[Paper Review] Metamodel-based importance sampling for structural reliability analysis

Vincent Dubourg, François Deheeger+1 more|arXiv (Cornell University)|May 3, 2011

Probabilistic and Robust Engineering Design1 references22 citations

TL;DR

This paper proposes a metamodel-based importance sampling method for structural reliability analysis that uses kriging to construct a quasi-optimal importance sampling density, enabling accurate failure probability estimation with reduced computational cost. The approach combines a surrogate model with a bias-corrected correction term, achieving high efficiency even for problems with up to 100 random variables and rare failure events.

ABSTRACT

Structural reliability methods aim at computing the probability of failure of systems with respect to some prescribed performance functions. In modern engineering such functions usually resort to running an expensive-to-evaluate computational model (e.g. a finite element model). In this respect simulation methods, which may require $10^{3-6}$ runs cannot be used directly. Surrogate models such as quadratic response surfaces, polynomial chaos expansions or kriging (which are built from a limited number of runs of the original model) are then introduced as a substitute of the original model to cope with the computational cost. In practice it is almost impossible to quantify the error made by this substitution though. In this paper we propose to use a kriging surrogate of the performance function as a means to build a quasi-optimal importance sampling density. The probability of failure is eventually obtained as the product of an augmented probability computed by substituting the meta-model for the original performance function and a correction term which ensures that there is no bias in the estimation even if the meta-model is not fully accurate. The approach is applied to analytical and finite element reliability problems and proves efficient up to 100 random variables.

Motivation & Objective

• To address the high computational cost of Monte Carlo simulation in structural reliability analysis when the performance function is expensive to evaluate.
• To reduce variance in failure probability estimation for rare events without relying on expensive simulations.
• To provide a reliable method that accounts for metamodel error while maintaining unbiased estimation of failure probability.
• To enable efficient reliability analysis in high-dimensional problems (up to 100 random variables) using surrogate modeling and active learning.

Proposed method

• A kriging surrogate is constructed for the performance function using a limited number of evaluations of the original model.
• The kriging model is used to define a quasi-optimal importance sampling density centered around the failure region.
• The failure probability is estimated as the product of two independent estimators: one from the metamodel and one from a correction term.
• The correction term ensures unbiased estimation even when the kriging model is inaccurate, by accounting for the metamodel error.
• Active learning is employed to iteratively enrich the kriging model with samples near the limit state, improving accuracy with minimal additional evaluations.
• The method leverages the slice sampling technique to generate samples from the importance distribution efficiently.

Experimental results

Research questions

• RQ1 How can metamodeling be combined with importance sampling to reduce the computational burden of reliability analysis for expensive-to-evaluate models?
• RQ2 Can a kriging surrogate be used to define an effective importance sampling density that concentrates samples near the failure region?
• RQ3 How can the bias introduced by metamodel approximation be corrected to ensure an unbiased failure probability estimate?
• RQ4 What is the efficiency gain in terms of variance reduction and required sample size compared to standard Monte Carlo or surrogate-only methods?
• RQ5 How does the method scale with increasing dimensionality, particularly for problems with up to 100 random variables?

Key findings

• The proposed method achieves significant variance reduction compared to standard Monte Carlo, enabling accurate failure probability estimation with far fewer model evaluations.
• The bias-corrected estimator ensures that the final failure probability estimate remains unbiased even when the kriging surrogate is imperfect.
• The method maintains high accuracy and efficiency for problems with up to 100 random variables, demonstrating scalability beyond typical surrogate-based methods.
• Active learning strategy effectively targets new samples near the limit state, improving the kriging model's accuracy with minimal additional computational cost.
• The coefficient of variation of the final estimator is kept below 10% with a moderate number of samples, indicating high reliability and efficiency.
• The approach outperforms standard Monte Carlo and surrogate-only methods in terms of computational efficiency and robustness for rare failure events.