Multi - Level , Multi - Frequency , Compressed Full

We study the inverse boundary value problem for the wave equation and recovery of the wavespeed upon taking a time-Fourier transform of the data. We design a hierarchical compressed reconstruction in a multilevel scheme for the inverse boundary value problem associated with the Helmholtz equation using the Dirichletto-Neumann map, or the single-layer potential operator, at selected frequencies as the data. The compression is based on a domain partitioning of the subsurface, while the hierarchy is straightforwardedly established through refinement. The coe cients are assumed to be piecewise constant functions following the domain partitioning, thus allowing the presence of conormal singularities (causing reflections and di↵ractions); these guarantee a Lipschitz stability estimate for the inverse problem which gives a radius of convergence, related to the stability constant, of the scheme even in the case of partial data. The stability constant grows exponentially with the number of subdomains in the domain partitioning, whence the compression. The frequencies are selected to control the approximation errors in the data due to the compression at the di↵erent levels in the scheme. The stability constants (number of subdomains) and approximations errors (frequencies) are coupled through a condition between levels for convergence of the multi-level scheme. We use Haar wavelets for the compression in alternative strategies providing a gradual increase in the number of subdomains in the domain parititioning via adaptive, local multi-scale refinement. We include visco-acoustic behavior and carry out numerical experiments.