Characterization of Eddy-current Testing Inverse Problems Using Adaptative Databases

One of the main challenges in Eddy Current Testing (ECT) is the solution of the inverse problem, i.e., the determination of the defect properties knowing the measured data. To this end, many approaches and mathematical tools have been proposed. The so-called adaptive database-method has recently been developed. Its main idea is to store corresponding input-output data pairs in a database and, by fitting an interpolator to these samples, to solve approximately the forward and inverse problems at a low computational cost. Such adaptive databases can be generated by meshing methods [1, 2] (inspired by the meshes of Finite Element Methods) or meshless approach as introduced in [3] and recast and extended in [4]. The sampling strategy of the database generation presented in [4] has been improved and will briefly be described, however the presentation will mainly be focused on the application of such adaptive databases for the characterization of the related inverse problem. For a formal description, let us define a defect model with p parameters by the vector x=[x1, x2, ..., xp] (usually position and dimensions of the defect), whereas the functional output (the measured data) can be the impedance variation ΔZ(s) of the probe coil, where s is related to the position of the coil. The input and the output are connected by the forward operator F: ΔZ(s)=F{x}. One can also assign a p dimensional region X, called the input space so that for all conceivable input vectors xεX holds. The co-domain of the operator F over X is the output space Y, i.e., F: X→Y. The goal of our adaptive sampling is to generate a set of n samples (xi,ΔZi(s)), i=1,2,...,n, such that the outputs ΔΖi(s) are as far as possible from each other and such that all conceivable outputs are closer to at least one stored sample than a pre-defined limit (in the sense of an appropriate norm). The output space is then evenly sampled leading to an “equidistant” output sampling. In the first version [4] the database generation procedure was an iterative-incremental loop, adding samples one-by-one according to some criteria based on the distances in the output space. In the present version a removing strategy has been added in order to make the sampling more uniform. Intuitively, such an incremental-iterative loop is supposed to converge to an output-equidistant database. As a first application, the combination of an optimal database and a simple interpolator (nearest neighbor for example) allows us to predict the solution ΔZ(s) for a sample x which does not belong to the database –in a very short time. But t he structure of such a database is also able to provide some meta-information about the involved forward operator F. As a matter of fact, in the regions of X where F varies rapidly, the sampling will be dense, whereas in the regions where F varies slowly the sampling will be sparse. With such information the related inverse problem can be characterized to some extent. On one hand, the ha l-0 04 93 72 5, v er si on 1 21 J un 2 01 0 Author manuscript, published in "15th International Workshop on Electromagnetic Non--Destructive Evaluation (ENDE'10), Szczecin : Poland (2010)"