A Low-rank solver for the Navier--Stokes equations with uncertain viscosity

TL;DR

This paper introduces an iterative low-rank approximation method for solving steady-state stochastic Navier--Stokes equations with uncertain viscosity, combining linearization, stochastic finite element discretization, and tensor GMRES to efficiently handle uncertainty.

Contribution

It develops a novel inexact low-rank nonlinear iteration method using tensor GMRES for efficient stochastic Navier--Stokes solutions with uncertain viscosity.

Findings

Effective low-rank approximation achieved in benchmark problems.

Method handles various statistical configurations of viscosity.

Numerical experiments demonstrate computational efficiency and accuracy.

Abstract

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.