Fast evaluation of far-field signals for time-domain wave propagation

Time-domain simulation of wave phenomena on a finite computational domain often requires a fictitious outer boundary. An important practical issue is the specification of appropriate boundary conditions on this boundary, often conditions of complete transparency. Attention to this issue has been paid elsewhere, and here we consider a different, although related, issue: far-field signal recovery. Namely, from smooth data recorded on the outer boundary we wish to recover the far-field signal which would reach arbitrarily large distances. These signals encode information about interior scatterers and often correspond to actual measurements. This article expresses far-field signal recovery in terms of time-domain convolutions, each between a solution multipole moment recorded at the boundary and a sum-of-exponentials kernel. Each exponential corresponds to a pole term in the Laplace transform of the kernel, a finite sum of simple poles. Greengard, Hagstrom, and Jiang have derived the large-$\ell$ (spherical-harmonic index) asymptotic expansion for the pole residues, and their analysis shows that, when expressed in terms of the exact sum-of-exponentials, large-$\ell$ signal recovery is plagued by cancellation errors. Nevertheless, through an alternative integral representation of the kernel and its subsequent approximation by a {\em smaller} number of exponential terms (kernel compression), we are able to alleviate these errors and achieve accurate signal recovery. We empirically examine scaling relations between the parameters which determine a compressed kernel, and perform numerical tests of signal "teleportation" from one radial value $r_1$ to another $r_2$, including the case $r_2=\infty$. We conclude with a brief discussion on application to other hyperbolic equations posed on non-flat geometries where waves undergo backscatter.