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

Journal Article · Thu Oct 31 00:00:00 EDT 2019 · SIAM/ASA Journal on Uncertainty Quantification

DOI:https://doi.org/10.1137/17M1151912· OSTI ID:1574809

Lee, Kookjin ^[1]; Elman, Howard C. ^[2]; Sousedík, Bedřich ^[3]

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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

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.

• Research Organization: Sandia National Laboratories (SNL-CA), Livermore, CA (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) (SC-21)
• Grant/Contract Number: AC04-94AL85000; SC0009301
• OSTI ID: 1574809
• Alternate ID(s): OSTI ID: 1598320
• Report Number(s): SAND2019--6127J; 676153
• Journal Information: SIAM/ASA Journal on Uncertainty Quantification, Journal Name: SIAM/ASA Journal on Uncertainty Quantification Journal Issue: 4 Vol. 7; ISSN 2166-2525
• Publisher: SIAM
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
Navier-Stokes equations
low-rank approximation
stochastic Galerkin method
stochastic Galerkin method
Navier–Stokes equations
low-rank approximation