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Characterization of Discontinuities in High-dimensional Stochastic Problems on Adaptive Sparse Grids

Summary: Characterization of Discontinuities in High-dimensional Stochastic
Problems on Adaptive Sparse Grids
Jakeman John
Richard Archibald
Dongbin Xiu
In this paper, we present a set of efficient algorithms for detection and identification of discontinuities
in high-dimensional space. The method is based on extension of polynomial annihilation for edge detection
in low dimensions. Compared to the earlier work, the present method poses significant improvements for
high-dimensional problems. The core of the algorithms relies on adaptive refinement of sparse grids. It
is demonstrated that in the commonly encountered cases where a discontinuity resides on a small subset
of the dimensions, the present method becomes "optimal", in the sense that the total number of points
required for function evaluations depends linearly on the dimensionality of the space. The details of the
algorithms will be presented and various numerical examples are utilized to demonstrate the efficacy of
the method.
Key Words: adaptive sparse grids, stochastic partial differential equations, multivariate edge detection,
generalized polynomial chaos method
1 Introduction
The behaviour of financial, chemical, mechanical, environmental and many other processes is often character-
ized by a large number of variables. The relationship between the variables that drive the system (inputs)


Source: Archibald, Richard - Computer Science and Mathematics Division, Oak Ridge National Laboratory


Collections: Computer Technologies and Information Sciences