Exploiting compression in solving discretized linear systems

Abstract

We propose a method for exploiting compression in computing the solution to a system of linear algebraic equations. The method is based on computing an approximate solution in a reduced space, and thus we seek a basis in which the solution has a compressed representation and can consequently be computed more efficiently. Although the method is completely general, it is especially effective for linear systems resulting from discretization of an underlying continuous problem, which will be our main focus. We address three primary issues: (1) how to compute an approximate solution to a given linear system using a given basis, (2) how to choose a basis that will yield significant compression, and (3) how to detect when the chosen basis is of sufficient dimension to provide a satisfactory approximation. While all three aspects have antecedents in previous ideas and methods, we combine, adapt, and extend them in a manner we believe to be novel for the purpose of solving discretized linear systems. We demonstrate that the resulting method can be competitive with—and often substantially outperforms—current standard methods and is effective for efficiently solving linear systems resulting from the discretization of major classes of continuous problems, including both differential equations and integral equations.