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

Journal Article · · SIAM Review
DOI:https://doi.org/10.1137/16M1071699· OSTI ID:1327571
 [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
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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.

Research Organization:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Organization:
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)
Grant/Contract Number:
AC05-00OR22725; SC0010678; 1854-V521-12; ERKJE45; ERKJ259; FA9550-15-1-0001; HR0011619523; 868-A017-15
OSTI ID:
1327571
Journal Information:
SIAM Review, Vol. 58, Issue 3; ISSN 0036-1445
Publisher:
Society for Industrial and Applied MathematicsCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 14 works
Citation information provided by
Web of Science

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Related Subjects

97 MATHEMATICS AND COMPUTING
discontinuity detection
hyperspherical coordinates
adaptive approximations
sparse grid interpolation
discrete projection
least squares
compressed sensing
hierarchical methods