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REAK: Reliability analysis through Error rate-based Adaptive Kriging

Zeyu Wang, Abdollah Shafieezadeh

arXiv:2002.01110v1stat.APcs.LG

TL;DR

Small failure probabilities in complex systems require costly simulations, creating a need for more efficient reliability analysis. REAK combines an error-rate bound derived with the Lindeberg-based Central Limit Theorem and adaptive effective sampling regions, and reports lower computational demand than AK-MCS and ISKRA in four examples.

  • Problem

    Reliability analysis for complex systems with small failure probabilities requires many costly simulations, while existing stopping criteria lack a direct link to failure-probability accuracy.

  • Method

    REAK derives a maximum error rate from wrong-sign Kriging estimates and integrates it with adaptively updated effective sampling regions and error-based stopping.

  • Results

    REAK reduced performance-function calls by 21%–59% versus AK-MCS and ISKRA across four examples at a 5% error-rate threshold.

  • Takeaways & Limitations

    REAK provides error-rate control while reducing computational demand, enabling a balance between accuracy and computational cost.

  • Takeaways & Limitations

    Very high-dimensional problems remain challenging because Kriging requires multiple inversions of large covariance matrices, and REAK assumes wrong-sign events are uncorrelated.

Abstract

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As models in various fields are becoming more complex, associated computational demands have been increasing significantly. Reliability analysis for these systems when failure probabilities are small is significantly challenging, requiring a large number of costly simulations. To address this challenge, this paper introduces Reliability analysis through Error rate-based Adaptive Kriging (REAK). An extension of the Central Limit Theorem based on Lindeberg condition is adopted here to derive the distribution of the number of design samples with wrong sign estimate and subsequently determine the maximum error rate for failure probability estimates. This error rate enables optimal establishment of effective sampling regions at each stage of an adaptive scheme for strategic generation of design samples. Moreover, it facilitates setting a target accuracy for failure probability estimation, which is used as stopping criterion for reliability analysis. These capabilities together can significantly reduce the number of calls to sophisticated, computationally demanding models. The application of REAK for four examples with varying extent of nonlinearity and dimension is presented. Results indicate that REAK is able to reduce the computational demand by as high as 50% compared to state-of-the-art methods of Adaptive Kriging with Monte Carlo Simulation (AK-MCS) and Improved Sequential Kriging Reliability Analysis (ISKRA).

1. INTRODUCTION

Reliability analysis must estimate failure probabilities while limiting costly evaluations of complex limit-state functions. The paper motivates REAK as a Kriging-based approach that links sampling and stopping decisions to error control.

  • Monte Carlo simulation can be accurate, but its large number of limit-state-function calls is costly for complex applications.
  • Surrogate models, especially Kriging, replace costly models using limited design points and support active learning through predictive uncertainty.
  • Existing adaptive methods use candidate samples and learning functions, but candidate-set size can be difficult to define, especially in high dimensions.
  • Conventional stopping criteria such as max(EFF) < 0.001 and min(U) >= 2 are arbitrary and lack a direct link to failure-probability accuracy.
  • REAK derives a maximum error rate from wrong-sign estimates using the Lindeberg-based Central Limit Theorem and uses it to control stopping and sampling.
  • Effective sampling regions prioritize important candidate samples and progressively reduce the region threshold, reducing unnecessary performance-function calls.

2. AK-MCS APPROACH

AK-MCS combines Kriging uncertainty with sequential learning to refine a surrogate and estimate failure probability using an expanding Monte Carlo candidate set. Its learning and stopping rules govern which points are evaluated and when the estimate is accepted.

  • 2.1 Kriging Model: Kriging models combine interpolation and regression while providing estimated responses and uncertainty for active selection of training points.
  • 2.1 Kriging Model: The ordinary Kriging model used here has constant basis function and regression coefficient, with a Gaussian-process residual component.
  • 2.2. Learning Function: AK-MCS learning functions select points for surrogate refinement using stochastic information, including proximity to the limit state and predictive variance.
  • 2.2. Learning Function: The EFF rule usually stops when max(EFF(x)) <= 0.001, while the U rule often stops at min(U(x)) >= 2, corresponding to a wrong-sign probability below 0.023.
  • 2.3. Reliability Estimation Using AK-MCS: AK-MCS begins with a small candidate set and increases it until the estimated failure probability satisfies a coefficient-of-variation threshold.

3. REAK: THE PROPOSED METHOD

REAK combines adaptive Kriging with effective sampling regions and a predicted maximum error rate to control reliability-analysis sampling and stopping. Its regions are refined by decreasing α, while error-rate estimation uses wrong-sign predictions and a Lindeberg-based approximation.

  • REAK combines effective sampling regions and maximum error-rate prediction with AK-MCS for reliability assessment.
  • Effective Sampling Regions: Effective sampling regions retain candidate samples whose probability density exceeds a threshold, excluding low-density points with negligible contribution to the estimated failure probability.
  • Effective Sampling Regions: REAK decreases α sequentially from a high initial value until the true error rate approaches but remains below the prescribed threshold.
  • Maximum Error Rate Estimation: The maximum error-rate estimate is derived by modeling wrong sign estimates as Bernoulli variables and applying a Central Limit Theorem based on the Lindeberg condition.
  • REAK Algorithm: The resulting conservative maximum error rate is integrated into REAK as a stopping criterion within the effective sampling regions.
  • REAK Algorithm: Unlike ISKRA’s fixed α, REAK adaptively decreases α while controlling the estimated error-rate constraint.

4. NUMERICAL INVESTIGATION

Across reliability problems with varying nonlinearity and dimension, REAK generally reduced performance-function calls while maintaining error-control capabilities. Its efficiency advantage was demonstrated against AK-MCS and ISKRA, although very high-dimensional problems remain challenging.

  • Four-Boundary Series System: 58.36 average calls for REAK at 𝜖_thr=0.05 were below 74.60 for ISKRA and 90.96 for AK-MCS; at 𝜖_thr=0.01, REAK required 79.84 calls versus 90.96 for AK-MCS.Average calls decreased as the error-rate threshold increased.
  • Four-Boundary Series System: REAK’s estimated maximum error rate was close to but larger than the true error rate and remained below the threshold, satisfying 𝜖̅ ≤ 𝜖̅_max ≤ 𝜖_thr.For ISKRA, the average true error rate was substantially below its coefficient or threshold; when α=0.05, 𝜖̅=0.002.
  • Four-Boundary Series System: The estimated maximum error exceeded the true error on average; for 𝜖_thr=0.01, its C.O.V. was 0.547 and the corresponding probability was 91.80%.Reported probabilities for other thresholds were 97.27% and 93.25%, close to the 95% confidence level.
  • Modified Rastrigin Function: For the modified Rastrigin function at 𝜖_thr=0.05, REAK used 230 training points versus 520 for AK-MCS and 559 for ISKRA.At 𝜖_thr=0.03, REAK used 276 calls versus 574 for ISKRA and 520 for AK-MCS; at 𝜖_thr=0.01, its estimated maximum error was 0.51% versus a true error of 0.43%.
  • Modified Rastrigin Function: In the modified Rastrigin simulations, successful estimates satisfying 𝜖̂_max ≥ 𝜖 were close to 95%; at 𝜖_thr=0.03, the probability was 94.55%.The authors attribute the reduction in calls and reliable maximum-error estimate to adaptive expansion of effective sampling regions with error-rate control.
  • Modified Cantilever Tube: For the high-dimensional structure, REAK reduced calls by 33% and 21% versus AK-MCS and ISKRA at 𝜖_thr=0.05, and by 30% and 17% at 𝜖_thr=0.03.At 𝜖_thr=0.01, REAK’s mean call count was larger than ISKRA’s; success probabilities that 𝜖̂_max ≥ 𝜖 were 99.97%, 99.58%, and 98.5% for thresholds 0.05, 0.03, and 0.01.
  • Limitations: REAK’s very high-dimensional performance is limited by Kriging parameter estimation, which requires multiple inversions of large covariance matrices.Dimension-reduction techniques are identified as a possible future improvement.

5. CONCLUSION

REAK combines error-rate control, effective sampling regions, and Kriging-based active learning to reduce performance-function calls while controlling failure-probability estimation error. Across four examples, it used fewer calls than AK-MCS and ISKRA and enabled balancing accuracy against computational demand.

  • REAK derives a maximum error rate for failure-probability estimates and integrates it with effective sampling regions for Kriging-based active learning.The method uses the Central Limit Theorem for non-identical random variables under Lindeberg’s condition.
  • REAK reduced performance-function calls by 52% and 50% in example 1, 56% and 59% in example 2, 51% and 50% in example 3, and 33% and 21% in example 4 versus AK-MCS and ISKRA, respectively.These reductions are reported for an error-rate threshold of 5%.
  • The maximum error rate remained close to but larger than the true error rate and converged toward it as the number of training samples increased.
  • Increasing the error-rate threshold decreased the number of performance-function calls, enabling a balance between accuracy and computational demand.
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