Limits to Nonlinear Inversion

For non-linear inverse problems, the mathematical structure of the mapping from model parameters to data is usually unknown or partly unknown. Absence of information about the mathematical structure of this function prevents us from presenting an analytical solution, so our solution depends on our ability to produce efficient search algorithms. Such algorithms may be completely problem-independent (which is the case for the so-called ’meta-heuristics’ or ’blind-search’ algorithms), or they may be designed with the structure of the concrete problem in mind.

We show that pure meta-heuristics are inefficient for large-scale, non-linear inverse problems, and that the ’no-free-lunch’ theorem holds. We discuss typical objections to the relevance of this theorem.

A consequence of the no-free-lunch theorem is that algorithms adapted to the mathematical structure of the problem perform more efficiently than pure meta-heuristics. We study problem-adapted inversion algorithms that exploit the knowledge of the smoothness of the misfit function of the problem. Optimal sampling strategies exist for such problems, but many of these problems remain hard.