Model Reduction Based on Proper Generalized Decomposition for the Stochastic Steady Incompressible Navier-Stokes Equations

In this paper we consider a proper generalized decomposition method to solve the steady incompressible Navier--Stokes equations with random Reynolds number and forcing term. The aim of such a technique is to compute a low-cost reduced basis approximation of the full stochastic Galerkin solution of the problem at hand. A particular algorithm, inspired by the Arnoldi method for solving eigenproblems, is proposed for an efficient greedy construction of a deterministic reduced basis approximation. This algorithm decouples the computation of the deterministic and stochastic components of the solution, thus allowing reuse of preexisting deterministic Navier--Stokes solvers. It has the remarkable property of only requiring the solution of $m$ uncoupled deterministic problems for the construction of an $m$-dimensional reduced basis rather than $M$ coupled problems of the full stochastic Galerkin approximation space, with $m \ll M$ (up to one order of magnitude for the problem at hand in this work).

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