High-dimensional and higher-order multifidelity Monte Carlo estimators
For uncertainty quantification practitioners, this work provides a theoretical and practical extension of multifidelity methods to broader classes of problems, though it is an incremental generalization of existing optimal scalar estimators.
The paper generalizes multifidelity Monte Carlo estimators to vector-valued quantities-of-interest, nonlinearly dependent models, and cases without closed-form error expressions, enabling optimal estimation of statistics like variance and sensitivity indices. The method is demonstrated on cardiac electrophysiology problems.
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.