A New Database Generation Method Combiningmaximin Method and Kriging Prediction

The numerical simulation of eddy-current non-destructive testing methods involves high complexity and expensive computational load. However, one needs to reach reliable solutions for these problems in order to be able, in particular, to solve the related inverse problem. To overcome this difficulty a new approach recently appeared. The main idea here is to build a problem-specific database (for a given ECT application), containing the calculated probe responses (“data”) for well-chosen defect configurations (at certain values of “defect parameters”). See for instance, [1, 2]. Once this database is built, the end-user of the ECT technique just has to use this database and a computationally cheap method yields the sought defect parameters from the database, knowing the measured data. However, the construction of such databases is not a simple task. The database should be “optimal” in some sense. Optimality is a rather general property; in our context; it means that the database achieves a prescribed precision when one applies a given interpolation method (based on the stored data points), while the size of the database remains as small as possible. Nowadays, the generation of optimal databases is quite challenging and an intensively studied field in non-destructive evaluation. To the best of our knowledge, the most preferred approach of database generation in this field so far is the application of mesh generation methods (mostly originated from finite-element meshing techniques), see for instance, [1, 2]. The present contribution provides a new methodology of optimal database generation. Let us introduce the following notation: the n entry of the database consists of a Ddimensional parameter vector pn (the defect is characterized by D parameters), and an Mdimensional data vector zn (zn can be complex, e.g., the impedance variation of a probe coil). The parameter vectors, which relate to all possible defects that one expects to find in the given experiment, span the parameter space P. The data vectors, which are related to the parameter vectors in the parameter space, span the data space Z. The basic idea of our method is to construct a database, whose data points zn (n=1, 2, ..., N) are as spread as possible over Z. In other words, the data space Z is uniformly sampled by the database. The proposed methodology is a combination of a maximin method (a well-known tool in decision theory, game theory, statistics) and a kriging prediction (a stochastic method for interpolating functions, based on some observed function values, by using a Gaussian process to model the functions). A close idea appears (in a quite different context) in [3]. However, the methodology presented in [3] does not work in the case of high-dimensional data spaces, like in our ECT application (note that M might be several hundred). Our algorithm is an incremental procedure, which means that the database is being built by inserting new entries (parameter and the corresponding data vector) one-by-one. The maximin theory is used to determine the next parameter vector to be inserted in order to get a new data vector in the data space which is as far as possible from its nearest neighbor. Since the distance between the data vector to be inserted and the data vectors have already been added is not known a priori, we use the kriging prediction of the distance relations in the data space. To illustrate the method, numerical simulations will be discussed. A comparison of the performances of the databases generated by this method and by the “classical” methods will be carried out. Preliminary results show that the proposed algorithm spreads the data vectors on the data space better than some conventional sampling methods. Another aim of these simulations is to explore and point out possible pitfalls and limitations of this database generation method. AcknowledgmentsThis research is partially supported by the French ANR through the “INDIAC”projectand by the DIGITEO cluster’s project N° 2008-15D. Laboratoire des Signaux et Systèmes etSUPELEC are founding members of the DIGITEO cluster. References1. Sz. Gyimóthy, I. Kiss, J. Pávó, “Adaptive sampling technique based on moving meshesfor building data-equidistant inversion databases for NDT,” OIPE 2008, Sept 14-17, 2008,Ilmenau (Germany), pp. 139-140.2. J. Pávó, Sz. Gyimóthy, “Adaptive inversion database for electromagnetic nondestructiveevaluation,” NDT&E; International, vol. 40, no. 3, 2007, pp. 192-202.3. R. Bettinger, P. Duchêne, L. Pronzato, E. Thierry, “Design of experiments for responsediversity,” J. Phys.: Conf. Ser. 135, 012017, 2008.