Enhanced alternating energy minimization methods for stochastic galerkin matrix equations

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In uncertainty quantification, it is commonly required to solve a forward model consist- ing 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.
Original languageEnglish
Pages (from-to)965-994
Number of pages30
JournalBIT Numerical Mathematics
Issue number3
Publication statusPublished - 3 Jan 2022


  • Alternating energy minimization
  • Low-rank approximation
  • Matrix equations
  • PDEs with random inputs
  • Stochastic Galerkin methods
  • Uncertainty quantification


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