A Low-Rank Solver for the Navier--Stokes Equations with Uncertain Viscosity

Abstract

We study an iterative low-rank approximation method for the solution of the steadystate 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.

Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations

“…31 Since the Picard iteration has a larger radius of convergence compared with the Newton iteration, our choice in this contribution is the Picard iteration. We apply the so-called Karush-Kuhn-Tucker procedure 31(chap8.2) to (6) and (7) to obtain the following linear optimality system: 36…”

“…A viable solution approach to optimization problems with stochastic constraints employs the spectral stochastic Galerkin finite element method (SGFEM). In particular, the steady‐state Navier‐Stokes equations were solved with SGFEM in References 4-7. However, this intrusive approachleads to the so‐called curse of dimensionality , in the sense that the number of expansion coefficients of the discrete solution grows exponentially in the number of random variables and quickly becomes intractable for direct calculation 2,8,9 …”

“…In the last decade, the use of SGFEM discretization with low‐rank approximations to solve high‐dimensional and parametrized problems was studied extensively (see References 2,7,17-19 and the references therein). In particular, References and approximate the solution in the canonical tensor format.…”

Interpolatory tensorial reduced order models for parametric dynamical systems

“…Despite an outstanding recent progress in numerical multi-linear algebra and, in particular, in understanding tensor decompositions (see, e.g., review articles [36,24,50]), tensor methods have yet to find their applications in reduced order modeling of dynamical systems. We mention two reports by Nouy [44,45], who reviewed tensor compressed formats and discussed their possible use for sparse function representation and reduced order modeling, as well as a series of publications on the treatment in compressed tensor formats of algebraic systems resulting from the stochastic Galerkin finite element method [10,6,7,38]. The authors of the survey [9] observe that "The combination of tensor calculus .…”

Inexact Methods for Symmetric Stochastic Eigenvalue Problems

“…The formulation of the Newton's method is closely related to that of [1], however we consider general parametrization of stochastic coefficients, the Jacobian matrices are symmetrized, and we propose a class of hierarchical preconditioners, which can be viewed as extensions of the hierarchical preconditioners used for the first method. We also note that we have recently successfully combined an inexact Newton-Krylov method with the stochastic Galerkin framework in a different context [19,34]. The performance of both methods is illustrated by numerical experiments, and the results are compared to that of MC and SC methods.…”