Multifidelity Monte Carlo estimation with adaptive low-fidelity models

Benjamin Peherstorfer

Research output: Contribution to journalArticle

Abstract

Multifidelity Monte Carlo (MFMC) estimation combines low- and high-fidelity models to speed up the estimation of statistics of the high-fidelity model outputs. MFMC optimally samples the lowand high-fidelity models such that the MFMC estimator has minimal mean-squared error (MSE) for a given computational budget. In the setup of MFMC, the low-fidelity models are static; i.e., they are given and fixed and cannot be changed and adapted. We introduce the adaptive MFMC (AMFMC) method that splits the computational budget between adapting the low-fidelity models to improve their approximation quality and sampling the low- and high-fidelity models to reduce the MSE of the estimator. Our AMFMC approach derives the quasi-optimal balance between adaptation and sampling in the sense that our approach minimizes an upper bound of the MSE, instead of the error directly. We show that the quasi-optimal number of adaptations of the low-fidelity models is bounded even in the limit of an infinite budget. This shows that adapting low-fidelity models in MFMC beyond a certain approximation accuracy is unnecessary and can even be wasteful. Our AMFMC approach trades off adaptation and sampling and so avoids overadaptation of the low- fidelity models. Besides the costs of adapting low-fidelity models, our AMFMC approach can also take into account the costs of the initial construction of the low-fidelity models ("offline costs"), which is critical if low-fidelity models are computationally expensive to build such as reduced models and data-fit surrogate models. Numerical results demonstrate that our adaptive approach can achieve orders of magnitude speedups compared to MFMC estimators with static low-fidelity models and compared to Monte Carlo estimators that use the high-fidelity model alone.

Original languageEnglish (US)
Pages (from-to)579-603
Number of pages25
JournalSIAM-ASA Journal on Uncertainty Quantification
Volume7
Issue number2
DOIs
StatePublished - Jan 1 2019

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Keywords

  • Model reduction
  • Monte Carlo
  • Multifidelity
  • Multilevel
  • Surrogate models
  • Uncertainty quantification

ASJC Scopus subject areas

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

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