A pre-processing procedure for the implementation of the greedy rank-one algorithm to solve high-dimensional linear systems

Abstract

[EN] Algorithms that use tensors are increasingly important due to the goodness of this operation when performing calculations with large amounts of data. Among them, we find the algorithms that search for the solution of a linear system in separated form, where the Greedy Rank-One Update method stands out, the starting point of the famous PGD family (from its acronym, Proper Generalized Decomposition).

When the matrices of these systems have the particular structure of a Laplacian-type matrix, the convergence of the previous methods is faster and more accurate. The Laplacian Decomposition Algorithm calculates the Laplacian matrix that best approximates a given square matrix. When the residue of this approximation is small, we will be able to solve the linear system associated with a Laplacian-type matrix and thus obtain an approximation of the solution of the original system, with a lower computational cost.

In this paper we prove that the discretization of a general Partial Differential Equation of the second order can be written as a linear system with a Laplacian-type matrix.