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Metamodel-based importance sampling for structural reliability analysis

V. Dubourg, F. Deheeger, B. Sudret

arXiv:1105.0562v2stat.MEstat.ML

TL;DR

Expensive performance functions make direct reliability simulation impractical, while surrogate error is difficult to quantify. The paper combines adaptive kriging with importance sampling and corrects the surrogate-based probability using the original limit-state function. The approach is reported efficient on analytical and finite-element problems with up to 100 random variables.

  • Problem

    Expensive performance functions make direct Monte Carlo simulation computationally demanding, while surrogate substitution error is usually difficult to quantify.

  • Method

    The method adaptively refines a kriging surrogate, uses its probabilistic prediction to construct a quasi-optimal importance-sampling density, and applies a correction factor from the original limit-state function.

  • Results

    The approach is reported efficient on several application examples, handling reasonably high numbers of random variables and multiple design points.

  • Takeaways & Limitations

    The method provides a reliability-analysis strategy that combines surrogate-based sampling with correction using the original performance function.

  • Takeaways & Limitations

    Sampling from the quasi-optimal instrumental density requires Markov chain Monte Carlo because its exact value is not known and inverse-transform simulation is inapplicable.

Abstract

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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.

1. Introduction

Structural reliability analysis estimates failure probabilities, but expensive performance functions make direct simulation impractical and surrogate error difficult to quantify. The paper proposes an adaptive kriging-based importance-sampling approach with an explicit correction factor.

  • Motivation: Reliability analysis estimates system safety through the failure probability of a performance function under a probabilistic model.Monte Carlo simulation evaluates the performance function through samples from the input random vector.
  • Motivation: Monte Carlo becomes intractable for rare events because its coefficient of variation increases sharply as the failure probability approaches zero.The same issue occurs for very frequent events when evaluating the complement probability.
  • Existing approaches: Surrogate models reduce evaluation cost, but their substitution error is often difficult or impossible to quantify.The cited alternatives include response surfaces, support vector machines, neural networks, and kriging.
  • Existing approaches: Importance sampling concentrates samples near the limit-state surface, while subset simulation estimates failure through products of conditional probabilities.Both approaches are described as robust but often too computationally demanding for industrial applications.
  • Contribution: The proposed hybrid method adaptively refines a kriging surrogate, derives a quasi-optimal importance-sampling density, and multiplies an augmented probability by a correction factor.The augmented term uses the surrogate, whereas the correction factor uses evaluations of the original performance function.

2. Probabilistic classification using margin metamodels

The paper uses kriging as a margin metamodel that represents both a predicted limit-state response and reducible epistemic uncertainty. That probabilistic prediction supports smoother failure classification, including uncertainty-aware treatment of misclassified points.

  • Probabilistic classification using margin metamodels: A metamodel approximates a model from scattered experimental-design observations, and here it is used to classify the failure-indicator response.The design of experiments is selected to capture information about the underlying input-output relationship.
  • Probabilistic classification using margin metamodels: A margin metamodel provides a probabilistic response prediction whose variance depends on the information in the design of experiments.This spread represents reducible epistemic uncertainty and decreases as observations are added.
  • Gaussian-process based prediction: Kriging represents the performance function through a Gaussian-process model with a regression mean and a stationary Gaussian-process component.The correlation structure is parameterized through an autocorrelation model and its length-scale parameters.
  • Gaussian-process based prediction: The kriging predictor supplies both a mean approximation of the limit-state function and an epistemic prediction variance.At experimental-design observations, interpolation makes the prediction variance zero.
  • Probabilistic classification function: The method uses the complete probabilistic kriging prediction rather than only its mean surrogate for reliability analysis.The probabilistic classification function is based on the Gaussian nature of the kriging predictor.
  • Probabilistic classification function: At x_0, the probabilistic classifier assigns a 60% failure probability under epistemic prediction uncertainty, smoothing a deterministic misclassification.The real performance function classifies x_0 as safe, while the mean kriging predictor classifies it as failed.

3. Metamodel-based importance sampling

The proposed estimator combines a kriging-based quasi-optimal importance sampling density with a correction factor to account for metamodel error. Its two terms are estimated independently, while sampling from the instrumental density requires MCMC because its normalizing constant is unknown.

  • 3.1. Importance sampling: Importance sampling concentrates samples near the failure event, but the exact optimal density is unavailable because it contains the unknown failure probability.The paper therefore constructs a quasi-optimal density using the probabilistic classification function.
  • 3.2. A metamodel-based approximation of the optimal instrumental PDF: The probabilistic classification function is used as a surrogate for the failure indicator when constructing the quasi-optimal instrumental PDF.This replaces the impractical optimal density with a density derived from the kriging prediction.
  • 3.3. The metamodel-based importance sampling estimator: The failure probability equals the product of an augmented failure probability and a correction factor.The augmented term uses the surrogate, while the correction factor compares the real indicator with the probabilistic classification function.
  • 3.3. The metamodel-based importance sampling estimator: The correction factor equals one for a fully accurate kriging prediction and otherwise adjusts for epistemic uncertainty from the finite DOE.This preserves the target failure probability despite metamodel inaccuracy.
  • 3.3. The metamodel-based importance sampling estimator: The two estimator terms use independent samples from the original PDF and the quasi-optimal instrumental PDF, making both estimates unbiased under the stated sampling procedure.Their coefficient of variation can be approximated from the respective coefficients of variation when these are reasonably small.
  • 3.4. Sampling from the quasi-optimal instrumental density: The quasi-optimal instrumental density has an unknown normalizing constant and cannot be sampled by inverse transformation in its multidimensional nonparametric form.The implementation consequently uses MCMC, specifically slice sampling, with thinning to reduce correlation between successive states.

4. Adaptive refinement of the probabilistic classification function

The refinement strategy adaptively improves the probabilistic classification function used to construct a quasi-optimal importance-sampling density. It replaces single-point optimization with sampling and clustering, while using cross-validation to determine when refinement is sufficient.

  • The instrumental PDF is adaptively refined so the quasi-optimal density converges toward its optimal counterpart.
  • A short state-of-the-art: The U-criterion targets regions where the kriging prediction’s sign is most uncertain, especially near the mean prediction’s limit-state surface.
  • A short state-of-the-art: Reliability-oriented refinement focuses on probabilistically significant regions because misclassified points there most affect estimator optimality.
  • The proposed strategy: Sampling the in-fill criterion with MCMC, clustering with K-means, and evaluating cluster centers produces a batch of new observations for kriging refinement.The procedure samples a large population, reduces it to K cluster centers, evaluates the performance function, and updates the experimental design.
  • The proposed strategy: The batch strategy addresses multiple local optima and permits distributed performance-function evaluations.K-means reduces sampled points to the most significant modes of the in-fill criterion.
  • Stopping criterion: Leave-one-out cross-validation estimates the correction factor and stops refinement when the design exceeds 30 points and the estimate is approximately 1.For high-dimensional or highly nonlinear problems, the design size is limited to m_max = 1000.

5. Algorithmic implementation

The implementation separates metamodel construction from two independent Monte Carlo estimates of the augmented failure probability and correction factor. User-selected evaluation, variation, and batch-size limits govern the procedure.

  • The algorithm requires a maximum number of performance-function evaluations, a target coefficient of variation, and a batch size K.
  • An initial probabilistic-classification approximation is built to provide variance reduction, although this step may be skipped when an existing design of experiments is available.
  • The two estimator calculations are independent and can run in parallel: Monte Carlo estimates the correction factor and augmented failure probability.The implementation targets δ_ε = δ_corr = δ_target/2 for the two estimators.

6. Application examples

The approach is tested on an analytical limit-state function across dimensions up to 100 and on a nonlinear finite-element shell-roof problem with 93 random variables. It agrees with reference methods while requiring fewer evaluations of the real performance function.

  • Analytical limit-state function – Influence of the dimension: The analytical example evaluates dimensions n = {2, 50, 100} against crude Monte Carlo, targeting a 2% coefficient of variation.The proposed estimator is reported in Table 1 alongside the reference simulation.
  • Analytical limit-state function – Influence of the dimension: The proposed estimator agrees well with crude Monte Carlo, while its correction factor increases as the dimension grows.For n = 2, the kriging surrogate almost exactly matches the performance function near the limit state; at larger dimensions, surrogate accuracy decreases.
  • Analytical limit-state function – Influence of the dimension: The real-function calls NDOE + Ncorr are much fewer than the Monte Carlo calls N for the same targeted coefficient of variation.Six refinement iterations are required in all cases, and K equals the number of input random variables.
  • Nonlinear stability analysis of an imperfect shell roof: The shell-roof example uses a nonlinear finite-element buckling model with four independent random fields and 93 independent random variables.The fields represent shape imperfection, Young’s modulus, yield stress, and shell thickness; each material or thickness field is represented through a Karhunen–Loève expansion.
  • Nonlinear stability analysis of an imperfect shell roof: The proposed importance-sampling scheme produces a shell-roof failure probability in full agreement with subset simulation.The augmented failure probability is 2.06 × 10^-4 with a 5.70% coefficient of variation, and the correction factor is 0.641 with a 12.49% coefficient of variation.
  • Nonlinear stability analysis of an imperfect shell roof: Multiple FORM identifies four most probable failure configurations associated with extreme demand and capacity-field values.These four modes are combined as a serial system to obtain a Multi-FORM approximation.

7. Conclusion

The paper addresses metamodel error and the cost of variance-reduction methods by combining adaptive kriging with importance sampling. Its correction-factor formulation preserves unbiased failure-probability estimation, and applications show efficiency for high-dimensional and multimodal problems.

  • Conclusion: The proposed hybrid strategy targets both unquantified metamodel error and the cost of variance-reduction methods for expensive black-box performance functions.These concerns motivate combining adaptive metamodeling with importance sampling.
  • Conclusion: A probabilistic classification based on kriging uncertainty defines a quasi-optimal importance-sampling density and separates estimation into an augmented probability and a correction factor.The augmented probability uses the metamodel, while the correction factor uses the original limit-state function.
  • Conclusion: The method is reported efficient on application examples, handling reasonably many random variables and multiple design points.Further work is proposed to incorporate the algorithm into reliability-based design optimization.

Appendix A. Calculation of the coefficient of variation

Appendix A derives the variance and coefficient of variation of the proposed estimator, which is the product of two independent unbiased estimators.

  • Calculation of the coefficient of variation: The final estimator is defined as the product of two unbiased independent estimators, with variances σ1^2 and σ2^2.The appendix begins by computing the variance of this product.
  • Calculation of the coefficient of variation: Independence and unbiasedness allow the product-estimator variance to be rewritten using the individual estimator moments.The derivation invokes the König-Huyghens theorem.
  • Calculation of the coefficient of variation: The coefficient of variation of the product estimator is expressed through the coefficients of variation δi = σi/pi of its two factors.The appendix then considers the practically relevant range of target coefficients of variation.

Appendix B. The slice sampling technique

Appendix B presents slice sampling by augmenting the target density with a scalar auxiliary variable. Marginalizing the auxiliary variable recovers the target distribution, and conditional sampling defines the procedure.

  • The slice sampling technique: Slice sampling introduces an auxiliary scalar U and a joint density J for the augmented vector (U, X).The construction represents sampling under the target density curve or surface.
  • The slice sampling technique: Marginalizing J with respect to U yields the target density p, so sampling X from p can be performed through the augmented distribution.The auxiliary variable is then ignored after sampling.
  • The slice sampling technique: The slice-sampling procedure uses conditional distributions derived from J, including uniform sampling of X over the slice S.The slice is defined by points whose target-density value is at least the sampled auxiliary level.
  • The slice sampling technique: Algorithmic improvements make slice sampling scalable to pseudo-densities even when the slice is difficult to identify in high dimensions.The method does not require explicit knowledge of the target density’s normalizing constant.
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