Minimal multi-element stochastic collocation for uncertainty quantification of discontinuous functions John D. Jakeman a,⇑,1 , Akil Narayan b,2 , Dongbin Xiu c,2 a Sandia National Laboratories, Albuquerque, NM 87185, United States b Department of Mathematics, University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, MA 02747, United States c Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States article info Article history: Received 22 February 2012 Received in revised form 12 February 2013 Accepted 13 February 2013 Available online 13 March 2013 Keywords: Uncertainty quantification Generalized polynomial chaos Stochastic collocation Multi-element Discontinuous functions abstract 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 decom- posed 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 effi- cient combination of the high-order polynomial annihilation technique along with adap- tive 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 gen- eralized 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. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction Uncertainty quantification (UQ) and stochastic computation have received much attention in recent years, with many re- search efforts devoted to the development of efficient numerical methods for complex systems. A particular focus is on the design of algorithms that are more efficient than the traditional Monte Carlo (MC) method, which, though extremely robust and easy to implement, can be prohibitively time-consuming for practical systems when high accuracy is required. 0021-9991/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcp.2013.02.035 ⇑ Corresponding author. Tel.: +1 5052849097. E-mail addresses: jdjakem@sandia.gov (J.D. Jakeman), akil.narayan@umassd.edu (A. Narayan), dxiu@purdue.edu (D. Xiu). 1 Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of Energys National Nuclear Security Administration under contract DE-AC04-94AL85000. 2 This was supported by AFOSR, DOE/NNSA, and NSF. Journal of Computational Physics 242 (2013) 790–808 Contents lists available at SciVerse ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp