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Coherent techniques for searches of gravitational-wave bursts effectivelycombine data from several detectors, taking into account differences in theirresponses. The efforts are now focused on the maximum likelihood principle asthe most natural way to combine data, which can also be used without priorknowledge of the signal. Recent studies however have shown that straightforwardapplication of the maximum likelihood method to gravitational waves withunknown waveforms can lead to inconsistencies and unphysical results such asdiscontinuity in the residual functional, or divergence of the variance of theestimated waveforms for some locations in the sky. So far the solutions tothese problems have been based on rather different physical arguments.Following these investigations, we now find that all these inconsistencies stemfrom rank deficiency of the underlying network response matrix. In this paperwe show that the detection of gravitational-wave bursts with a network ofinterferometers belongs to the category of ill-posed problems. We then applythe method of Tikhonov regularization to resolve the rank deficiency andintroduce a minimal regulator which yields a well-conditioned solution to theinverse problem for all locations on the sky.Comment: 12 pages, 5 figure

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