerical approach is introduced, inconditionnally stable with respect to the mesh steps. A generalized finite(More)
We study a mixed–state Schrödinger–Poisson–Slater system (SPSS). This system combines the nonlinear and nonlocal Coulomb interaction with a local potential nonlinearity known as the ”Slater exchange term” which models a fermionic effect. The origin of the model is explained and related models are also proposed. Existence, uniqueness and regularity of(More)
Explicit, unconditionally stable, high-order schemes for the approximation of some firstand second-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,(More)
This paper is intended to constitute a review of some mathematical theories incorporating quantum corrections to the Schrödinger-Poisson (SP) system. More precisely we shall focus our attention in the electrostatic Poisson potential with corrections of power type. The SP system is a simple model used for the study of quantum transport in semiconductor(More)
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(More)