Efficient adaptive Kriging-based reliability analysis combining new learning function and error-based stopping criterion


The Kriging-based reliability analysis is extensively adopted in engineering structural reliability analysis for its capacity to achieve accurate failure probability estimation with high efficiency. Generally, the Kriging-based reliability analysis is an active-learning process that mainly includes three aspects: (1) the determination of the design space; (2) the rule of choosing new samples, i.e., the learning function; and (3) the stopping criterion of the active-learning process. In this work, a new learning function and an error-based stopping criterion are proposed to enhance the efficiency of the active-learning Kriging-based reliability analysis. First, the reliability-based lower confidence bounding (RLCB) function is proposed to select the update points, which can balance the exploration and exploitation through the probability density-based weight. Second, an improved stopping criterion based on the relative error estimation of the failure probability is developed to avoid the pre-mature and late-mature of the active-learning Kriging-based reliability analysis method. Specifically, the samples that have large probabilities to change their safety statuses are identified. The estimated relative error caused by these samples is derived as the stopping criterion. To verify the performance of the proposed RLCB function and the error-based stopping criterion, four examples with different complexities are tested. Results show that the RLCB function is competitive compared with state-of-the-art learning functions, especially for highly non-linear problems. Meanwhile, the new stopping criterion reduces the computational resource of the active-learning process compared with the state-of-the-art stopping criteria.

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Probability density function


Cumulative density function


Limit state function


Limit state


Kriging-based reliability analysis


Active-learning Kriging-based reliability analysis


Error-based stopping criterion


Low confidence bounding function


Reliability-based low confidence bounding function


Error-based stopping criterion using bootstrap confidence estimation

Cov :

Coefficient of variation


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This research has been supported by the National Natural Science Foundation of China (NSFC) under Grant No. 51805179, the National Defense Innovation Program under Grant No. 18-163-00-TS-004-033-01, and the Research Funds of the Maritime Defense Technologies Innovation.

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Correspondence to Jun Liu.

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Yi, J., Zhou, Q., Cheng, Y. et al. Efficient adaptive Kriging-based reliability analysis combining new learning function and error-based stopping criterion. Struct Multidisc Optim (2020). https://doi.org/10.1007/s00158-020-02622-3

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  • Reliability analysis
  • Kriging model
  • Learning function
  • Reliability-based lower confidence bounding
  • Error-based stopping criterion