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Adaptive Locally Weighted Projection Regression Method for Uncertainty Quantification

Published online by Cambridge University Press:  03 June 2015

Peng Chen
Materials Process Design and Control Laboratory, 101 Frank H. T. Rhodes Hall, Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853-3801, USA
Nicholas Zabaras*
Materials Process Design and Control Laboratory, 101 Frank H. T. Rhodes Hall, Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853-3801, USA
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We develop an efficient, adaptive locally weighted projection regression (ALWPR) framework for uncertainty quantification (UQ) of systems governed by ordinary and partial differential equations. The algorithm adaptively selects the new input points with the largest predictive variance and decides when and where to add new local models. It effectively learns the local features and accurately quantifies the uncertainty in the prediction of the statistics. The developed methodology provides predictions and confidence intervals at any query input and can deal with multi-output cases. Numerical examples are presented to show the accuracy and efficiency of the ALWPR framework including problems with non-smooth local features such as discontinuities in the stochastic space.

Research Article
Copyright © Global Science Press Limited 2013

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[1]Grigoriu, M., Stochastic Systems: Uncertainty Quantification and Propagation (Springer Series in Reliability Engineering Reliability Engineering), 2012.Google Scholar
[2]Cao, Y. and Zang, T. A., An efficient Monte Carlo method for optimal control problems with uncertainty, Comput. Opt. Appl., 26(3) (2003), 219230.Google Scholar
[3]Mathelin, L. and Zang, T. A., Stochastic approaches to uncertainty quantification in CFD simulations, Numer. Algorithms, 38 (2005), 209236.Google Scholar
[4]Mathelin, L., Zang, T. A. and Bataille, F., Uncertainty propagation for a turbulent compressible nozzle flow using stochastic methods, AIAA J., 42(8) (2004), 16691676.Google Scholar
[5]Poroseva, S. V. and Letschert, J., Application of evidence theory to quantify uncertainty in hurricane/typhoon track forecasts, Meteorology and Atmospheric Physics, 97 (2007), 149169.Google Scholar
[6]Ghanem, R. G. and Spanos, P. D., Stochastic Finite Elements: A Spectral Approach, 2003.Google Scholar
[7]Xiu, D. and Karniadakis, G. E., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2) (2002), 619644.Google Scholar
[8]Wan, X. and Karniadakis, G. E., An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, J. Comput. Phys., 209(2) (2005), 617642.Google Scholar
[9]Wan, X. and Karniadakis, G. E., Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput., 28(3) (2006), 901928.Google Scholar
[10]Foo, J. and Karniadakis, G. E., Multi-element probabilistic collocation method in high dimensions, J. Comput. Phys., 229(5) (2010), 15361557.Google Scholar
[11]Babuska, I., Nobile, F. and Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Rev., 52 (2) (2010), 317355.Google Scholar
[12]Nobile, F., Tempone, R. and Webster, C. G., A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46(5) (2008), 23092345.Google Scholar
[13]Smolyak, S. A., Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Mathematics, Doklady, 4 (1963), 240243.Google Scholar
[14]Ma, X. and Zabaras, N., An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J. Comput. Phys., 228(8) (2009), 30843113.Google Scholar
[15]Bilinois, I. and Zabaras, N., Multi-output local Gaussian process regression: application to uncertainty quantification, J. Comput. Phys., 231 (2012), 57185746.Google Scholar
[16]Gramacy, R. B. and Lee, H. K. H., Bayesian treed Gaussian Process models with anapplication to computer modeling, 103 (2008), 11191130.Google Scholar
[17]Atkeson, C. G., Moore, A. W. and Schaal, S., Locally weighted learning, Artificial Intelligence Rev., 11(1) (1997), 1173.Google Scholar
[18]Schaal, S. and Atkeson, C. G., Constructive incremental learning from only local information, Neural Computation, 10(8) (1998), 20472084.Google Scholar
[19]Vijayakumar, S., A. DSouza and S. Schaal, Incremental online learning in high dimensions, Neural Computation, 17(12) (2005), 26022634.Google Scholar
[20]Hoffman, H., Schaal, S. and Vijayakumar, S., Local dimensionality reduction for non-parametric regression, Neural Process Lett., 29 (2009), 109131.Google Scholar
[21]Nievergelt, Y., Total least squares: state-of-the-art regression in numerical analysis, SIAM Rev., 36(2) (1994), 258264.Google Scholar
[22]Hawkins, D. M., On the investigation of alternative regressions by principal component analysis, 22(3) (1973), 275286.Google Scholar
[23]Jolliffe, I. T., A note on the use of principal components in regression, Journal of the Royal Statistical Society, Series C (Applied Statistics), 31(3) (1982), 300303.Google Scholar
[24]Draper, N. R. and van Nostrand, R. C., Ridge regression and James-Stein estimation: review and comments, Technometrics, 21 (4) (1979), 451466.Google Scholar
[25]Swindel, B. F., Geometry of ridge regression illustrated, The American Statistician, 35(1) (1981), 1215.Google Scholar
[26]Frank, I. E. and Friedman, J. H., A statistical view of some chemometrics regression tools, Technometrics, 35(2) (1993), 109135.Google Scholar
[27]Wold, S., Ruhe, A., Wold, H. and III Dunn, W. J., The collinearity problem in linear regression. the partial least squares (pls) approach to generalized inverses, SIAM J. Sci. Stat. Comput., 5(3) (1984), 735743.Google Scholar
[28]Abdi, H., Partial least squares regression and projection on latent structure regression (PLS Regression), Wiley Interdisciplinary Reviews: Computational Statistics, 2(1) (2010), 97106.Google Scholar
[29]Box, G. E., Hunter, W. G., Hunter, J. S. and Hunter, W. G., Statistics for Experimenters: Design, Innovation and Discovery, 2nd Edition, 2005.Google Scholar
[30]Rubens, N., Kaplan, D. and Sugiyama, M., Recommender Systems Handbook: Active Learning in Recommender Systems, Springer, 2011.Google Scholar
[31]Koutsourelakis, P. and Bilionis, E., Scalable Bayesian reduced-order models for simulating high-dimensional multiscale dynamical systems, Multiscale Modeling and Simulation, 9(1) (2011), 449485.Google Scholar
[32]Maljovec, D., Wang, B., Kupresanin, A., Johannesson, G., Pascucci, V. and Bremer, P.Adaptive Sampling with Topological Scores, Int. J.Uncertainty Quantification, (accepted), 2012.Google Scholar
[33]Galass, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Alken, P., Booth, M. and Rossi, F., GNU Scientific Library Reference Manual, 2009.Google Scholar
[34]Udawalpola, R. and Berggren, M., Optimization of an acoustic horn with respect to efficiency and directivity, Int. J.Numer. Methods Eng., 73(11) (2008), 15711606.Google Scholar
[35]Udawalpola, R., Wadbro, E. and Berggren, M., Optimization of a variable mouth acoustic horn, Internat. J. Numer. Methods Eng., (2011), 591606.Google Scholar
[36]Wadbro, E., Udawalpola, R. and Berggren, M., Shape and topology optimization of anacoustic horn-lens combination, J. Comput. Appl. Math., 234 (2010), 17811787.Google Scholar
[37]Hu, X., Lin, G., Hou, T. Y. and Yan, P., An adaptive ANOVA-based data-driven stochastic method for elliptic PDE with random coefficients, Technical Report, Applied and Computational Mathematics, California Institute of Technology, January 11, 2012.Google Scholar
[38]Hecht, F., Pironneau, O., Morice, J., Hyaric, A. L. and Ohtsuka, K., Free FEM++ Manual.Google Scholar
[39]Sherman, J. and Morrison, W.J., Adjustment of an Inverse Matrix Corresponding to Changes in the Elements of a Given Column or a Given Row of the Original Matrix, Annals Math. Statist., 20 (1949), 620624.Google Scholar