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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2017
- DOI:
- 10.48550/arXiv.1710.05812
- arXiv:
- arXiv:1710.05812
- Bibcode:
- 2017arXiv171005812L
- Keywords:
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- Mathematics - Numerical Analysis
- E-Print:
- SIAM/ASA Journal on Uncertainty Quantification, 7(4), 1275-1300, 2019