A priori estimates of attraction basins for nonlinear least squares, with application to Helmholtz seismic inverse problem

Abstract

In this paper, we provide an a priori optimizability analysis of nonlinear least squares problems that are solved by local optimization algorithms. We define attraction (convergence) basins where the misfit functional is guaranteed to have only one local—and hence global—stationary point, provided the data error is below some tolerable error level. We use geometry in the data space (strictly quasiconvex sets) in order to compute the size of the attraction basin (in the parameter space) and the associated tolerable error level (in the data space). These estimates are defined a priori, i.e. they do not involve any least squares minimization problems, and only depend on the forward map. This methodology is applied to the comparison of the optimizability properties of two methods for the seismic inverse problem for a time-harmonic wave equation: the full waveform inversion (FWI) and its migration-based travel time (MBTT) reformulation. Computing the size of the attraction basins for the two approaches allows us to quantify the benefits of the latter, which can alleviate the requirement of low-frequency data for reconstruction of the background velocity model.