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

## TL;DR

This paper introduces a low-rank iterative solver for stochastic unsteady Navier-Stokes equations, significantly reducing computational costs by using tensor representations and effective preconditioning.

## Contribution

It develops a novel low-rank Krylov subspace method with mean-based preconditioning for efficient all-at-once stochastic Navier-Stokes simulations.

## Key findings

- Achieves significant reductions in storage and computational costs.
- Requires only a small number of linear iterations per Picard step.
- Demonstrates efficiency on a 2D flow model.

## 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.