Nonintrusive approximation of parametrized limits of matrix power algorithms -- application to matrix inverses and log-determinants
For scientists and engineers needing fast approximations of matrix functions in parametric settings, this offers a nonintrusive alternative, though it is incremental over existing EIM-based methods.
The paper proposes a nonintrusive method using the Empirical Interpolation Method to approximate limits of matrix power algorithms, such as matrix inverses and log-determinants, under affine parameter dependence. Numerical tests show the algorithm performs well compared to other nonintrusive techniques.
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.