Interpolation of inverse operators for preconditioning parameter-dependent equations

TL;DR

This paper introduces a novel interpolation-based preconditioning method for parameter-dependent matrices, utilizing Frobenius norm approximations and adaptive strategies to enhance large system solutions.

Contribution

It presents a new approach for constructing preconditioners through inverse matrix interpolation, incorporating random matrix-based Frobenius norm estimations and adaptive point selection strategies.

Findings

Effective preconditioners for large parameter-dependent systems.

High-probability quasi-optimal Frobenius norm estimators.

Adaptive interpolation points improve model order reduction.

Abstract

We propose a method for the construction of preconditioners of parameter-dependent matrices for the solution of large systems of parameter-dependent equations. The proposed method is an interpolation of the matrix inverse based on a projection of the identity matrix with respect to the Frobenius norm. Approximations of the Frobenius norm using random matrices are introduced in order to handle large matrices. The resulting statistical estimators of the Frobenius norm yield quasi-optimal projections that are controlled with high probability. Strategies for the adaptive selection of interpolation points are then proposed for different objectives in the context of projection-based model order reduction methods: the improvement of residual-based error estimators, the improvement of the projection on a given reduced approximation space, or the recycling of computations for sampling based model order reduction methods.