To be or not to be intrusive? the solution of parametric and stochastic equations - Proper generalized decomposition

Abstract

A numerical method is proposed to compute a low-rank Galerkin approximation to the solution of a parametric or stochastic equation in a nonintrusive 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 nonintrusiveness, 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 nonintrusive 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.