High-dimensional and higher-order multifidelity Monte Carlo estimators

TL;DR

This paper extends multifidelity Monte Carlo methods to handle high-dimensional, vector-valued, and nonlinearly dependent models, providing a generalized framework for efficient uncertainty quantification.

Contribution

It introduces a generalized estimator framework for complex models, enabling optimal coefficient determination through an analytical solution and numerical estimation.

Findings

Enhanced efficiency in high-dimensional model estimation

Successful application to cardiac electrophysiology models

Analytical and numerical methods improve estimator performance

Abstract

Multifidelity Monte Carlo methods rely on a hierarchy of possibly less accurate but statistically correlated simplified or reduced models, in order to accelerate the estimation of statistics of high-fidelity models without compromising the accuracy of the estimates. This approach has recently gained widespread attention in uncertainty quantification. This is partly due to the availability of optimal strategies for the estimation of the expectation of scalar quantities-of-interest. In practice, the optimal strategy for the expectation is also used for the estimation of variance and sensitivity indices. However, a general strategy is still lacking for vector-valued problems, nonlinearly statistically-dependent models, and estimators for which a closed-form expression of the error is unavailable. The focus of the present work is to generalize the standard multifidelity estimators to the above cases. The proposed generalized estimators lead to an optimization problem that can be solved analytically and whose coefficients can be estimated numerically with few runs of the high- and low-fidelity models. We analyze the performance of the proposed approach on a selected number of experiments, with a particular focus on cardiac electrophysiology, where a hierarchy of physics-based low-fidelity models is readily available.