A greedy approach to multiscale methods

In this paper we study the problem of compute the solution of a linear system in a separable representation form. It arises in the discretized equations appearing in various physical domains, such as kinetic theory, statistical mechanics, quantum mechanics, and in nanoscience and nanotechnology among others. In particular, we use the fact that tensors of order 3 or higher have best rank-1 approximation. This fact allow to us to propose an iterative method based in the so-called by the signal processing community as the Matching Pursuit Algorithm, also known as Projection Pursuit by the statistics community or as a Pure Greedy Algorithm in the approximation theory community. We also give some numerical examples and describe its relationship with the Finite Element Method for High-Dimensional Partial Differential Equations based on the tensorial product of one-dimensional bases. We illustrate this situation taking as a model problem the multidimensional Poisson equation with homogeneous Dirichlet boundary condition.