Quantifying uncertain system outputs via the multi-level Monte Carlo method -- distribution and robustness measures

TL;DR

This paper develops an adaptive multi-level Monte Carlo method for accurately estimating distributional and risk measures of complex random differential models, with novel error estimators and demonstrated efficiency.

Contribution

It introduces new computable error estimators for distribution and risk measure estimation, enabling optimal tuning of the MLMC hierarchy in an adaptive framework.

Findings

Efficient and robust adaptive MLMC algorithm demonstrated on numerical test cases.

Novel error estimators improve the accuracy of distribution and risk measure estimates.

Adaptive continuation-MLMC outperforms non-adaptive methods in complex models.

Abstract

In this work, we consider the problem of estimating the probability distribution, the quantile or the conditional expectation above the quantile, the so called conditional-value-at-risk, of output quantities of complex random differential models by the MLMC method. We follow the approach of (reference), which recasts the estimation of the above quantities to the computation of suitable parametric expectations. In this work, we present novel computable error estimators for the estimation of such quantities, which are then used to optimally tune the MLMC hierarchy in a continuation type adaptive algorithm. We demonstrate the efficiency and robustness of our adaptive continuation-MLMC in an array of numerical test cases.