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Adaptive Locally Weighted Projection Regression Method for Uncertainty Quantification

  • Peng Chen (a1) and Nicholas Zabaras (a1)

Abstract

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.

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Corresponding author

*Corresponding author.Email:nzabaras@gmail.com

References

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[1]Grigoriu, M., Stochastic Systems: Uncertainty Quantification and Propagation (Springer Series in Reliability Engineering Reliability Engineering), 2012.
[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.
[3]Mathelin, L. and Zang, T. A., Stochastic approaches to uncertainty quantification in CFD simulations, Numer. Algorithms, 38 (2005), 209236.
[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.
[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.
[6]Ghanem, R. G. and Spanos, P. D., Stochastic Finite Elements: A Spectral Approach, 2003.
[7]Xiu, D. and Karniadakis, G. E., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2) (2002), 619644.
[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.
[9]Wan, X. and Karniadakis, G. E., Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput., 28(3) (2006), 901928.
[10]Foo, J. and Karniadakis, G. E., Multi-element probabilistic collocation method in high dimensions, J. Comput. Phys., 229(5) (2010), 15361557.
[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.
[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.
[13]Smolyak, S. A., Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Mathematics, Doklady, 4 (1963), 240243.
[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.
[15]Bilinois, I. and Zabaras, N., Multi-output local Gaussian process regression: application to uncertainty quantification, J. Comput. Phys., 231 (2012), 57185746.
[16]Gramacy, R. B. and Lee, H. K. H., Bayesian treed Gaussian Process models with anapplication to computer modeling, 103 (2008), 11191130.
[17]Atkeson, C. G., Moore, A. W. and Schaal, S., Locally weighted learning, Artificial Intelligence Rev., 11(1) (1997), 1173.
[18]Schaal, S. and Atkeson, C. G., Constructive incremental learning from only local information, Neural Computation, 10(8) (1998), 20472084.
[19]Vijayakumar, S., A. DSouza and S. Schaal, Incremental online learning in high dimensions, Neural Computation, 17(12) (2005), 26022634.
[20]Hoffman, H., Schaal, S. and Vijayakumar, S., Local dimensionality reduction for non-parametric regression, Neural Process Lett., 29 (2009), 109131.
[21]Nievergelt, Y., Total least squares: state-of-the-art regression in numerical analysis, SIAM Rev., 36(2) (1994), 258264.
[22]Hawkins, D. M., On the investigation of alternative regressions by principal component analysis, 22(3) (1973), 275286.
[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.
[24]Draper, N. R. and van Nostrand, R. C., Ridge regression and James-Stein estimation: review and comments, Technometrics, 21 (4) (1979), 451466.
[25]Swindel, B. F., Geometry of ridge regression illustrated, The American Statistician, 35(1) (1981), 1215.
[26]Frank, I. E. and Friedman, J. H., A statistical view of some chemometrics regression tools, Technometrics, 35(2) (1993), 109135.
[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.
[28]Abdi, H., Partial least squares regression and projection on latent structure regression (PLS Regression), Wiley Interdisciplinary Reviews: Computational Statistics, 2(1) (2010), 97106.
[29]Box, G. E., Hunter, W. G., Hunter, J. S. and Hunter, W. G., Statistics for Experimenters: Design, Innovation and Discovery, 2nd Edition, 2005.
[30]Rubens, N., Kaplan, D. and Sugiyama, M., Recommender Systems Handbook: Active Learning in Recommender Systems, Springer, 2011.
[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.
[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.
[33]Galass, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Alken, P., Booth, M. and Rossi, F., GNU Scientific Library Reference Manual, 2009.
[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.
[35]Udawalpola, R., Wadbro, E. and Berggren, M., Optimization of a variable mouth acoustic horn, Internat. J. Numer. Methods Eng., (2011), 591606.
[36]Wadbro, E., Udawalpola, R. and Berggren, M., Shape and topology optimization of anacoustic horn-lens combination, J. Comput. Appl. Math., 234 (2010), 17811787.
[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.
[38]Hecht, F., Pironneau, O., Morice, J., Hyaric, A. L. and Ohtsuka, K., Free FEM++ Manual.
[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.

Keywords

Adaptive Locally Weighted Projection Regression Method for Uncertainty Quantification

  • Peng Chen (a1) and Nicholas Zabaras (a1)

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