A multilevel Monte Carlo method for computing failure probabilities

TL;DR

This paper introduces a multilevel Monte Carlo approach for efficiently estimating failure probabilities in complex numerical models with uncertain inputs, significantly reducing computational costs while maintaining accuracy.

Contribution

The paper develops a novel multilevel Monte Carlo method that combines quantile estimation with failure probability computation, improving efficiency over traditional methods.

Findings

Cost is proportional to a single model realization

Significant computational savings demonstrated in experiments

Method maintains accuracy while reducing costs

Abstract

We propose and analyze a method for computing failure probabilities of systems modeled as numerical deterministic models (e.g., PDEs) with uncertain input data. A failure occurs when a functional of the solution to the model is below (or above) some critical value. By combining recent results on quantile estimation and the multilevel Monte Carlo method we develop a method which reduces computational cost without loss of accuracy. We show how the computational cost of the method relates to error tolerance of the failure probability. For a wide and common class of problems, the computational cost is asymptotically proportional to solving a single accurate realization of the numerical model, i.e., independent of the number of samples. Significant reductions in computational cost are also observed in numerical experiments.