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Title: Hyperspherical Sparse Approximation Techniques for High-Dimensional Discontinuity Detection

Abstract

This work proposes a hyperspherical sparse approximation framework for detecting jump discontinuities in functions in high-dimensional spaces. The need for a novel approach results from the theoretical and computational inefficiencies of well-known approaches, such as adaptive sparse grids, for discontinuity detection. Our approach constructs the hyperspherical coordinate representation of the discontinuity surface of a function. Then sparse approximations of the transformed function are built in the hyperspherical coordinate system, with values at each point 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. Several approaches are used to approximate the transformed discontinuity surface in the hyperspherical system, including adaptive sparse grid and radial basis function interpolation, discrete least squares projection, and compressed sensing approximation. Moreover, hierarchical acceleration techniques are also incorporated to further reduce the overall complexity. In conclusion, rigorous complexity analyses of the new methods are provided, as are several numerical examples that illustrate the effectiveness of our approach.

Authors:
 [1];  [1];  [2];  [2]
  1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Dept. of Computational and Applied Mathematics
  2. Florida State Univ., Tallahassee, FL (United States). Dept. of of Scientific Computing
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE Office of Science (SC); Defense Advanced Research Projects Agency (DARPA); US Air Force Office of Scientific Research (AFOSR)
OSTI Identifier:
1327571
Grant/Contract Number:
AC05-00OR22725; SC0010678; 1854-V521-12; ERKJE45; ERKJ259; FA9550-15-1-0001; HR0011619523; 868-A017-15
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
SIAM Review
Additional Journal Information:
Journal Volume: 58; Journal Issue: 3; Journal ID: ISSN 0036-1445
Publisher:
Society for Industrial and Applied Mathematics
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; discontinuity detection; hyperspherical coordinates; adaptive approximations; sparse grid interpolation; discrete projection; least squares; compressed sensing; hierarchical methods

Citation Formats

Zhang, Guannan, Webster, Clayton G., Gunzburger, Max, and Burkardt, John. Hyperspherical Sparse Approximation Techniques for High-Dimensional Discontinuity Detection. United States: N. p., 2016. Web. doi:10.1137/16M1071699.
Zhang, Guannan, Webster, Clayton G., Gunzburger, Max, & Burkardt, John. Hyperspherical Sparse Approximation Techniques for High-Dimensional Discontinuity Detection. United States. doi:10.1137/16M1071699.
Zhang, Guannan, Webster, Clayton G., Gunzburger, Max, and Burkardt, John. Thu . "Hyperspherical Sparse Approximation Techniques for High-Dimensional Discontinuity Detection". United States. doi:10.1137/16M1071699. https://www.osti.gov/servlets/purl/1327571.
@article{osti_1327571,
title = {Hyperspherical Sparse Approximation Techniques for High-Dimensional Discontinuity Detection},
author = {Zhang, Guannan and Webster, Clayton G. and Gunzburger, Max and Burkardt, John},
abstractNote = {This work proposes a hyperspherical sparse approximation framework for detecting jump discontinuities in functions in high-dimensional spaces. The need for a novel approach results from the theoretical and computational inefficiencies of well-known approaches, such as adaptive sparse grids, for discontinuity detection. Our approach constructs the hyperspherical coordinate representation of the discontinuity surface of a function. Then sparse approximations of the transformed function are built in the hyperspherical coordinate system, with values at each point 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. Several approaches are used to approximate the transformed discontinuity surface in the hyperspherical system, including adaptive sparse grid and radial basis function interpolation, discrete least squares projection, and compressed sensing approximation. Moreover, hierarchical acceleration techniques are also incorporated to further reduce the overall complexity. In conclusion, rigorous complexity analyses of the new methods are provided, as are several numerical examples that illustrate the effectiveness of our approach.},
doi = {10.1137/16M1071699},
journal = {SIAM Review},
number = 3,
volume = 58,
place = {United States},
year = {Thu Aug 04 00:00:00 EDT 2016},
month = {Thu Aug 04 00:00:00 EDT 2016}
}

Journal Article:
Free Publicly Available Full Text
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Citation Metrics:
Cited by: 3works
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  • 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 newmore » 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.« less
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