Preconditioners for parameter-dependent equations and projection-based model order reduction methods

We consider large systems of parameter-dependent equations, e.g. arising from the discretization of PDEs with random coefficients. We propose an interpolation of matrix inverse based on a projection of the identity matrix with respect to the Frobenius norm. The use of randomized linear algebra allow for handling large matrices. Adaptive interpolation strategies are then proposed for two different objectives in the context of projection-based model reduction methods: the improvement of residual-based error estimators and the improvement of Galerkin-type projections on a given reduced approximation space. We finally present a preconditioned version of the standard greedy algorithm for Reduced Basis methods. The preconditioner is used for the computation of Petrov-Galerkin projections onto successive reduced approximation spaces, and also for the computation of the error estimator which is used in the selection of successive elements of the reduced basis. The preconditioner can be updated at each iteration using as a new interpolation point the point selected by the greedy algorithm. The resulting algorithm presents asymptotically the performance of an ideal greedy algorithm applied to the solution itself. For a given accuracy of the generated reduced order model, this approach allows reducing both offline and online computational costs of standard reduced basis methods.