# Reduced Basis Solvers for Stochastic Galerkin Matrix Equations

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UNQW03 - Reducing dimensions and cost for UQ in complex systems

In the applied mathematics community, reduced basis methods are typically used to reduce the computational cost of applying sampling methods to parameter-dependent partial differential equations (PDEs). When dealing with PDE models in particular, repeatedly running computer models (eg finite element solvers) for many choices of the input parameters, is computationally infeasible. The cost of obtaining each sample of the numerical solution is instead sought by projecting the so-called high fidelity problem into a reduced (lower-dimensional) space. The choice of reduced space is crucial in balancing cost and overall accuracy. In this talk, we do not consider sampling methods. Rather, we consider stochastic Galerkin finite element methods (SGFEMs) for parameter-dependent PDEs. Here, the idea is to approximate the solution to the PDE model as a function of the input parameters. We combine finite element approximation in physical space, with global polynomial approximation on the parameter domain. In the statistics community, the term intrusive polynomial chaos approximation is often used. Unlike samping methods, which require the solution of many deterministic problems, SGFE Ms yield a single very large linear system of equations with coefficient matrices that have a characteristic Kronecker product structure. By reformulating the systems as multiterm linear matrix equations, we have developed [see: C.E. Powell, D. Silvester, V.Simoncini, An efficient reduced basis solver for stochastic Galerkin matrix equations, SIAM J . Comp. Sci. 39(1), pp A141 -A163 (2017)] a memory-efficient solution algorithm which generalizes ideas from rational Krylov subspace approximation (which are known in the linear algebra community). The new approach determines a low-rank approximation to the solution matrix by performing a projection onto a reduced space that is iteratively augmented with problem-specific basis vectors. Crucially, it requires far less memory than standard iterative methods applied to the Kronecker formulation of the linear systems. For test problems consisting of elliptic PDEs, and indefinite problems with saddle point structure, we are able to solve systems of billions of equations on a standard desktop computer quickly and efficiently.

This talk is part of the Isaac Newton Institute Seminar Series series.

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