Rare event estimation with sequential directional importance sampling (SDIS)

TL;DR

This paper introduces SDIS, a sequential directional importance sampling method that efficiently estimates rare event probabilities by combining Monte Carlo and importance sampling with adaptive parameter tuning.

Contribution

The paper presents a novel SDIS approach that improves rare event probability estimation using sequential importance sampling and adaptive parameter selection.

Findings

SDIS outperforms existing methods in various test cases.

The method effectively estimates small failure probabilities.

Adaptive parameter choice reduces variance in estimates.

Abstract

In this paper, we propose a sequential directional importance sampling (SDIS) method for rare event estimation. SDIS expresses a small failure probability in terms of a sequence of auxiliary failure probabilities, defined by magnifying the input variability. The first probability in the sequence is estimated with Monte Carlo simulation in Cartesian coordinates, and all the subsequent ones are computed with directional importance sampling in polar coordinates. Samples from the directional importance sampling densities used to estimate the intermediate probabilities are drawn in a sequential manner through a resample-move scheme. The latter is conveniently performed in Cartesian coordinates and directional samples are obtained through a suitable transformation. For the move step, we discuss two Markov Chain Monte Carlo (MCMC) algorithms for application in low and high-dimensional problems. Finally, an adaptive choice of the parameters defining the intermediate failure probabilities is proposed and the resulting coefficient of variation of the failure probability estimate is analyzed. The proposed SDIS method is tested on five examples in various problem settings, which demonstrate that the method outperforms existing sequential sampling reliability methods.