Reduced basis methods for nonlocal diffusion problems with random input data

Qingguang Guan, Max Gunzburger, Clayton G. Webster, Guannan Zhang

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

The construction, analysis, and application of reduced-basis methods for uncertainty quantification problems involving nonlocal diffusion problems with random input data is the subject of this work. Because of the lack of sparsity of discretized nonlocal models relative to analogous local partial differential equation models, the need for reduced-order modeling is much more acute in the nonlocal setting. In this effort, we develop reduced-basis approximations for nonlocal diffusion equations with affine random coefficients. Efficiency estimates of the proposed greedy reduced-basis methods are provided. Numerical examples are used to illustrate the effect varying various model parameters have on the efficiency and accuracy of the reduced-basis method relative to sparse-grid interpolation using the full finite element method. It is shown that the proposed reduced-basis approach can indeed provide substantial savings over standard sparse-grid methods.

Original languageEnglish
Pages (from-to)746-770
Number of pages25
JournalComputer Methods in Applied Mechanics and Engineering
Volume317
DOIs
StatePublished - Apr 15 2017

Funding

This material is based upon work supported in part by the U.S. Air Force of Scientific Research under grants 1854-V521-12 and FA9550-15-1-0001; the U.S. Defense Advanced Research Projects Agency, Defense Sciences Office under contract and award HR0011619523 and 1868-A017-15; the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contracts and awards ERKJ259, ERKJE45, ERKJ314, DE-SC0009324, and DE-SC0010678; and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC. for the U.S. Department of Energy under contract DE-AC05-00OR22725.

Keywords

  • Finite element methods
  • Nonlocal diffusion
  • Reduced-basis methods
  • Uncertainty quantification

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