$\mathcal{H}_2$ Pseudo-Optimal Reduction of Structured DAEs by Rational Interpolation

TL;DR

This paper extends $ abla_2$ pseudo-optimal model reduction techniques to structured differential-algebraic equations (DAEs), providing algorithms that guarantee stability and adaptively optimize reduced models using rational interpolation.

Contribution

It introduces a novel $ abla_2$ inner product framework for DAEs and develops an adaptive rational Krylov algorithm ensuring stability and optimality in model reduction.

Findings

The new method guarantees stability preservation.

It adaptively selects interpolation frequencies.

Numerical examples demonstrate effectiveness.

Abstract

In this contribution, we extend the concept of $\mathcal{H}_2$ inner product and $\mathcal{H}_2$ pseudo-optimality to dynamical systems modeled by differential-algebraic equations (DAEs). To this end, we derive projected Sylvester equations that characterize the $\mathcal{H}_2$ inner product in terms of the matrices of the DAE realization. Using this result, we extend the $\mathcal{H}_2$ pseudo-optimal rational Krylov algorithm for ordinary differential equations to the DAE case. This algorithm computes the globally optimal reduced-order model for a given subspace of $\mathcal{H}_2$ defined by poles and input residual directions. Necessary and sufficient conditions for $\mathcal{H}_2$ pseudo-optimality are derived using the new formulation of the $\mathcal{H}_2$ inner product in terms of tangential interpolation conditions. Based on these conditions, the cumulative reduction procedure combined with the adaptive rational Krylov algorithm, known as CUREd SPARK, is extended to DAEs. Important properties of this procedure are that it guarantees stability preservation and adaptively selects interpolation frequencies and reduced order. Numerical examples are used to illustrate the theoretical discussion. Even though the results apply in theory to general DAEs, special structures will be exploited for numerically efficient implementations.