Matrix-free Krylov iteration for implicit convolution of numerically low-rank data

Evaluating the response of a linear shift-invariant system is a problem that occurs frequently in a wide variety of science and engineering problems. Calculating the system response via a convolution may be done efficiently with Fourier transforms. When one must compute the response of one system to m input signals, or the response of m systems to one signal, it may be the case that one may approximate all system responses without having to compute all m Fourier transforms. This can lead to substantial computational savings. Rather than process each point individually, one may only process basis vectors that span the output data space. However, to get a low-error approximation, it is necessary that the output vectors have low numerical rank if they were assembled into a matrix. We develop theory that shows how the singular value decay of a matrix ź A that is a product of a convolution operator ź and an arbitrary matrix A depends in a linear fashion on the singular value decays of ź and A . We propose gap-rank, a measure of the relative numerical rank of a matrix. We show that convolution cannot destroy the numerical low-rank-ness of ź A data with only modest assumptions. We then develop a new method that exploits low-rank problems with block Golub-Kahan iteration in a Krylov subspace to approximate the low-rank problem. Our method can exploit parallelism in both the individual convolutions and the linear algebra operations in the block Golub-Kahan algorithm. We present numerical examples from signal and image processing that show the low error and scalability of our method.