Multifidelity preconditioning of the cross-entropy method for rare event simulation and failure probability estimation

Benjamin Peherstorfer, Boris Kramer, Karen Willcox

Research output: Contribution to journalArticlepeer-review

Abstract

Accurately estimating rare event probabilities with Monte Carlo can become costly if for each sample a computationally expensive high-fidelity model evaluation is necessary to approximate the system response. Variance reduction with importance sampling significantly reduces the number of required samples if a suitable biasing density is used. This work introduces a multifidelity approach that leverages a hierarchy of low-cost surrogate models to efficiently construct biasing densities for importance sampling. Our multifidelity approach is based on the cross-entropy method that derives a biasing density via an optimization problem. We approximate the solution of the optimization problem at each level of the surrogate-model hierarchy, reusing the densities found on the previous levels to precondition the optimization problem on the subsequent levels. With the preconditioning, an accurate approximation of the solution of the optimization problem at each level can be obtained from a few model evaluations only. In particular, at the highest level, only a few evaluations of the computationally expensive high-fidelity model are necessary. Our numerical results demonstrate that our multifidelity approach achieves speedups of several orders of magnitude in a thermal and a reacting-flow example compared to the single-fidelity cross-entropy method that uses a single model alone.

Original languageEnglish (US)
Pages (from-to)737-761
Number of pages25
JournalSIAM-ASA Journal on Uncertainty Quantification
Volume6
Issue number2
DOIs
StatePublished - 2018

Keywords

  • Cross-entropy method
  • Failure probability estimation
  • Importance sampling
  • Monte Carlo
  • Multifidelity
  • Multilevel
  • Rare event simulation
  • Reduced models
  • Surrogate models
  • Variance reduction

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Statistics, Probability and Uncertainty
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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