Athena Picarelli

Simone Cacace, Emiliano Cristiani, Maurizio Falcone et al.

In this paper we present a new algorithm for the solution of Hamilton-Jacobi-Bellman equations related to optimal control problems. The key idea is to divide the domain of computation into subdomains which are shaped by the optimal dynamics of the underlying control problem. This can result in a rather complex geometrical subdivision, but it has the advantage that every subdomain is invariant with respect to the optimal dynamics, and then the solution can be computed independently in each subdomain. The features of this dynamics-dependent domain decomposition can be exploited to speed up the computation and for an efficient parallelization, since the classical transmission conditions at the boundaries of the subdomains can be avoided. For their properties, the subdomains are patches in the sense introduced by Ancona and Bressan [ESAIM Control Optim. Calc. Var., 4 (1999), pp. 445-471]. Several examples in two and three dimensions illustrate the properties of the new method.

Olivier Bokanowski, Athena Picarelli, Christoph Reisinger

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.

Olivier Bokanowski, Athena Picarelli, Christoph Reisinger

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.