A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with residual norm corresponding to the error in a specified solution norm. We introduce and analyze an iterative algorithm that is able to provide a controlled approximation of the optimal approximation of the solution in a given low-rank subset, without any a priori information on this solution. We also introduce a weak greedy algorithm which uses this perturbed minimal residual method for the computation of successive greedy corrections in small tensor subsets. We prove its convergence under some conditions on the parameters of the algorithm. The residual norm can be designed such that the resulting low-rank approximations are quasi-optimal with respect to particular norms of interest, thus yielding to goal-oriented order reduction strategies for the approximation of high-dimensional problems. The proposed numerical method is applied to the solution of a stochastic partial differential equation which is discretized using standard Galerkin methods in tensor product spaces

A unified approach for the solution of the Fokker-Planck equation

This paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the Fokker-Planck equation. The unified features of the discrete singular convolution algorithm are discussed. It is demonstrated that different implementations of the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. Three benchmark stochastic systems, the repulsive Wong process, the Black-Scholes equation and a genuine nonlinear model, are employed to illustrate the robustness and to test accuracy of the present approach for the solution of the Fokker-Planck equation via a time-dependent method. An additional example, the incompressible Euler equation, is used to further validate the present approach for more difficult problems. Numerical results indicate that the present unified approach is robust and accurate for solving the Fokker-Planck equation.Comment: 19 page

Multigrid methods for two-player zero-sum stochastic games

We present a fast numerical algorithm for large scale zero-sum stochastic games with perfect information, which combines policy iteration and algebraic multigrid methods. This algorithm can be applied either to a true finite state space zero-sum two player game or to the discretization of an Isaacs equation. We present numerical tests on discretizations of Isaacs equations or variational inequalities. We also present a full multi-level policy iteration, similar to FMG, which allows to improve substantially the computation time for solving some variational inequalities.Comment: 31 page