A Computational Approach to Edge Detection
IEEE Transactions on Pattern Analysis and Machine Intelligence
Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
An introduction to support Vector Machines: and other kernel-based learning methods
An introduction to support Vector Machines: and other kernel-based learning methods
Computing
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Uncertainty propagation using Wiener-Haar expansions
Journal of Computational Physics
Polynomial Fitting for Edge Detection in Irregularly Sampled Signals and Images
SIAM Journal on Numerical Analysis
Why Are High-Dimensional Finance Problems Often of Low Effective Dimension?
SIAM Journal on Scientific Computing
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
SIAM Journal on Scientific Computing
Adaptive Edge Detectors for Piecewise Smooth Data Based on the minmod Limiter
Journal of Scientific Computing
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Determining the locations and discontinuities in the derivatives of functions
Applied Numerical Mathematics
SIAM Journal on Numerical Analysis
The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications
Journal of Computational Physics
Discontinuity detection in multivariate space for stochastic simulations
Journal of Computational Physics
Journal of Computational Physics
Numerical approach for quantification of epistemic uncertainty
Journal of Computational Physics
Dimension-wise integration of high-dimensional functions with applications to finance
Journal of Complexity
Journal of Computational Physics
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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 discontinuity 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.