The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications

Jasmine Foo, Xiaoliang Wan, George Em Karniadakis

Research output: Contribution to journalArticle

161 Scopus citations


Stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters. In this work, we present and examine the multi-element probabilistic collocation method (ME-PCM), which is a generalized form of the probabilistic collocation method. In the ME-PCM, the parametric space is discretized and a collocation/cubature grid is prescribed on each element. Both full and sparse tensor product grids based on Gauss and Clenshaw-Curtis quadrature rules are considered. We prove analytically and observe in numerical tests that as the parameter space mesh is refined, the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness. In addition, the L2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided. Numerical examples demonstrating adaptive ME-PCM are shown, including low-regularity problems and long-time integration. We test the ME-PCM on two-dimensional Navier-Stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions. While the convergence rate of ME-PCM deteriorates in 50 dimensions, the error in the mean and variance is two orders of magnitude lower than the error obtained with the Monte Carlo method using only a small number of samples (e.g., 100). The computational cost of ME-PCM is found to be favorable when compared to the cost of other methods including stochastic Galerkin, Monte Carlo and quasi-random sequence methods.

Original languageEnglish (US)
Pages (from-to)9572-9595
Number of pages24
JournalJournal of Computational Physics
Issue number22
StatePublished - Nov 20 2008


  • Domain decomposition
  • Sparse grids
  • Stochastic partial differential equations

Fingerprint Dive into the research topics of 'The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications'. Together they form a unique fingerprint.

  • Cite this