# Multifidelity probability estimation via fusion of estimators

## TL;DR

This paper introduces a multifidelity approach that fuses multiple estimators to efficiently and accurately estimate failure probabilities in complex models, reducing computational cost while maintaining precision.

## Contribution

It develops a novel unbiased fusion method for importance sampling estimators that optimally combines multiple models to minimize variance and computational effort.

## Key findings

- Fused estimator achieves lower variance than individual estimators.
- Method reduces computational cost by 65% in a turbulent flow model.
- Asymptotic analysis confirms optimality of the fusion approach.

## Abstract

This paper develops a multifidelity method that enables estimation of failure probabilities for expensive-to-evaluate models via information fusion and importance sampling. The presented general fusion method combines multiple probability estimators with the goal of variance reduction. We use low-fidelity models to derive biasing densities for importance sampling and then fuse the importance sampling estimators such that the fused multifidelity estimator is unbiased and has mean-squared error lower than or equal to that of any of the importance sampling estimators alone. By fusing all available estimators, the method circumvents the challenging problem of selecting the best biasing density and using only that density for sampling. A rigorous analysis shows that the fused estimator is optimal in the sense that it has minimal variance amongst all possible combinations of the estimators. The asymptotic behavior of the proposed method is demonstrated on a convection-diffusion-reaction partial differential equation model for which $10^5$ samples can be afforded. To illustrate the proposed method at scale, we consider a model of a free plane jet and quantify how uncertainties at the flow inlet propagate to a quantity of interest related to turbulent mixing. Compared to an importance sampling estimator that uses the high-fidelity model alone, our multifidelity estimator reduces the required CPU time by 65\% while achieving a similar coefficient of variation.