Multi-Index Stochastic Collocation for random PDEs

Abdul Lateef Haji Ali, Fabio Nobile, Lorenzo Tamellini, Raul Tempone

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

In this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most effective mixed differences to include in the MISC estimator: such optimization is a crucial step and allows us to build a method that, provided with sufficient solution regularity, is potentially more effective than other multi-level collocation methods already available in literature. We then provide a complexity analysis that assumes decay rates of product type for such mixed differences, showing that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one dimensional problem. We show the effectiveness of MISC with some computational tests, comparing it with other related methods available in the literature, such as the Multi-Index and Multilevel Monte Carlo, Multilevel Stochastic Collocation, Quasi Optimal Stochastic Collocation and Sparse Composite Collocation methods.
Original languageEnglish (US)
Pages (from-to)95-122
Number of pages28
JournalComputer Methods in Applied Mechanics and Engineering
Volume306
DOIs
StatePublished - Mar 28 2016

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Mechanics of Materials
  • Mechanical Engineering
  • Computational Mechanics
  • Computer Science Applications

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