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 problem, and only depend on the forward map. The methodology is applied to the comparison of the optimizability properties of two methods for the seismic inverse problem for a timeharmonic wave equation: the Full Waveform Inversion (FWI) and its Migration Based Travel Time (MBTT) reformulation. Computation of the size of attraction basins for the two approaches allows to quantify the benefits of the latter, which can alleviate the requirement of low-frequency data for the reconstruction of the background velocity model.

Full waveform inversion for bore reconstruction of woodwind-like instruments

“…As this relation involves complex operators, the complex derivative must be carefully defined. The general principles, presented in detail by Barucq et al [32] are recalled in Appendix A. In particular, as n 2 R N n and F ðnÞ 2 R, the following relation can be shown [32]…”

A Variational Adjoint Approach on Wide-Angle Parabolic Equation for Refractivity Inversion

“…In practice, the two attributes reflect on the problem (7) as nonconvexity and nonsmoothness of the optimization landscape. Therefore, from our perspective, looking at the cost function landscape [43] can guide the community to build the right strategy. RFC is much more complex than (7), so optimization landscape analysis of ( 7) is valuable as an intermediate step for understanding why RFC is still an open problem.…”

“…Unlike previous work, the obtained gradient of the cost function is computed and validated successfully using a finite differences approximation. We also include the optimization landscapes in our analysis in order to visualize the inversion complexity and to explain how much the inversion is sensitive to initial guess for a given setup [43]. The method is presented for tomographic approach as an intermediate step towards a full effective RFC system.…”

Eigenvector models for solving the seismic inverse problem for the Helmholtz equation

“…For each combination of frequency and associated number of eigenvectors N in the decomposition, 30 iterations are performed (n iter in Algorithm 2). Barucq et al, 2019b). We illustrate the performance of the reconstruction with the results in the data-space: in Figure 18, we show the time-domain seismograms for the true, starting and reconstructed velocity models.…”

“…For future work, it seems that the lack of background velocity information would not be overcome by the decomposition, possibly resulting in artifacts. Therefore, we envision the use of multiple basis to parametrize the velocity (e.g., using the background/reflectivity decomposition idea of Clément et al (2001); Barucq et al (2019b), with a dedicated smooth eigenvector basis to represent the background, and another to represent the reflectors).…”