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.L 2 (D) ⊗ L 2 (Γ) := q : D × Γ → R | E q 2 L 2 (D) < ∞ .

The physics informed neural networks for the unsteady Stokes problems

“…Partial differential equations (PDEs) are used to model a variety of phenomena in natural science. As a broad‐spectrum and significant topic in science and engineering, the Stokes problem has attracted a lot of attention 1‐16 . Common to most of the PDEs encountered in practical applications is that they cannot be solved analytically but require various approximation techniques.…”

Stochastic Discontinuous Galerkin Methods with Low--Rank Solvers for Convection Diffusion Equations

“…We address this issue by using low-rank Krylov subspace methods, which reduce both the storage requirements and the computational complexity by exploiting a Kroneckerproduct structure of system matrices, see, e.g., [6,26,40]. Similar approaches have been used to solve steady stochastic diffusion equations [12,27,34], unsteady stochastic diffusion equations [7], and stochastic Navier-Stokes equations [15,28]. In the aforementioned studies, randomness is generally defined in the diffusion parameter however we here consider the randomness both in diffusion or convection parameters.…”

“…Now, we first find a bound for the first term of (B.1). By the coercivity of the bilinear form (17a), the Galerkin orthogonality, an integration by parts over convective term in the bilinear form (15), and the assumption on the convective term ∇ • b = 0, we obtain…”