To be or not to be intrusive? The solution of parametric and stochastic equations --- Proper Generalized Decomposition

TL;DR

This paper introduces a non-intrusive numerical method for efficiently approximating solutions to parametric and stochastic equations using low-rank tensor decompositions and an alternating minimization scheme.

Contribution

It presents a novel non-intrusive approach combining low-rank tensor approximation with quasi-Newton minimization for parametric equations.

Findings

Effective low-rank approximation demonstrated on numerical examples.

Non-intrusive method avoids Hessian computations, reducing complexity.

Applicable to nonlinear, convex functional problems.

Abstract

A numerical method is proposed to compute a low-rank Galerkin approximation to the solution of a parametric or stochastic equation in a non-intrusive fashion. The considered nonlinear problems are associated with the minimization of a parameterized differentiable convex functional. We first introduce a bilinear parameterization of fixed-rank tensors and employ an alternating minimization scheme for computing the low-rank approximation. In keeping with the idea of non-intrusiveness, at each step of the algorithm the minimizations are carried out with a quasi-Newton method to avoid the computation of the Hessian. The algorithm is made non-intrusive through the use of numerical integration. It only requires the evaluation of residuals at specific parameter values. The algorithm is then applied to two numerical examples.