Low-Rank Solution Methods for Stochastic Eigenvalue Problems

Howard C. Elman; Tengfei Su

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arXiv · 1803.03717

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

Howard C. Elman·Tengfei Su

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Abstract

We study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data. With the stochastic Galerkin approach, the solutions are represented as generalized polynomial chaos expansions. A low-rank variant of the inverse subspace iteration algorithm is presented for computing one or several minimal eigenvalues and corresponding eigenvectors of parameter-dependent matrices. In the algorithm, the iterates are approximated by low-rank matrices, which leads to significant cost savings. The algorithm is tested on two benchmark problems, a stochastic diffusion problem with some poorly separated eigenvalues, and an operator derived from a discrete stochastic Stokes problem whose minimal eigenvalue is related to the inf-sup stability constant. Numerical experiments show that the low-rank algorithm produces accurate solutions compared to the Monte Carlo method, and it uses much less computational time than the original algorithm without low-rank approximation.

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BibTeX RIS

Howard C. Elman, Tengfei Su. 2018-03-09. Low-Rank Solution Methods for Stochastic Eigenvalue Problems. https://arxiv.org/abs/1803.03717

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