TY - JOUR

T1 - Enhanced alternating energy minimization methods for stochastic galerkin matrix equations

AU - Lee, Kookjin

AU - Elman, Howard C.

AU - Powell, Catherine E.

AU - Lee, Dongeun

N1 - Funding Information:
This work was supported by the U.S. Department of Energy Office of Advanced Scientific Computing Research, Applied Mathematics program under award DE-SC0009301 and by the U.S. National Science Foundation under grant DMS1819115
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature B.V.

PY - 2022

Y1 - 2022

N2 - In uncertainty quantification, it is commonly required to solve a forward model consisting of a partial differential equation (PDE) with a spatially varying uncertain coefficient that is represented as an affine function of a set of random variables, or parameters. Discretizing such models using stochastic Galerkin finite element methods (SGFEMs) leads to very high-dimensional discrete problems that can be cast as linear multi-term matrix equations (LMTMEs). We develop efficient computational methods for approximating solutions of such matrix equations in low rank. To do this, we follow an alternating energy minimization (AEM) framework, wherein the solution is represented as a product of two matrices, and approximations to each component are sought by solving certain minimization problems repeatedly. Inspired by proper generalized decomposition methods, the iterative solution algorithms we present are based on a rank-adaptive variant of AEM methods that successively computes a rank-one solution component at each step. We introduce and evaluate new enhancement procedures to improve the accuracy of the approximations these algorithms deliver. The efficiency and accuracy of the enhanced AEM methods is demonstrated through numerical experiments with LMTMEs associated with SGFEM discretizations of parameterized linear elliptic PDEs.

AB - In uncertainty quantification, it is commonly required to solve a forward model consisting of a partial differential equation (PDE) with a spatially varying uncertain coefficient that is represented as an affine function of a set of random variables, or parameters. Discretizing such models using stochastic Galerkin finite element methods (SGFEMs) leads to very high-dimensional discrete problems that can be cast as linear multi-term matrix equations (LMTMEs). We develop efficient computational methods for approximating solutions of such matrix equations in low rank. To do this, we follow an alternating energy minimization (AEM) framework, wherein the solution is represented as a product of two matrices, and approximations to each component are sought by solving certain minimization problems repeatedly. Inspired by proper generalized decomposition methods, the iterative solution algorithms we present are based on a rank-adaptive variant of AEM methods that successively computes a rank-one solution component at each step. We introduce and evaluate new enhancement procedures to improve the accuracy of the approximations these algorithms deliver. The efficiency and accuracy of the enhanced AEM methods is demonstrated through numerical experiments with LMTMEs associated with SGFEM discretizations of parameterized linear elliptic PDEs.

KW - Alternating energy minimization

KW - Low-rank approximation

KW - Matrix equations

KW - PDEs with random inputs

KW - Stochastic Galerkin methods

KW - Uncertainty quantification

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U2 - 10.1007/s10543-021-00903-x

DO - 10.1007/s10543-021-00903-x

M3 - Article

AN - SCOPUS:85122242203

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

ER -