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arXiv · 1405.0875

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

Loïc Giraldi·Dishi Liu·Hermann G. Matthies·Anthony Nouy
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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.

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Loïc Giraldi, Dishi Liu, Hermann G. Matthies, Anthony Nouy. 2014-05-05. To be or not to be intrusive? The solution of parametric and stochastic equations --- Proper Generalized Decomposition. https://doi.org/10.1137/140969063

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