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# 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:

- 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.
- Univ. of Maryland, College Park, MD (United States). Dept. of Computer Science, and Inst. for Advanced Computer Studies
- 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}

}