Nonintrusive approximation of parametrized limits of matrix power algorithms -- application to matrix inverses and log-determinants

TL;DR

This paper introduces a nonintrusive method using the Empirical Interpolation Method to approximate parametrized matrix limits like inverses and log-determinants, with proven error bounds and competitive numerical performance.

Contribution

It develops a novel nonintrusive approximation technique for matrix limits based on EIM, applicable under affine parameter dependence, with theoretical error bounds and empirical validation.

Findings

The proposed method achieves accurate approximations of matrix inverses and log-determinants.

It outperforms classical nonintrusive techniques in test cases.

The approach provides theoretical error bounds for the approximations.

Abstract

We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones.