Kriging-based generation of optimal databases as forward and inverse surrogate models

Abstract

Numerical methods are used to simulate the mathematical models of a wide range of engineering problems. The precision provided by such simulators is usually fine but at the price of computational cost. In some applications this cost might be crucial. This leads us to consider cheap surrogate models in order to reduce the computation time still meeting the precision requirements. Among all available surrogate models, we deal herein with the generation of an "optimal" database of pre-calculated results combined with a simple interpolator. A database generation approach is investigated which is intending to achieve an optimal sampling. Such databases can be used for the approximate solution of both forward and inverse problems. Their structure carries some meta-information about the involved physical problem. In the case of the inverse problem, an approach for predicting the uncertainty of the solution (due to the applied surrogate model and/or the uncertainty of the measured data) is presented. All methods are based on kriging-a stochastic tool for function approximation. Illustrative examples are drawn from eddy-current nondestructive evaluation.

Characterization of the Inverse Problem in Critical Dimension Measurement of Diffraction Gratings

“…This uniform output distribution is achieved by means of a sequential adaptive sampling that is not discussed herein. The distribution of the input samples corresponding to such uniform output distribution bears valuable information on the conditioning of the inverse problem, as pointed out in [16,17]. A central contribution of the present paper is to apply this…”

“…The measured data are then used to reconstruct the conductivity distribution inside the specimen, which bears information on a possible structural degradation (crack, void, etc.). In our work, the method developed for ECT inverse problems in [16] is used. This algorithm consists in the adaptive generation of a database of corresponding input parameter -output data pairs: This database D is called optimal in the sense that the output samples y i are uniformly spread out over the output space, i.e., over the domain spanned by all conceivable outputs of the simulation.…”

Metamodel-based Markov-Chain-Monte-Carlo parameter inversion applied in eddy current flaw characterization

“…For statistical parameter inversion, thousands of forward evaluations could be required, hence, in effect overwhelming the algorithm if we employ MoM directly within the inversion. To overcome this problem, data-fitting surrogate models, also called metamodels [7,21,22,30,40], are proposed.…”

“…Denoting D = {x j ,f (x j )}, j = 1, 2, · · · , J as the metamodel database,f (x j ) is the MoM simulator and J is the total number of database pairs. Details on this adaptive database training are found in [7,21]. In brief, a global approximation is achieved,…”

Metamodel-Based Nested Sampling for Model Selection in Eddy-Current Testing

“…For a model selection, the computational cost is particularly crucial since it requires at least one order-of-magnitude more of likelihood evaluations compared to parameter inversion. Data-fitting surrogate models have been proposed [21], [23], [24] to reduce the computational cost in the inversion. Here, we use the same idea to overcome the computational cost problem.…”