2016/05/17 by Kookjin Lee, Howard C. Elman, Lee, Kookjin +1
Earth and Planetary Sciences · Mathematics · #FOS: Mathematics #Geophysics and Gravity Measurements #Numerical Analysis (math.NA) #Statistical and numerical algorithms #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1605.05297
openalex publication_date 2016/05/17 · openalex created_date 2022/10/01 · openalex updated_date 2026/08/01
In this study, we consider the numerical solution of large systems of linear\nequations obtained from the stochastic Galerkin formulation of stochastic\npartial differential equations. We propose an iterative algorithm that exploits\nthe Kronecker product structure of the linear systems. The proposed algorithm\nefficiently approximates the solutions in low-rank tensor format. Using\nstandard Krylov subspace methods for the data in tensor format is\ncomputationally prohibitive due to the rapid growth of tensor ranks during the\niterations. To keep tensor ranks low over the entire iteration process, we\ndevise a rank-reduction scheme that can be combined with the iterative\nalgorithm. The proposed rank-reduction scheme identifies an important subspace\nin the stochastic domain and compresses tensors of high rank on-the-fly during\nthe iterations. The proposed reduction scheme is a multilevel method in that\nthe important subspace can be identified inexpensively in a coarse spatial grid\nsetting. The efficiency of the proposed method is illustrated by numerical\nexperiments on benchmark problems.\n