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

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.
 [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:
OSTI Identifier:
Grant/Contract Number:
AC05-00OR22725; SC0010678; 1854-V521-12; ERKJE45; ERKJ259; FA9550-15-1-0001; HR0011619523; 868-A017-15;
Accepted Manuscript
Journal Name:
SIAM Review
Additional Journal Information:
Journal Volume: 58; Journal Issue: 3; Journal ID: ISSN 0036-1445
Society for Industrial and Applied Mathematics
Research Org:
Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Sponsoring Org:
ORNL LDRD Director's R&D; USDOE Office of Science (SC); U.S. Defense Advanced Research Projects Agency (DARPA); U.S. Air Force of Scientific Research
Country of Publication:
United States
97 MATHEMATICS AND COMPUTING discontinuity detection; hyperspherical coordinates; adaptive approximations; sparse grid interpolation; discrete projection; least squares; compressed sensing; hierarchical methods