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Multilevel quasi-Monte Carlo for optimization under uncertainty

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Abstract

This paper considers the problem of optimizing the average tracking error for an elliptic partial differential equation with an uncertain lognormal diffusion coefficient. In particular, the application of the multilevel quasi-Monte Carlo (MLQMC) method to the estimation of the gradient is investigated, with a circulant embedding method used to sample the stochastic field. A novel regularity analysis of the adjoint variable is essential for the MLQMC estimation of the gradient in combination with the samples generated using the circulant embedding method. A rigorous cost and error analysis shows that a randomly shifted quasi-Monte Carlo method leads to a faster rate of decay in the root mean square error of the gradient than the ordinary Monte Carlo method, while considering multiple levels substantially reduces the computational effort. Numerical experiments confirm the improved rate of convergence and show that the MLQMC method outperforms the multilevel Monte Carlo method and single level quasi-Monte Carlo method.

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Acknowledgements

PG is grateful to the DFG RTG1953 “Statistical Modeling of Complex Systems and Processes” for funding of this research. AVB is funded by PhD fellowship 72661 by the research foundation Flanders (FWO - Fonds Wetenschappelijk Onderzoek Vlaanderen).

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, ÖAW, Altenbergerstrasse 69, 4040, Linz, Austria

    Philipp A. Guth

  2. Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001, Leuven, Belgium

    Andreas Van Barel

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  1. Philipp A. Guth
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  2. Andreas Van Barel
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Correspondence to Philipp A. Guth.

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Guth, P.A., Van Barel, A. Multilevel quasi-Monte Carlo for optimization under uncertainty. Numer. Math. 154, 443–484 (2023). https://doi.org/10.1007/s00211-023-01364-w

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