Linear and nonlinear dimension reduction strategies for multifidelity uncertainty propagation of nonparametric distributions

We consider the problem of estimating the expectation of quantities of interest of computationally expensive models when the distribution of the random inputs is either known analytically or provided through samples. Our goal is to accelerate the evaluation of these statistics by resorting to models with varying degree of accuracy and cost. More specifically, we assume that a cheaper low-fidelity surrogate of the high-fidelity model is available and is characterized, in general, by a different number of random inputs or different distributions; we refer to this situation as dissimilar parametrization. Under these conditions, we focus on the derivation of new multifidelity estimators as, for instance, multifidelity Monte Carlo estimators. To deal with the dissimilar parametrization and retain an efficient evaluation of statistics, we propose two different methodologies that rely on constructing a shared subspace of important directions employing normalizing flows and either linear or nonlinear dimensionality reduction techniques such as active subspaces and autoencoders, respectively. Both approaches lead to an increased correlation between low-fidelity and high-fidelity models and consequently to a multifidelity estimator with reduced variance. Numerical examples highlight the advantages of our approaches with respect to the standard single-fidelity Monte Carlo method.