Large-Scale Computation of ${\mathcal L}_\infty$-Norms by a Greedy Subspace Method

TL;DR

This paper introduces a greedy subspace method for efficiently computing the ${\mathcal L}_\infty$-norm of large-scale matrix-valued functions, with proven local superlinear convergence and demonstrated effectiveness on numerical examples.

Contribution

The paper presents a novel subspace projection algorithm that reduces computational complexity while accurately approximating the ${\mathcal L}_\infty$-norm for large-scale systems, with proven convergence properties.

Findings

The method achieves accurate norm computations for large-scale systems.

The algorithm exhibits locally superlinear convergence.

Numerical examples demonstrate efficiency and accuracy.

Abstract

We are concerned with the computation of the ${\mathcal L}_\infty$-norm for an ${\mathcal L}_\infty$-function of the form $H(s) = C(s) D(s)^{-1} B(s)$, where the middle factor is the inverse of a meromorphic matrix-valued function, and $C(s),\, B(s)$ are meromorphic functions mapping to short-and-fat and tall-and-skinny matrices, respectively. For instance, transfer functions of descriptor systems and delay systems fall into this family. We focus on the case where the middle factor is large-scale. We propose a subspace projection method to obtain approximations of the function $H$ where the middle factor is of much smaller dimension. The ${\mathcal L}_\infty$-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimal points on the imaginary axis where the ${\mathcal L}_\infty$-norm of the reduced function is attained. The subspace method is designed so that certain Hermite interpolation properties hold between the largest singular values of the original and reduced functions. This leads to a locally superlinearly convergent algorithm with respect to the subspace dimension, which we prove and illustrate on various numerical examples.