Testing multidimensional integration routines
Proc. of international conference on Tools, methods and languages for scientific and engineering computation
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
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Polynomial Fitting for Edge Detection in Irregularly Sampled Signals and Images
SIAM Journal on Numerical Analysis
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
Journal of Computational Physics
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
Determining the locations and discontinuities in the derivatives of functions
Applied Numerical Mathematics
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
Padé-Legendre approximants for uncertainty analysis with discontinuous response surfaces
Journal of Computational Physics
A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties
Journal of Computational Physics
Multi-element probabilistic collocation method in high dimensions
Journal of Computational Physics
Journal of Computational Physics
Numerical Methods for Stochastic Computations: A Spectral Method Approach
Numerical Methods for Stochastic Computations: A Spectral Method Approach
Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids
Journal of Computational Physics
Uncertainty Quantification given Discontinuous Model Response and a Limited Number of Model Runs
SIAM Journal on Scientific Computing
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We propose a multi-element stochastic collocation method that can be applied in high-dimensional parameter space for functions with discontinuities lying along manifolds of general geometries. The key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized. On each of the resulting elements the function is smooth and can be approximated using high-order methods with fast convergence properties. The decomposition strategy is in direct contrast to the traditional multi-element approaches which define the sub-domains by repeated splitting of the axes in the parameter space. Such methods are more prone to the curse-of-dimensionality because of the fast growth of the number of elements caused by the axis based splitting. The present method is a two-step approach. Firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth. The detector uses an efficient combination of the high-order polynomial annihilation technique along with adaptive sparse grids, and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low-dimensional. After partitioning, an adaptive technique based on the least orthogonal interpolant is used to construct a generalized Polynomial Chaos surrogate on each element. The adaptive technique reuses all information from the partitioning and is variance-suppressing. We present numerous numerical examples that illustrate the accuracy, efficiency, and generality of the method. When compared against standard locally-adaptive sparse grid methods, the present method uses many fewer number of collocation samples and is more accurate.