Stochastic Least-Squares Petrov-Galerkin Method for Parameterized Linear Systems

TL;DR

This paper introduces a novel stochastic least-squares Petrov-Galerkin method for parameterized linear systems, which optimally minimizes residuals and errors in weighted norms, outperforming traditional spectral methods.

Contribution

The paper proposes a new LSPG method that minimizes weighted residuals and errors, with adaptability to different norms and output quantities, and provides theoretical analysis and numerical validation.

Findings

Weighted LSPG outperforms spectral methods in minimizing target norms.

The method can be tailored to minimize errors in various weighted norms.

Theoretical error bounds and optimality are established.

Abstract

We consider the numerical solution of parameterized linear systems where the system matrix, the solution, and the right-hand side are parameterized by a set of uncertain input parameters. We explore spectral methods in which the solutions are approximated in a chosen finite-dimensional subspace. It has been shown that the stochastic Galerkin projection technique fails to minimize any measure of the solution error [20]. As a remedy for this, we propose a novel stochastic least-squares Petrov--Galerkin (LSPG) method. The proposed method is optimal in the sense that it produces the solution that minimizes a weighted l2-norm of the residual over all solutions in a given finite-dimensional subspace. Moreover, the method can be adapted to minimize the solution error in different weighted l2-norms by simply applying a weighting function within the least-squares formulation. In addition, a goal-oriented semi-norm induced by an output quantity of interest can be minimized by defining a weighting function as a linear functional of the solution. We establish optimality and error bounds for the proposed method, and extensive numerical experiments show that the weighted LSPG methods outperforms other spectral methods in minimizing corresponding target weighted norms.