Multifidelity Monte Carlo Estimator for High-Dimensional Quantities of Interest

Zhibo Dai; Hamdi A. Tchelepi; Daniel M. Tartakovsky

cs.LGOct 18, 2026

Multifidelity Monte Carlo Estimator for High-Dimensional Quantities of Interest

Authors: Zhibo Dai, Hamdi A. Tchelepi, Daniel M. Tartakovsky

Organizations: Department of Energy Science and Engineering, Stanford University, Stanford, CA 94305 USA

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.

Uncertainty Quantification

Uncertainty quantification estimates how reliable predictions and scientific inferences are, and how uncertainty propagates into decisions.

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Bifidelity Parameter Estimation Using Conditional Diffusion Models

Caroline Tatsuoka, Minglei Yang, Dongbin Xiu, Guannan Zhang

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA · Fusion Energy Science Division, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA · Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

Simulation-Based Inference Surrogate Modeling Uncertainty Quantification Bayesian Inverse Problems Multifidelity Modeling Conditional Diffusion Models

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We present a bifidelity method for uncertainty quantification of parameter estimates in complex systems, leveraging generative models trained to sample the target conditional distribution. In the Bayesian inference setting, traditional parameter estimation methods rely on repeated simulations of potentially expensive forward models to determine the posterior distribution of the parameter values, which may result in computationally intractable workflows. Furthermore, methods such as Markov Chain Monte Carlo (MCMC) necessitate rerunning the entire algorithm for each new data observation, further increasing the computational burden. Hence, we propose a novel method for efficiently obtaining posterior distributions of parameter estimates for high-fidelity models given data observations of interest. The method first constructs a low-fidelity, conditional generative model capable of amortized Bayesian inference and hence rapid posterior density approximation over a wide-range of data observations. When higher accuracy is needed for a specific data observation, the method employs adaptive refinement of the density approximation. It uses outputs from the low-fidelity generative model to refine the parameter sampling space, ensuring efficient use of the computationally expensive high-fidelity solver. Subsequently, a high-fidelity, unconditional generative model is trained to achieve greater accuracy in the target posterior distribution. Both low- and high- fidelity generative models enable efficient sampling from the target posterior and do not require repeated simulation of the high-fidelity forward model. We demonstrate the effectiveness of the proposed method on several numerical examples, including cases with multi-modal densities, as well as an application in plasma physics for a runaway electron simulation model.

Abstract

Uncertainty Quantification

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