Rank deficiency and Tikhonov regularization in the inverse problem for gravitational-wave bursts

TL;DR

This paper addresses the ill-posed inverse problem in gravitational-wave burst detection by identifying rank deficiency as the core issue and applying Tikhonov regularization to obtain stable, physically meaningful solutions across all sky locations.

Contribution

It reveals that rank deficiency causes inconsistencies in maximum likelihood methods and introduces a Tikhonov regularization approach to stabilize solutions in gravitational-wave burst analysis.

Findings

Identifies rank deficiency as the cause of unphysical results.

Applies Tikhonov regularization to improve solution stability.

Provides a minimal regulator for well-conditioned inverse solutions.

Abstract

Coherent techniques for searches of gravitational-wave bursts effectively combine data from several detectors, taking into account differences in their responses. The efforts are now focused on the maximum likelihood principle as the most natural way to combine data, which can also be used without prior knowledge of the signal. Recent studies however have shown that straightforward application of the maximum likelihood method to gravitational waves with unknown waveforms can lead to inconsistencies and unphysical results such as discontinuity in the residual functional, or divergence of the variance of the estimated waveforms for some locations in the sky. So far the solutions to these problems have been based on rather different physical arguments. Following these investigations, we now find that all these inconsistencies stem from rank deficiency of the underlying network response matrix. In this paper we show that the detection of gravitational-wave bursts with a network of interferometers belongs to the category of ill-posed problems. We then apply the method of Tikhonov regularization to resolve the rank deficiency and introduce a minimal regulator which yields a well-conditioned solution to the inverse problem for all locations on the sky.