TY - JOUR

T1 - Discontinuity detection in multivariate space for stochastic simulations

AU - Archibald, Rick

AU - Gelb, Anne

AU - Saxena, Rishu

AU - Xiu, Dongbin

N1 - Funding Information:
The submitted manuscript has been authored by contractors [UT-Battelle LLC, manager of Oak Ridge National Laboratory (ORNL)] of the U.S. Government under Contract No. DE-AC05-00OR22725. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. Anne Gelb was partially supported by NSF DMS-0510813, NSF FRG DMS-0652833, NSF RUI-0608844 and NSF SCREMS DMS-0421846. Dongbin Xiu was partially supported by AFOSR FA9550-08-1-0353, DOE DE-FC52-08NA28617 and NSF CAREER DMS-0645035.

PY - 2009/4/20

Y1 - 2009/4/20

N2 - Edge detection has traditionally been associated with detecting physical space jump discontinuities in one dimension, e.g. seismic signals, and two dimensions, e.g. digital images. Hence most of the research on edge detection algorithms is restricted to these contexts. High dimension edge detection can be of significant importance, however. For instance, stochastic variants of classical differential equations not only have variables in space/time dimensions, but additional dimensions are often introduced to the problem by the nature of the random inputs. The stochastic solutions to such problems sometimes contain discontinuities in the corresponding random space and a prior knowledge of jump locations can be very helpful in increasing the accuracy of the final solution. Traditional edge detection methods typically require uniform grid point distribution. They also often involve the computation of gradients and/or Laplacians, which can become very complicated to compute as the number of dimensions increases. The polynomial annihilation edge detection method, on the other hand, is more flexible in terms of its geometric specifications and is furthermore relatively easy to apply. This paper discusses the numerical implementation of the polynomial annihilation edge detection method to high dimensional functions that arise when solving stochastic partial differential equations.

AB - Edge detection has traditionally been associated with detecting physical space jump discontinuities in one dimension, e.g. seismic signals, and two dimensions, e.g. digital images. Hence most of the research on edge detection algorithms is restricted to these contexts. High dimension edge detection can be of significant importance, however. For instance, stochastic variants of classical differential equations not only have variables in space/time dimensions, but additional dimensions are often introduced to the problem by the nature of the random inputs. The stochastic solutions to such problems sometimes contain discontinuities in the corresponding random space and a prior knowledge of jump locations can be very helpful in increasing the accuracy of the final solution. Traditional edge detection methods typically require uniform grid point distribution. They also often involve the computation of gradients and/or Laplacians, which can become very complicated to compute as the number of dimensions increases. The polynomial annihilation edge detection method, on the other hand, is more flexible in terms of its geometric specifications and is furthermore relatively easy to apply. This paper discusses the numerical implementation of the polynomial annihilation edge detection method to high dimensional functions that arise when solving stochastic partial differential equations.

KW - Generalized polynomial chaos method

KW - Multivariate edge detection

KW - Stochastic partial differential equations

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U2 - 10.1016/j.jcp.2009.01.001

DO - 10.1016/j.jcp.2009.01.001

M3 - Article

AN - SCOPUS:60149086517

SN - 0021-9991

VL - 228

SP - 2676

EP - 2689

JO - Journal of Computational Physics

JF - Journal of Computational Physics

IS - 7

ER -