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Title: A Low-Rank Solver for the Navier--Stokes Equations with Uncertain Viscosity

Abstract

In this work, we study an iterative low-rank approximation method for the solution of the steady-state stochastic Navier--Stokes equations with uncertain viscosity. The method is based on linearization schemes using Picard and Newton iterations and stochastic finite element discretizations of the linearized problems. For computing the low-rank approximate solution, we adapt the nonlinear iterations to an inexact and low-rank variant, where the solution of the linear system at each nonlinear step is approximated by a quantity of low rank. This is achieved by using a tensor variant of the GMRES method as a solver for the linear systems. We explore the inexact low-rank nonlinear iteration with a set of benchmark problems, using a model of flow over an obstacle, under various configurations characterizing the statistical features of the uncertain viscosity, and we demonstrate its effectiveness by extensive numerical experiments.

Authors:
 [1]; ORCiD logo [2]; ORCiD logo [3]
  1. Univ. of Maryland, College Park, MD (United States). Dept. of Computer Science; Sandia National Lab. (SNL-CA), Livermore, CA (United States).Extreme-scale Data Science and Analytics Dept.
  2. Univ. of Maryland, College Park, MD (United States). Dept. of Computer Science, and Inst. for Advanced Computer Studies
  3. Univ. of Maryland Baltimore County (UMBC), Baltimore, MD (United States). Dept. of Mathematics and Statistics
Publication Date:
Research Org.:
Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1574809
Report Number(s):
SAND2019-6127J
Journal ID: ISSN 2166-2525; 676153
Grant/Contract Number:  
AC04-94AL85000; C-SC0009301
Resource Type:
Accepted Manuscript
Journal Name:
SIAM/ASA Journal on Uncertainty Quantification
Additional Journal Information:
Journal Volume: 7; Journal Issue: 4; Journal ID: ISSN 2166-2525
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; stochastic Galerkin method; Navier-Stokes equations; low-rank approximation

Citation Formats

Lee, Kookjin, Elman, Howard C., and Sousedík, Bedřich. A Low-Rank Solver for the Navier--Stokes Equations with Uncertain Viscosity. United States: N. p., 2019. Web. doi:10.1137/17M1151912.
Lee, Kookjin, Elman, Howard C., & Sousedík, Bedřich. A Low-Rank Solver for the Navier--Stokes Equations with Uncertain Viscosity. United States. doi:10.1137/17M1151912.
Lee, Kookjin, Elman, Howard C., and Sousedík, Bedřich. Thu . "A Low-Rank Solver for the Navier--Stokes Equations with Uncertain Viscosity". United States. doi:10.1137/17M1151912.
@article{osti_1574809,
title = {A Low-Rank Solver for the Navier--Stokes Equations with Uncertain Viscosity},
author = {Lee, Kookjin and Elman, Howard C. and Sousedík, Bedřich},
abstractNote = {In this work, we study an iterative low-rank approximation method for the solution of the steady-state stochastic Navier--Stokes equations with uncertain viscosity. The method is based on linearization schemes using Picard and Newton iterations and stochastic finite element discretizations of the linearized problems. For computing the low-rank approximate solution, we adapt the nonlinear iterations to an inexact and low-rank variant, where the solution of the linear system at each nonlinear step is approximated by a quantity of low rank. This is achieved by using a tensor variant of the GMRES method as a solver for the linear systems. We explore the inexact low-rank nonlinear iteration with a set of benchmark problems, using a model of flow over an obstacle, under various configurations characterizing the statistical features of the uncertain viscosity, and we demonstrate its effectiveness by extensive numerical experiments.},
doi = {10.1137/17M1151912},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
number = 4,
volume = 7,
place = {United States},
year = {2019},
month = {10}
}

Journal Article:
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This content will become publicly available on October 31, 2020
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