Langevin Bi-fidelity Importance Sampling for Failure Probability Estimation

Estimating failure probability is a key task in the field of uncertainty quantification. In this domain, importance sampling has proven to be an effective estimation strategy; however, its efficiency heavily depends on the choice of the biasing distribution. An improperly selected biasing distribution can significantly increase estimation error. One approach to address this challenge is to leverage a less expensive, lower-fidelity surrogate. Building on the accessibility to such a model and its derivative on the random uncertain inputs, we introduce an importance sampling-based estimator, termed the Langevin bi-fidelity importance sampling (L-BF-IS), which uses score-function-based sampling algorithms to generate new samples and substantially reduces the mean square error (MSE) of failure probability estimation. The proposed method demonstrates lower estimation error, especially in high-dimensional input spaces and when limited high-fidelity evaluations are available. The L-BF-IS estimator's effectiveness is validated through experiments with two synthetic functions and two real-world applications governed by partial differential equations. These real-world applications involve a composite beam, which is represented using a simplified Euler-Bernoulli equation as a low-fidelity surrogate, and a steady-state stochastic heat equation, for which a pre-trained neural operator serves as the low-fidelity surrogate.