A nonintrusive method to approximate linear systems with nonlinear parameter dependence

TL;DR

This paper introduces a nonintrusive approach using Empirical Interpolation to approximate parameter-dependent linear systems, enabling efficient reduced basis methods without modifying underlying code.

Contribution

It presents a novel nonintrusive procedure for separated representation of system matrices based on Empirical Interpolation, applicable to boundary-value and scattering problems.

Findings

Effective approximation of system matrices demonstrated on boundary-value problems.

Applicable to high-dimensional acoustic scattering problems.

Outperforms traditional intrusive methods in ease of implementation.

Abstract

We consider a family of linear systems $A_\mu \alpha=C$ with system matrix $A_\mu$ depending on a parameter $\mu$ and for simplicity parameter-independent right-hand side $C$. These linear systems typically result from the finite-dimensional approximation of a parameter-dependent boundary-value problem. We derive a procedure based on the Empirical Interpolation Method to obtain a separated representation of the system matrix in the form $A_\mu\approx\sum_{m}\beta_m(\mu)A_{\mu_m}$ for some selected values of the parameter. Such a separated representation is in particular useful in the Reduced Basis Method. The procedure is called nonintrusive since it only requires to access the matrices $A_{\mu_m}$. As such, it offers a crucial advantage over existing approaches that instead derive separated representations requiring to enter the code at the level of assembly. Numerical examples illustrate the performance of our new procedure on a simple one-dimensional boundary-value problem and on three-dimensional acoustic scattering problems solved by a boundary element method.