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Optimization of scaling parameter in kriging and polyharmonic approximation methods

Technische Universiteit Eindhoven
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  • Computer Science
  • Earth Science
  • Mathematics


Where innovation starts Department of Mathematics and Computer Science Centre for Analysis, Scientific Computing and Applications Den Dolech 2, 5612 AZ Eindhoven P.O. Box 513, 5600 MB Eindhoven The Netherlands Author Mohit Kumar Date August 23, 2011 Optimization of Scaling Parameter in Kriging and Polyharmonic Approximation Methods Author: Mohit Kumar Supervisors: Dr. M. E. Hochstenbach (TU/e) Dr. Rik van Laarhoven (ASML) Technische Universiteit Eindhoven University of Technology 1 Abstract Yieldstar is a meteorological tool developed by ASML, for the purpose of validating the litho- graphic process, i.e., to check if the integrated circuits (ICs) are fabricated according to (height,width, etc.) the desired specifications. This thesis first examines the construction of an approximation F of a multi-variable function f . The function f is related to the inverse of a Maxwell operator, the variables are typically material constants and geometric parameters. By calculating F rather than f , considerable CPU time is saved-in exchange for an approxi- mation error which is desired to be as small as possible. Yieldstar currently employs Kriging model as a computationally inexpensive approximation F of computationally expensive dis- crete Maxwell solver f . In this thesis, we provide with a new method called Polyharmonic method to make approximation model F . Kriging and Polyharmonic models are tested for different test function in different dimensions. In the study we find that Polyharmonic works better than Kriging. Both approximation models contain a shape parameter which controls the prediction accuracy of the model. The main goal of present thesis is to come up with an algorithm or method which finds the optimal value of shape parameter corresponding to which the prediction accuracy of the model is maximum. Due to non-availability of derivative information about f , derivative free optimization methods are examined. We see that most commonly used derivative free

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