Low-Rank Solution Methods for Stochastic Eigenvalue Problems
This work addresses the computational bottleneck of solving stochastic eigenvalue problems for engineers and scientists, offering a more efficient method for parameter-dependent matrices.
The paper presents a low-rank variant of the inverse subspace iteration algorithm for computing minimal eigenvalues and eigenvectors of parameter-dependent matrices in stochastic eigenvalue problems, achieving accurate solutions with significantly reduced computational time compared to the original algorithm and Monte Carlo method.
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.