Well-Posed Questions and Exploration of the Space of Parameters in Linear and Non Linear Inversion

摘要

Solving inverse problems is defining convenient routes, in the space of parameters, towards those called solutions of the problem. Thus methods for solving inverse problems are primarily related to the existence of routes in a space rather than to statistical, algebraic, or other considerations. After a brief survey of the local routes which yield for instance least square inversion, global routes obtained by using geometric transforms are dealt with. One shows on examples how to construct these routes, how to use them, what questions can be answered which escape the range of local routes. These methods apply at least to all inverse problems related with exact solutions of nonlinear partial differential equations, and in particular to the inverse problems of wave equations. They yield efficient algorithms. The case in which localization of the desired solutions is only obtained through some extreme points of a domain is then reviewed. In this method of analysis, one replaces the ill-posed problem of finding solutions that correspond to actual measurements by a set of well-posed questions whose answers are inequalities on theoretical measurements of the set of solutions- and enable decision in applied problems. Several applications in engineering and geophysical research were given in the last few years.