Chance-constrained programming : a tool for solving linear eddy current inverse problem
Résumé
This paper deals with the application of chance-constrained programming to eddy current inverse problems, linearized in the Born approximation framework. It suggests two probabilistic formulations of this linear inverse problem: the first one uses joint chance constraints and the second one is based upon individual chance constraints; the latter allows a layer-by-layer reconstruction suitable for taking attenuation into account. The paper illustrates how structural properties of chance constraints are a powerful tool for dealing with the ill-posed nature of the problem. Two situations are considered: i) the first one concerns a flaw made of a simple conductivity contrast and an underlying random process belonging to the class of elliptically symmetric distributions; in that case, a deterministic equivalence of the problem can be found provided that a suitable level of probability is chosen, ii) the second situation concerns a notch; a tractable approximate algorithm based upon the Hoeffding inequality is suggested for solving the inverse problem in that case.