A globally convergent convexification algorithm for the inverse problem of

M. V. Klibanov ; A. A. Timonov

Numerical methods and programming, Tome 4 (2003) no. 1, pp. 52-81

Résumé

A globally convergent convexification algorithm for the numerical solution of the inverse problem of electromagnetic frequency sounding in one dimension is presented. This algorithm is based on the concept of convexification of a multiextremal objective function proposed recently by the authors. A key point in the proposed algorithm is that unlike conventional layer-stripping algorithms, it provides the stable approximate solution via minimization of a finite sequence of strictly convex objective functions resulted from applying the nonlinear weighted least squares method with Carleman's weight functions. The other advantage of the proposed algorithm is that its convergence to the “exact” solution does not depend on a starting vector. Thus, the uncertainty inherent to the local methods, such as the gradient or Newton-like methods, is eliminated. The 1-D inverse model of magnetotelluric sounding is selected to exemplify the convexification approach. Based on the localizing property of Carleman's weight functions, it is proven that the distance between the approximate and “exact” solutions is small if the approximation error is small. The results of computational experiments with several realistic and synthetic marine shallow water configurations are presented to demonstrate the computational feasibility of the proposed algorithm.

Détail

Mots-clés : convexification
Keywords: magnetotelluric sounding, electromagnetic frequency sounding, gradient methods, Newton-like methods, regularization method.