Computational Aspects of Stochastic Collocation with Multifidelity Models

In this paper we discuss a numerical approach for the stochastic collocation method with multifidelity simulation models. The method we consider was recently proposed in [A. Narayan, C. Gittelson, and D. Xiu, SIAM J. Sci. Comput., 36 (2014), pp. A495--A521] to combine the computational efficiency of low-fidelity models with the high accuracy of high-fidelity models. This method is able to produce more accurate results at a much reduced simulation cost. The purpose of this paper includes (1) a presentation of the detailed implementation of the method developed by [A. Narayan, C. Gittelson, and D. Xiu, SIAM J. Sci. Comput., 36 (2014), pp. A495--A521], which is largely theoretical; (2) an adaptation of that method to handle multifidelity scenarios that are of more practical interest; (3) a closer examination of the method via a set of more comprehensive benchmark examples including several two-dimensional stochastic PDEs with high-dimensional random parameters; and (4) a more detailed investigation of the st...

[1]  Houman Owhadi,et al.  A non-adapted sparse approximation of PDEs with stochastic inputs , 2010, J. Comput. Phys..

[2]  Dongbin Xiu,et al.  Stochastic Collocation Methods on Unstructured Grids in High Dimensions via Interpolation , 2012, SIAM J. Sci. Comput..

[3]  Leo Wai-Tsun Ng,et al.  Multifidelity Uncertainty Quantification Using Non-Intrusive Polynomial Chaos and Stochastic Collocation , 2012 .

[4]  Stephen Roberts,et al.  Stochastic galerkin and collocation methods for quantifying uncertainty in differential equations: a review , 2009 .

[5]  Loic Le Gratiet,et al.  Bayesian Analysis of Hierarchical Multifidelity Codes , 2011, SIAM/ASA J. Uncertain. Quantification.

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

[7]  C. Schwab,et al.  Sparse high order FEM for elliptic sPDEs , 2009 .

[8]  Xiang Ma,et al.  An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations , 2009, J. Comput. Phys..

[9]  Shun Zhang,et al.  On the Use of ANOVA Expansions in Reduced Basis Methods for Parametric Partial Differential Equations , 2016, J. Sci. Comput..

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

[11]  Nitin Agarwal,et al.  A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties , 2009, J. Comput. Phys..

[12]  Baskar Ganapathysubramanian,et al.  Sparse grid collocation schemes for stochastic natural convection problems , 2007, J. Comput. Phys..

[13]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[14]  D. Xiu.  Efficient collocational approach for parametric uncertainty analysis , 2007 .

[15]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[16]  Andrew J. Majda,et al.  Absorbing Boundary Conditions for Numerical Simulation of Waves , 1977 .

[17]  M. Eldred.  Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design , 2009 .

[18]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[19]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[20]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[21]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[22]  Dongbin Xiu,et al.  A Stochastic Collocation Algorithm with Multifidelity Models , 2014, SIAM J. Sci. Comput..

[23]  George E. Karniadakis,et al.  The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications , 2008, J. Comput. Phys..