RECURSIVE CO-KRIGING MODEL FOR DESIGN OF COMPUTER EXPERIMENTS WITH MULTIPLE LEVELS OF FIDELITY

We consider in this paper the problem of building a fast-running approximation—also called surrogate model—of a complex computer code. The co-kriging based surrogate model is a promising tool to build such an approximation when the complex computer code can be run at different levels of accuracy. We present here an original approach to perform a multi-fidelity co-kriging model which is based on a recursive formulation. We prove that the predictive mean and the variance of the presented approach are identical to the ones of the original co-kriging model. However, our new approach allows to obtain original results. First, closed-form formulas for the universal co-kriging predictive mean and variance are given. Second, a fast cross-validation procedure for the multi-fidelity co-kriging model is introduced. Finally, the proposed approach has a reduced computational complexity compared to the previous one. The multi-fidelity model is successfully applied to emulate a hydrodynamic simulator.

[1]  Peter Z. G. Qian,et al.  CONSTRUCTION OF NESTED SPACE-FILLING DESIGNS , 2009, 0909.0598.

[2]  Michael L. Stein,et al.  Interpolation of spatial data , 1999 .

[3]  David Wooff,et al.  Bayes Linear Statistics: Theory and Methods , 2007 .

[4]  Dave Higdon,et al.  Combining Field Data and Computer Simulations for Calibration and Prediction , 2005, SIAM J. Sci. Comput..

[5]  R. Stocki,et al.  A method to improve design reliability using optimal Latin hypercube sampling , 2005 .

[6]  A. O'Hagan,et al.  Predicting the output from a complex computer code when fast approximations are available , 2000 .

[7]  G. Matheron.  Principles of geostatistics , 1963 .

[8]  Hao Zhang,et al.  Kriging and cross‐validation for massive spatial data , 2009 .

[9]  Runze Li,et al.  Design and Modeling for Computer Experiments , 2005 .

[10]  Alexander I. J. Forrester,et al.  Multi-fidelity optimization via surrogate modelling , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[11]  Jon Lee.  Maximum entropy sampling , 2001 .

[12]  T. J. Mitchell,et al.  Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments , 1991 .

[13]  Serge Gauthier,et al.  A second-order turbulence model for gaseous mixtures induced by Richtmyer—Meshkov instability , 2005 .

[14]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[15]  Peter Z. G. Qian,et al.  Bayesian Hierarchical Modeling for Integrating Low-Accuracy and High-Accuracy Experiments , 2008, Technometrics.

[16]  D. Harville.  Matrix Algebra From a Statistician's Perspective , 1998 .

[17]  Anthony O'Hagan,et al.  Diagnostics for Gaussian Process Emulators , 2009, Technometrics.

[18]  Michael Goldstein,et al.  Constructing partial prior specifications for models of complex physical systems , 1998 .

[19]  James O. Berger,et al.  A Framework for Validation of Computer Models , 2007, Technometrics.

[20]  A. O'Hagan,et al.  Bayesian calibration of computer models , 2001 .

[21]  H. D. Patterson,et al.  Recovery of inter-block information when block sizes are unequal , 1971 .

[22]  H. Jeffreys,et al.  The Theory of Probability , 1896 .

[23]  M. Stein.  Large sample properties of simulations using latin hypercube sampling , 1987 .

[24]  Sonja Kuhnt,et al.  Design and analysis of computer experiments , 2010 .

[25]  Michael Goldstein,et al.  Small Sample Bayesian Designs for Complex High-Dimensional Models Based on Information Gained Using Fast Approximations , 2009, Technometrics.

[26]  D. Krige.  A statistical approach to some basic mine valuation problems on the Witwatersrand, by D.G. Krige, published in the Journal, December 1951 : introduction by the author , 1951 .

[27]  Olivier Dubrule,et al.  Cross validation of kriging in a unique neighborhood , 1983 .