Stochastic Least-Squares Petrov--Galerkin Method for Parameterized Linear Systems

Lee, Kookjin; Carlberg, Kevin; Elman, Howard C.

doi:10.1137/17M1110729

Title: Stochastic Least-Squares Petrov--Galerkin Method for Parameterized Linear Systems

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Abstract

We consider the numerical solution of parameterized linear systems where the system matrix, the solution, and the right-hand side are parameterized by a set of uncertain input parameters. We explore spectral methods in which the solutions are approximated in a chosen finite- dimensional subspace. It has been shown that the stochastic Galerkin projection technique fails to minimize any measure of the solution error [20]. As a remedy for this, we propose a novel stochastic least-squares Petrov–Galerkin (LSPG) method. The proposed method is optimal in the sense that it produces the solution that minimizes a weighted l2-norm of the residual over all solutions in a given finite-dimensional subspace. Moreover, the method can be adapted to minimize the solution error in different weighted l2-norms by simply applying a weighting function within the least-squares formulation. In addition, a goal-oriented semi-norm induced by an output quantity of interest can be minimized by defining a weighting function as a linear functional of the solution. We establish optimality and error bounds for the proposed method, and extensive numerical experiments show that the weighted LSPG methods outperforms other spectral methods in minimizing corresponding target weighted norms.

Authors:

Lee, Kookjin ^[1]; Carlberg, Kevin ^[2]; Elman, Howard C. ^[3]

Univ. of Maryland, College Park, MD (United States). Dept. of Computer Science
Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Univ. of Maryland, College Park, MD (United States). Dept. of Computer Science and Inst. for Advanced Computer Studies

Publication Date:: Thu Mar 29 00:00:00 EDT 2018

Research Org.:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of Maryland, College Park, MD (United States)

Sponsoring Org.:: USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)

OSTI Identifier:: 1464179

Alternate Identifier(s):: OSTI ID: 1598443

Report Number(s):: SAND-2017-0035J
Journal ID: ISSN 2166-2525; 650185

Grant/Contract Number:: AC04-94AL85000; SC0009301; DMS1418754

Resource Type:: Accepted Manuscript

Journal Name:: SIAM/ASA Journal on Uncertainty Quantification

Additional Journal Information:: Journal Volume: 6; Journal Issue: 1; Journal ID: ISSN 2166-2525

Publisher:: SIAM

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; stochastic Galerkin; least-squares Petrov-Galerkin projection; residual minimization; spectral pro- jection; stochastic Galerkin, least-squares Petrov–Galerkin projection, residual minimiza- tion, spectral projection

Citation Formats


                    Lee, Kookjin, Carlberg, Kevin, and Elman, Howard C. Stochastic Least-Squares Petrov--Galerkin Method for Parameterized Linear Systems.  United States: N. p., 2018. 
Web.  doi:10.1137/17M1110729.

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                    Lee, Kookjin, Carlberg, Kevin, & Elman, Howard C. Stochastic Least-Squares Petrov--Galerkin Method for Parameterized Linear Systems.  United States.  https://doi.org/10.1137/17M1110729

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                    Lee, Kookjin, Carlberg, Kevin, and Elman, Howard C. Thu .  
"Stochastic Least-Squares Petrov--Galerkin Method for Parameterized Linear Systems".  United States.  https://doi.org/10.1137/17M1110729.  https://www.osti.gov/servlets/purl/1464179.

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@article{osti_1464179,

  title        = {Stochastic Least-Squares Petrov--Galerkin Method for Parameterized Linear Systems},

  author       = {Lee, Kookjin and Carlberg, Kevin and Elman, Howard C.},

  abstractNote = {We consider the numerical solution of parameterized linear systems where the system matrix, the solution, and the right-hand side are parameterized by a set of uncertain input parameters. We explore spectral methods in which the solutions are approximated in a chosen finite- dimensional subspace. It has been shown that the stochastic Galerkin projection technique fails to minimize any measure of the solution error [20]. As a remedy for this, we propose a novel stochastic least-squares Petrov–Galerkin (LSPG) method. The proposed method is optimal in the sense that it produces the solution that minimizes a weighted l2-norm of the residual over all solutions in a given finite-dimensional subspace. Moreover, the method can be adapted to minimize the solution error in different weighted l2-norms by simply applying a weighting function within the least-squares formulation. In addition, a goal-oriented semi-norm induced by an output quantity of interest can be minimized by defining a weighting function as a linear functional of the solution. We establish optimality and error bounds for the proposed method, and extensive numerical experiments show that the weighted LSPG methods outperforms other spectral methods in minimizing corresponding target weighted norms.},

  doi          = {10.1137/17M1110729},

  journal      = {SIAM/ASA Journal on Uncertainty Quantification},

  number       = 1,

  volume       = 6,

  place        = {United States},

  year         = {Thu Mar 29 00:00:00 EDT 2018},

  month        = {Thu Mar 29 00:00:00 EDT 2018}

}

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