Multi-element probabilistic collocation method in high dimensions

Jasmine Foo; George Em Karniadakis

doi:10.1016/j.jcp.2009.10.043

Multi-element probabilistic collocation method in high dimensions

Jasmine Foo, George Em Karniadakis

Mathematics

Research output: Contribution to journal › Article › peer-review

173 Scopus citations

Abstract

We combine multi-element polynomial chaos with analysis of variance (ANOVA) functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity. Specifically, we employ the multi-element probabilistic collocation method MEPCM [1] and so we refer to the new method as MEPCM-A. We investigate the dependence of the convergence of MEPCM-A on two decomposition parameters, the polynomial order μ and the effective dimension ν, with ν ≪ N, and N the nominal dimension. Numerical tests for multi-dimensional integration and for stochastic elliptic problems suggest that ν ≥ μ for monotonic convergence of the method. We also employ MEPCM-A to obtain error bars for the piezometric head at the Hanford nuclear waste site under stochastic hydraulic conductivity conditions. Finally, we compare the cost of MEPCM-A against Monte Carlo in several hundred dimensions, and we find MEPCM-A to be more efficient for up to 600 dimensions for a specific multi-dimensional integration problem involving a discontinuous function.

Original language	English (US)
Pages (from-to)	1536-1557
Number of pages	22
Journal	Journal of Computational Physics
Volume	229
Issue number	5
DOIs	https://doi.org/10.1016/j.jcp.2009.10.043
State	Published - Mar 1 2010

Bibliographical note

Funding Information:
We would like to thank Prof. Xiaoliang Wan for his help with the Hanford site simulations. In addition, we would like to thank Prof. Christoph Schwab and Dr. Marcel Beri at ETH, Zurich, for their helpful suggestions. We thank the anonymous referees for their comments and suggestions which greatly improved the manuscript. This work was partially supported by DOE, ONR and ONR/ESRDC Consortium, and computations were performed on the DoD/HPCM supercomputers. Jasmine Foo would like to acknowledge the support of DOE fellowship (CSGF) under Grant DE-FG02-97ER25308 and the Krell institute and travel support by NSF Grant OISE-0456114 .

Keywords

Domain decomposition
Sparse grids
Stochastic partial differential equations

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abstract = "We combine multi-element polynomial chaos with analysis of variance (ANOVA) functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity. Specifically, we employ the multi-element probabilistic collocation method MEPCM [1] and so we refer to the new method as MEPCM-A. We investigate the dependence of the convergence of MEPCM-A on two decomposition parameters, the polynomial order μ and the effective dimension ν, with ν ≪ N, and N the nominal dimension. Numerical tests for multi-dimensional integration and for stochastic elliptic problems suggest that ν ≥ μ for monotonic convergence of the method. We also employ MEPCM-A to obtain error bars for the piezometric head at the Hanford nuclear waste site under stochastic hydraulic conductivity conditions. Finally, we compare the cost of MEPCM-A against Monte Carlo in several hundred dimensions, and we find MEPCM-A to be more efficient for up to 600 dimensions for a specific multi-dimensional integration problem involving a discontinuous function.",

keywords = "Domain decomposition, Sparse grids, Stochastic partial differential equations",

author = "Jasmine Foo and Karniadakis, {George Em}",

note = "Funding Information: We would like to thank Prof. Xiaoliang Wan for his help with the Hanford site simulations. In addition, we would like to thank Prof. Christoph Schwab and Dr. Marcel Beri at ETH, Zurich, for their helpful suggestions. We thank the anonymous referees for their comments and suggestions which greatly improved the manuscript. This work was partially supported by DOE, ONR and ONR/ESRDC Consortium, and computations were performed on the DoD/HPCM supercomputers. Jasmine Foo would like to acknowledge the support of DOE fellowship (CSGF) under Grant DE-FG02-97ER25308 and the Krell institute and travel support by NSF Grant OISE-0456114 .",

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N2 - We combine multi-element polynomial chaos with analysis of variance (ANOVA) functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity. Specifically, we employ the multi-element probabilistic collocation method MEPCM [1] and so we refer to the new method as MEPCM-A. We investigate the dependence of the convergence of MEPCM-A on two decomposition parameters, the polynomial order μ and the effective dimension ν, with ν ≪ N, and N the nominal dimension. Numerical tests for multi-dimensional integration and for stochastic elliptic problems suggest that ν ≥ μ for monotonic convergence of the method. We also employ MEPCM-A to obtain error bars for the piezometric head at the Hanford nuclear waste site under stochastic hydraulic conductivity conditions. Finally, we compare the cost of MEPCM-A against Monte Carlo in several hundred dimensions, and we find MEPCM-A to be more efficient for up to 600 dimensions for a specific multi-dimensional integration problem involving a discontinuous function.

AB - We combine multi-element polynomial chaos with analysis of variance (ANOVA) functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity. Specifically, we employ the multi-element probabilistic collocation method MEPCM [1] and so we refer to the new method as MEPCM-A. We investigate the dependence of the convergence of MEPCM-A on two decomposition parameters, the polynomial order μ and the effective dimension ν, with ν ≪ N, and N the nominal dimension. Numerical tests for multi-dimensional integration and for stochastic elliptic problems suggest that ν ≥ μ for monotonic convergence of the method. We also employ MEPCM-A to obtain error bars for the piezometric head at the Hanford nuclear waste site under stochastic hydraulic conductivity conditions. Finally, we compare the cost of MEPCM-A against Monte Carlo in several hundred dimensions, and we find MEPCM-A to be more efficient for up to 600 dimensions for a specific multi-dimensional integration problem involving a discontinuous function.

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Multi-element probabilistic collocation method in high dimensions

Abstract

Bibliographical note

Keywords

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