The expected Improvement Global Optimiza
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 EI algorithm starts with an initialization part: at some well-chosen points, the costfunction is evaluated (direct problem solved). Then, the kriging prediction of the cost function at
unobserved points is performed, based on the already computed points. The next evaluation point
in the iterative loop is always chosen according to the EI criterion: the parameter point, which
promises the best (i.e. the smallest) cost-function value, is selected and the cost function is
evaluated there. The EI criterion has a simple analytical expression, since the modeling process is
Gaussian. The kriging prediction is updated after every new evaluation. The process is iteratively
repeated until the required stopping criterion is reached.
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.
2. S. Bilicz, E. Vazquez, M. Lambert, Sz. Gyimóthy, J. Pávó, “Characterization of a 3D defect
using the Expected Improvement algorithm,” IGTE 2008, Sept. 21-24, 2008, Graz (Austria),
pp. 56-60.
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 EI algorithm starts with an initialization part: at some well-chosen points, the costfunction is evaluated (direct problem solved). Then, the kriging prediction of the cost function at
unobserved points is performed, based on the already computed points. The next evaluation point
in the iterative loop is always chosen according to the EI criterion: the parameter point, which
promises the best (i.e. the smallest) cost-function value, is selected and the cost function is
evaluated there. The EI criterion has a simple analytical expression, since the modeling process is
Gaussian. The kriging prediction is updated after every new evaluation. The process is iteratively
repeated until the required stopping criterion is reached.
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.
2. S. Bilicz, E. Vazquez, M. Lambert, Sz. Gyimóthy, J. Pávó, “Characterization of a 3D defect
using the Expected Improvement algorithm,” IGTE 2008, Sept. 21-24, 2008, Graz (Austria),
pp. 56-60.