Discontinuity detection in multivariate space for stochastic simulations

Rick Archibald, Anne Gelb, Rishu Saxena, Dongbin Xiu

Research output: Contribution to journalArticlepeer-review

24 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)2676-2689
Number of pages14
JournalJournal of Computational Physics
Volume228
Issue number7
DOIs
StatePublished - Apr 20 2009

Keywords

  • Generalized polynomial chaos method
  • Multivariate edge detection
  • Stochastic partial differential equations

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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