THE “EXPECTED IMPROVEMENT”
GLOBAL OPTIMIZATION ALGORITHM FOR THE SOLUTION OF
EDDY-CURRENT TESTING INVERSE PROBLEMS
S. BILICZ1,2, E. VAZQUEZ3, M. LAMBERT1, Sz. GYIMÓTHY2 and J. PÁVÓ2
1
Département de Recherche en Électromagnétisme, Laboratoire des
Signaux et Systèmes UMR8506 (CNRS-SUPELEC-Univ Paris-Sud), Gif-sur-Yvette, France
2
Budapest University of Technology and Economics, Hungary
3
École Supérieure d'Électricité (SUPELEC), Gif-sur-Yvette, France
Abstract
A classical way to solve the inverse problem of defect characterization is to construct an iterative loop which tries to achieve the best resemblance between the measured data and the output of an appropriate simulator of the considered experiment. If the similarity of these signals is best, one can say that a solution for the defect characterization problem has been found (in a certain sense), the current input parameters of the simulator are assumed to approximate the parameters of the real defect.
The described method leads us to a solution through sequential runs of a forward simulator. A vast of contributions have dealt with different strategies for this step-by-step iteration loop. However, the domain is still challenging, there is no “best” method. The strategies have to face with two main difficulties. First, the complexity of the inverse problem can be a pitfall (for instance, if one defines the resemblance of output signals in terms of a cost-function, it may have several local minima which make the optimization problem unkind). Second, the applied simulator often involves difficult field computation tasks, thus the simulation is computationally expensive. Consequently, a reliable global optimization method is needed, moreover, the number of simulations should be kept as small as possible.
Our contribution presents an optimization strategy which has been quite well-known for ten years or so in different domains of engineering but unexplored in eddy-current testing inversion. The so-called “Expected Improvement” algorithm (EI) [1] is a global optimization tool designed for expensive-to-evaluate functions. Our main aim is to introduce this approach to the nondestructive testing community. It is based on the kriging interpolation of the cost function. Kriging [1] is originated from geostatistics, from the 60s. Its main idea is to model the function by a Gaussian process. The interpolation is based on some observed function values and provides the best linear unbiased prediction (BLUP) of the modeling process. An essential property of kriging is that beyond the mere prediction, some information about the uncertainty of the prediction is also provided.
The performance of the method will be illustrated by some test cases, using synthetic data obtained in the eddy current application of the characterization of a defect (one single volumetric defect [2] or double thin defects) affecting a plate from the measurement of the variation of impedance of a pancake coil moving above the top of the plate will be shown.
Acknowledgments
We would like to thank the Pôle de compétitivité “SYSTEM@TIC PARIS REGION” for its financial support.
References
1. D. Jones, “A taxonomy of global optimization methods based on response surfaces,” Journal of Global Optimization, vol. 21, pp. 345–383, 2001.