A Low-rank Solver for the Stochastic Unsteady Navier-Stokes Problem

Abstract

We study a low-rank iterative solver for the unsteady Navier-Stokes equations for incompressible flows with a stochastic viscosity. The equations are discretized using the stochastic Galerkin method, and we consider an all-at-once formulation where the algebraic systems at all the time steps are collected and solved simultaneously. The problem is linearized with Picard's method. To efficiently solve the linear systems at each step, we use low-rank tensor representations within the Krylov subspace method, which leads to significant reductions in storage requirements and computational costs. Combined with effective mean-based preconditioners and the idea of inexact solve, we show that only a small number of linear iterations are needed at each Picard step. The proposed algorithm is tested with a model of flow in a two-dimensional symmetric step domain with different settings to demonstrate the computational efficiency.

References in corpus (5)

• The density-matrix renormalization group
• Alternating minimal energy methods for linear systems in higher dimensions. Part II: Faster algorithm and application to nonsymmetric systems
• Alternating minimal energy methods for linear systems in higher dimensions. Part I: SPD systems
• Stochastic Galerkin methods for the steady-state Navier-Stokes equations
• Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods

Cited by in corpus (3)

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