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This paper advances Multifidelity Monte Carlo methods by extending the framework to vector-valued quantities of interest through matrix-based control variates, offering closed-form solutions for optimal coefficients. The research demonstrates practical variance reduction across physics-informed problems like advection-diffusion and two-phase flow, highlighting how pilot data reuse can lower mean-squared error in high-dimensional estimation tasks.
Multifidelity Monte Carlo Estimator for High-Dimensional Quantities of Interest
arXiv:2610.25016v1 Announce Type: new Abstract: Multifidelity Monte Carlo (MFMC) reduces the cost of estimating the statistics of quantities of interest (QoIs) computed via expensive high-fidelity model by using cheaper low-fidelity models as control variates. We extend MFMC to vector-valued QoIs by replacing the scalar control-variate coefficients with matrices and minimizing a weighted mean-squared error. The optimal full-matrix coefficients are available in closed form, and their variance reduction is summarized by an effective correlation that takes the place of the squared Pearson correlation in the closed-form MFMC design. Restricting the coefficients to block-diagonal, diagonal, and scalar classes gives a nested hierarchy of estimators with increasing variance reduction and increasing pilot-data demand, with closed-form coefficients for compatible weightings. For a scalar QoI obtained by linearly projecting the state, correcting the full state before projecting is never worse at the population level. Experiments on a Gaussian problem, a reduced-order model of an advection--diffusion problem, and a coarse-grid two-phase flow problem show that the best class depends on the pilot size relative to the number of covariance quantities it must estimate, and that reusing the pilot samples lowers the mean-squared error.
Abstract:Multifidelity Monte Carlo (MFMC) reduces the cost of estimating the statistics of quantities of interest (QoIs) computed via expensive high-fidelity model by using cheaper low-fidelity models as control variates. We extend MFMC to vector-valued QoIs by replacing the scalar control-variate coefficients with matrices and minimizing a weighted mean-squared error. The optimal full-matrix coefficients are available in closed form, and their variance reduction is summarized by an effective correlation that takes the place of the squared Pearson correlation in the closed-form MFMC design. Restricting the coefficients to block-diagonal, diagonal, and scalar classes gives a nested hierarchy of estimators with increasing variance reduction and increasing pilot-data demand, with closed-form coefficients for compatible weightings. For a scalar QoI obtained by linearly projecting the state, correcting the full state before projecting is never worse at the population level. Experiments on a Gaussian problem, a reduced-order model of an advection--diffusion problem, and a coarse-grid two-phase flow problem show that the best class depends on the pilot size relative to the number of covariance quantities it must estimate, and that reusing the pilot samples lowers the mean-squared error.
Machine Learning (cs.LG)
arXiv:2610.25016 [cs.LG]
(or arXiv:2610.25016v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.25016 Focus to learn more arXiv-issued DOI via DataCite
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