A Low-Rank Solver for the Navier--Stokes Equations with Uncertain Viscosity
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2019-10-31
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Lee, Kookjin; Elman, Howard C.; Sousedik, Bedrich; A Low-Rank Solver for the Navier--Stokes Equations with Uncertain Viscosity; SIAM/ASA Journal on Uncertainty Quantification, 7(4), 1275-1300, 31 October, 2019; https://doi.org/10.1137/17M1151912
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© 2019, Society for Industrial and Applied Mathematics and American Statistical Association.
© 2019, Society for Industrial and Applied Mathematics and American Statistical Association.
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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.