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

Olivier Bokanowski (1) , Athena Picarelli (2) , Christoph Reisinger (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 MI - Mathematical Institute [Oxford]

In this work, we present and analyse a class of " filtered " numerical schemes for second order Hamilton-Jacobi-Bellman (HJB) equations. The work follows the ideas recently introduced by Oberman and Froese [14] and applied by Bokanowski et al. [5] and Oberman and Salvador [24] to first order HJB 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 converge 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.

Mots clés: Filtered schemes; Hamilton-Jacobi-Bellman equations

Domaines: Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Olivier Bokanowski, Athena Picarelli, Christoph Reisinger. High-order filtered schemes for time-dependent second order HJB equations. 2016. ⟨hal-01368202v1⟩