A hyper-spherical adaptive sparse-grid method for high-dimensional discontinuity detection

Zhang, Guannan¹  Webster, Clayton G¹  Gunzburger, Max D¹  Burkardt, John V¹ 

[1] ORNL

关键词: discontinuity detection;    hyper-spherical coordinate system;    adaptive sparse grid;    rare event;    hierarchical acceleration;   

DOI  :  10.2172/1126551
RP-ID  :  ORNL/TM-2014/36
PID  :  OSTI ID: 1126551
Others  :  Other: KJ0401000
Others  :  ERKJE45

美国|英语

来源: SciTech Connect

【 摘 要 】

This work proposes and analyzes a hyper-spherical adaptive hi- erarchical sparse-grid method for detecting jump discontinuities of functions in high-dimensional spaces is proposed. The method is motivated by the the- oretical and computational inefficiencies of well-known adaptive sparse-grid methods for discontinuity detection. Our novel approach constructs a func- tion representation of the discontinuity hyper-surface of an N-dimensional dis- continuous quantity of interest, by virtue of a hyper-spherical transformation. Then, a sparse-grid approximation of the transformed function is built in the hyper-spherical coordinate system, whose value at each point is estimated by solving a one-dimensional discontinuity detection problem. Due to the smooth- ness of the hyper-surface, the new technique can identify jump discontinuities with significantly reduced computational cost, compared to existing methods. Moreover, hierarchical acceleration techniques are also incorporated to further reduce the overall complexity. Rigorous error estimates and complexity anal- yses of the new method are provided as are several numerical examples that illustrate the effectiveness of the approach.