Budget-limited distribution learning in multifidelity problems

Multifidelity methods are widely used for estimation of quantities of interest (QoIs) in uncertainty quantification using simulation codes of differing costs and accuracies. Many methods approximate numerical-valued statistics that represent only limited information of the QoIs. In this paper, we generalize the ideas in \cite{xu2021bandit} to develop a multifidelity method that approximates the distribution of scalar-valued QoI. Under a linear model hypothesis, we propose an exploration-exploitation strategy to reconstruct the full distribution, not just statistics, of a scalar-valued QoI using samples from a subset of low-fidelity regressors. We derive an informative asymptotic bound for the mean 1-Wasserstein distance between the estimator and the true distribution, and use it to adaptively allocate computational budget for parametric estimation and non-parametric approximation of the probability distribution. Assuming the linear model is correct, we prove that such a procedure is consistent and converges to the optimal policy (and hence optimal computational budget allocation) under an upper bound criterion as the budget goes to infinity. As a corollary, we obtain convergence of the approximated distribution in the mean 1-Wasserstein metric. The major advantages of our approach are that convergence to the full distribution of the output is attained under appropriate assumptions, and that the procedure and implementation require neither a hierarchical model setup, knowledge of cross-model information or correlation, nor \textit{a priori} known model statistics. Numerical experiments are provided in the end to support our theoretical analysis.