Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids

John D. Jakeman; Richard Archibald; Dongbin Xiu

doi:10.1016/j.jcp.2011.02.022

Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids

John D. Jakeman, Richard Archibald, Dongbin Xiu

Data Analysis and Machine Learning

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

48 Scopus citations

Abstract

In this paper we present a set of efficient algorithms for detection and identification of discontinuities in high dimensional space. The method is based on extension of polynomial annihilation for discontinuity detection in low dimensions. Compared to the earlier work, the present method poses significant improvements for high dimensional problems. The core of the algorithms relies on adaptive refinement of sparse grids. It is demonstrated that in the commonly encountered cases where a discontinuity resides on a small subset of the dimensions, the present method becomes "optimal", in the sense that the total number of points required for function evaluations depends linearly on the dimensionality of the space. The details of the algorithms will be presented and various numerical examples are utilized to demonstrate the efficacy of the method.

Original language	English
Pages (from-to)	3977-3997
Number of pages	21
Journal	Journal of Computational Physics
Volume	230
Issue number	10
DOIs	https://doi.org/10.1016/j.jcp.2011.02.022
State	Published - May 10 2011

Funding

The submitted manuscript has been authored by contractors [UT-Battelle LLC, manager of Oak Ridge National Laboratory (ORNL)] of the US Government under Contract No. DE-AC05-00OR22725. Accordingly, the US Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for US Government purposes. Dongbin Xiu was partially supported by AFOSR FA9550-08-1-0353, DOE/NNSA DE-FC52-08NA28617, and NSF CAREER DMS-0645035 and IIS-0914447.

Funders	Funder number
DOE/NNSA	DE-FC52-08NA28617
US Government
National Science Foundation	IIS-0914447, DMS-0645035
Air Force Office of Scientific Research	FA9550-08-1-0353
Oak Ridge National Laboratory

Keywords

Adaptive sparse grids
Generalized polynomial chaos method
High-dimensional approximation
Multivariate discontinuity detection
Stochastic partial differential equations

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10.1016/j.jcp.2011.02.022

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Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids

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