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Convergence analysis of multifidelity Monte Carlo estimation

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Abstract

The multifidelity Monte Carlo method provides a general framework for combining cheap low-fidelity approximations of an expensive high-fidelity model to accelerate the Monte Carlo estimation of statistics of the high-fidelity model output. In this work, we investigate the properties of multifidelity Monte Carlo estimation in the setting where a hierarchy of approximations can be constructed with known error and cost bounds. Our main result is a convergence analysis of multifidelity Monte Carlo estimation, for which we prove a bound on the costs of the multifidelity Monte Carlo estimator under assumptions on the error and cost bounds of the low-fidelity approximations. The assumptions that we make are typical in the setting of similar Monte Carlo techniques. Numerical experiments illustrate the derived bounds.

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Acknowledgements

The first and the third author were supported in part by the AFOSR MURI on multi-information sources of multi-physics systems under Award Number FA9550-15-1-0038, program manager Jean-Luc Cambier, and by the United States Department of Energy Applied Mathematics Program, Awards DE-FG02-08ER2585 and DE-SC0009297, as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center. The second author was supported by the US Department of Energy Office of Science grant DE-SC0009324 and the Air Force Office of Scientific Grant FA9550-15-1-0001. Some of the numerical examples were computed on the computer cluster of the Munich Centre of Advanced Computing.

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Authors and Affiliations

  1. Department of Mechanical Engineering and Wisconsin Institute for Discovery, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Benjamin Peherstorfer

  2. Department of Scientific Computing, Florida State University, 400 Dirac Science Library, Tallahassee, FL, 32306-412, USA

    Max Gunzburger

  3. Department of Aeronautics and Astronautics, MIT, Cambridge, MA, 02139, USA

    Karen Willcox

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  1. Benjamin Peherstorfer
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  3. Karen Willcox
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Correspondence to Benjamin Peherstorfer.

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Peherstorfer, B., Gunzburger, M. & Willcox, K. Convergence analysis of multifidelity Monte Carlo estimation. Numer. Math. 139, 683–707 (2018). https://doi.org/10.1007/s00211-018-0945-7

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  • DOI: https://doi.org/10.1007/s00211-018-0945-7

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