$\mathcal{L}_2$-optimal Reduced-order Modeling Using Parameter-separable Forms

TL;DR

This paper introduces a unifying, data-driven framework for $ abla$-optimal reduced-order modeling of linear systems, enabling efficient approximation using output samples and gradient-based optimization.

Contribution

It develops a gradient-based, non-intrusive algorithm for $ abla$-optimal reduced-order modeling using parameter-separable forms, applicable to both continuous and discrete cost functions.

Findings

Effective in constructing reduced models from output data

Applicable to a wide range of cost functions

Numerical examples demonstrate the method's efficacy

Abstract

We provide a unifying framework for $\mathcal{L}_2$-optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive the gradients of the $\mathcal{L}_2$ cost function with respect to the reduced matrices, which then allows a non-intrusive, data-driven, gradient-based descent algorithm to construct the optimal approximant using only output samples. By choosing an appropriate measure, the framework covers both continuous (Lebesgue) and discrete cost functions. We show the efficacy of the proposed algorithm via various numerical examples. Furthermore, we analyze under what conditions the data-driven approximant can be obtained via projection.