A Hyperspherical Adaptive Sparse-Grid Method for High-Dimensional Discontinuity Detection
Abstract
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.
- Authors:
-
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Florida State Univ., Tallahassee, FL (United States)
- Publication Date:
- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
- OSTI Identifier:
- 1302912
- Grant/Contract Number:
- AC05-00OR22725; SC0010678
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM Journal on Numerical Analysis
- Additional Journal Information:
- Journal Volume: 53; Journal Issue: 3; Journal ID: ISSN 0036-1429
- Publisher:
- Society for Industrial and Applied Mathematics
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; discontinuity detection; hyperspherical coordinate system; adaptive sparse grid; rare events; hierarchical acceleration
Citation Formats
Zhang, Guannan, Webster, Clayton G., Gunzburger, Max D., and Burkardt, John V. A Hyperspherical Adaptive Sparse-Grid Method for High-Dimensional Discontinuity Detection. United States: N. p., 2015.
Web. doi:10.1137/140971531.
Zhang, Guannan, Webster, Clayton G., Gunzburger, Max D., & Burkardt, John V. A Hyperspherical Adaptive Sparse-Grid Method for High-Dimensional Discontinuity Detection. United States. https://doi.org/10.1137/140971531
Zhang, Guannan, Webster, Clayton G., Gunzburger, Max D., and Burkardt, John V. Wed .
"A Hyperspherical Adaptive Sparse-Grid Method for High-Dimensional Discontinuity Detection". United States. https://doi.org/10.1137/140971531. https://www.osti.gov/servlets/purl/1302912.
@article{osti_1302912,
title = {A Hyperspherical Adaptive Sparse-Grid Method for High-Dimensional Discontinuity Detection},
author = {Zhang, Guannan and Webster, Clayton G. and Gunzburger, Max D. and Burkardt, John V.},
abstractNote = {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.},
doi = {10.1137/140971531},
journal = {SIAM Journal on Numerical Analysis},
number = 3,
volume = 53,
place = {United States},
year = {Wed Jun 24 00:00:00 EDT 2015},
month = {Wed Jun 24 00:00:00 EDT 2015}
}
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