Stochastic Least-Squares Petrov--Galerkin Method for Parameterized Linear Systems
- 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
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.
- Research Organization:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of Maryland, College Park, MD (United States)
- Sponsoring Organization:
- USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
- Grant/Contract Number:
- AC04-94AL85000; SC0009301; DMS1418754
- OSTI ID:
- 1464179
- Alternate ID(s):
- OSTI ID: 1598443
- Report Number(s):
- SAND-2017-0035J; 650185
- Journal Information:
- SIAM/ASA Journal on Uncertainty Quantification, Vol. 6, Issue 1; ISSN 2166-2525
- Publisher:
- SIAMCopyright Statement
- Country of Publication:
- United States
- Language:
- English
Web of Science
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