Adaptive Radial-Based Importance Sampling for Efficient Estimation of Low Failure Probabilities in Geotechnical Random-Field Problems

Crude Monte Carlo Simulation (MCS) is consistent and unbiased for estimating failure probabilities in geotechnical reliability analysis, but its computational cost scales inversely with Pf: for small failure probabilities (Pf < 10⁻³), tens of thousands of slope stability evaluations may be required to achieve acceptable estimation accuracy. This paper presents Adaptive Radial-Based Importance Sampling (ARBIS) as an efficient alternative. ARBIS relocates sampling effort from the entire standard normal space to a neighbourhood of the design point the point on the limit-state surface G(u) = 0 closest to the origin using a weighted proposal distribution. For the soil-nailed slope case study considered, ARBIS achieves equivalent Pf accuracy with fewer than one-tenth the number of model evaluations required by crude MCS, while simultaneously providing tighter confidence bounds on the Pf estimate. The efficiency advantage is most pronounced precisely in the regime small Pf, high dimensionality from RF discretisation where crude MCS is most costly.