A Hyperspherical Adaptive Sparse-Grid Method for High-Dimensional Discontinuity Detection

Zhang, Guannan; Webster, Clayton G.; Gunzburger, Max D.; Burkardt, John V.

doi:10.1137/140971531

Title: A Hyperspherical Adaptive Sparse-Grid Method for High-Dimensional Discontinuity Detection

Journal Article · Wed Jun 24 00:00:00 EDT 2015 · SIAM Journal on Numerical Analysis

DOI:https://doi.org/10.1137/140971531· OSTI ID:1302912

Zhang, Guannan ^[1]; Webster, Clayton G. ^[1]; Gunzburger, Max D. ^[2]; Burkardt, John V. ^[2]

Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Florida State Univ., Tallahassee, FL (United States)

This study proposes and analyzes a hyperspherical adaptive hierarchical sparse-grid method for detecting jump discontinuities of functions in high-dimensional spaces. The method is motivated by the theoretical and computational inefficiencies of well-known adaptive sparse-grid methods for discontinuity detection. Our novel approach constructs a function representation of the discontinuity hypersurface of an N-dimensional discontinuous quantity of interest, by virtue of a hyperspherical transformation. Then, a sparse-grid approximation of the transformed function is built in the hyperspherical coordinate system, whose value at each point is estimated by solving a one-dimensional discontinuity detection problem. Due to the smoothness of the hypersurface, the new technique can identify jump discontinuities with significantly reduced computational cost, compared to existing methods. In addition, hierarchical acceleration techniques are also incorporated to further reduce the overall complexity. Rigorous complexity analyses of the new method are provided as are several numerical examples that illustrate the effectiveness of the approach.

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Research Organization:: Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC05-00OR22725; SC0010678

OSTI ID:: 1302912

Journal Information:: SIAM Journal on Numerical Analysis, Vol. 53, Issue 3; ISSN 0036-1429

Publisher:: Society for Industrial and Applied MathematicsCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 6 works

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97 MATHEMATICS AND COMPUTING
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