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Reliability-based design optimization using kriging surrogates and subset simulation
V. Dubourg, B. Sudret, J. -M. Bourinet
TL;DR
RBDO with expensive performance models needs both computational efficiency and error quantification beyond what direct simulation or most-probable-failure-point methods provide. The paper combines kriging surrogates, adaptive refinement, and subset simulation within gradient-based optimization, and reports competitive results with only a few dozen real performance-function evaluations. Its efficiency depends on the augmented reliability space, while kriging efficiency can decline as the number of variables and experiments grows.
Problem
RBDO is difficult for expensive performance models because direct simulation is unaffordable and most-probable-failure-point approaches do not quantify failure-probability estimation error.
Method
The strategy adaptively refines kriging surrogates, propagates surrogate error to failure-probability estimates, uses subset simulation for reliability and sensitivities, and reuses surrogates in an augmented reliability space.
Results
Convergence is achieved with only a few dozen evaluations of the real performance functions, while the proposed error measure quantifies and sequentially minimizes surrogate error on optimal failure probability.
Takeaways & Limitations
The proposed surrogate-based strategy is competitive with FORM-based counterparts while providing an empirical error measure for the reliability quantity of interest.
Takeaways & Limitations
Kriging efficiency decreases as the number of variables grows and when the design of experiments contains more than a few thousand experiments.
Abstract
from arXiv · showhide
The aim of the present paper is to develop a strategy for solving reliability-based design optimization (RBDO) problems that remains applicable when the performance models are expensive to evaluate. Starting with the premise that simulation-based approaches are not affordable for such problems, and that the most-probable-failure-point-based approaches do not permit to quantify the error on the estimation of the failure probability, an approach based on both metamodels and advanced simulation techniques is explored. The kriging metamodeling technique is chosen in order to surrogate the performance functions because it allows one to genuinely quantify the surrogate error. The surrogate error onto the limit-state surfaces is propagated to the failure probabilities estimates in order to provide an empirical error measure. This error is then sequentially reduced by means of a population-based adaptive refinement technique until the kriging surrogates are accurate enough for reliability analysis. This original refinement strategy makes it possible to add several observations in the design of experiments at the same time. Reliability and reliability sensitivity analyses are performed by means of the subset simulation technique for the sake of numerical efficiency. The adaptive surrogate-based strategy for reliability estimation is finally involved into a classical gradient-based optimization algorithm in order to solve the RBDO problem. The kriging surrogates are built in a so-called augmented reliability space thus making them reusable from one nested RBDO iteration to the other. The strategy is compared to other approaches available in the literature on three academic examples in the field of structural mechanics.
1 Introduction
The paper targets RBDO with expensive performance models by combining surrogate modeling and simulation-based reliability analysis. It formulates design variables as distribution hyperparameters and reviews nested, decoupled, single-loop, simulation, and surrogate-based alternatives.
- Problem formulation: RBDO embeds uncertain structural performance in probabilistic failure constraints while minimizing an objective over admissible design variables.The performance functions may involve expensive black-box models such as finite-element simulations.
- Problem formulation: The proposed formulation treats design variables θ as hyperparameters of the random vector X to simplify reliability-sensitivity computation.Deterministic design variables can instead be modeled as artificially random with sufficiently small variance.
- Problem formulation: The cost function remains deterministic in the formulation, so randomness in the cost induced by X is not fully represented.The authors note that this simplification is common because usual cost models are inexpensive to simulate.
- Related approaches: Nested RBDO becomes computationally difficult when reliability analysis relies on simulation for complex performance functions.Alternative families include sample-average refinement, decoupled reliability and optimization, single-loop reformulations, and stochastic subset optimization.
- Related approaches: The investigated surrogate-based approach replaces expensive performance functions with faster surrogates built and refined from evaluations of the real model.The paper presents surrogate methods as more flexible than classical first- or second-order reliability approximations.
2 The kriging surrogate
Kriging provides a probabilistic emulator of an expensive deterministic simulator, with prediction uncertainty that supports adaptive refinement. The paper develops its Gaussian-process construction, parameter estimation, interpolation property, and uncertainty caveats.
- Kriging concept: In this application, kriging replaces expensive performance functions with faster emulators sharing the simulator’s input and output spaces.The modeled performance functions are the time-consuming quantities used in RBDO.
- Kriging concept: Kriging represents prediction uncertainty as epistemic uncertainty that varies with the available knowledge at each input.This uncertainty reflects lack of knowledge rather than randomness in the deterministic simulator output.
- Gaussian-process model: Universal kriging combines a regression trend with a stationary zero-mean Gaussian process and autocovariance model.The model parameters and regression basis are inferred from observations collected at an experimental design.
- Prediction properties: The kriging predictor is an exact interpolator: at observed points, its variance is zero and its prediction equals the observation.This property holds for the stated regression and regular autocorrelation models.
- Parameter estimation: Maximum likelihood estimation is used in the applications to infer autocovariance parameters through a numerically tractable global optimization problem.Bayesian estimation can quantify additional parameter uncertainty but is difficult to incorporate in the proposed nested procedure without suitable prior information.
- Uncertainty caveat: The kriging predictive variance underestimates total uncertainty because it omits covariance-model selection and parameter-estimation uncertainty.Bayesian kriging accounts for this additional uncertainty but is difficult to propagate through the predictor.
3 Design of experiments for the kriging surrogate
The kriging design-of-experiments strategy targets epistemic uncertainty near the limit-state surface and adaptively adds multiple informative observations until reliability estimates meet an accuracy criterion.
- Refinement principle: Kriging refinement focuses on the vicinity of the limit-state surface, where prediction uncertainty affects reliability estimation.The margin contains points with uncertain predicted signs, so reducing its spread improves reliability accuracy.
- Population-based refinement: The proposed strategy adds a set of improvement points simultaneously because its refinement criteria are highly multimodal.This can also exploit computational platforms able to evaluate several performance-function simulations concurrently.
- Refinement principle: The epistemic uncertainty margin is weighted by a probability density to define a refinement criterion over the input space.The weighting density can be the original input PDF or a uniform PDF over a sufficiently large confidence region.
- Population-based refinement: MCMC sampling generates candidates concentrated near refinement-criterion modes, and K-means clustering selects a smaller population of improvement points.The reduced population is intended to cover the uncertainty margin while supporting multiple additions to the DOE.
- Stopping criterion: Reliability refinement stops when the spread between generalized reliability-index estimates is sufficiently small, typically using ǫβ = 10^-1–10^-2.Simulation-estimation uncertainty can be incorporated through lower and upper 95% confidence bounds.
4 The proposed adaptive surrogate-based RBDO strategy
The proposed RBDO strategy nests adaptively refined kriging surrogates within gradient-based optimization and builds reusable models in an augmented reliability space.
- Nested RBDO strategy: The method nests the kriging surrogate and adaptive refinement procedure inside a classical nested RBDO algorithm.Reliability and reliability-sensitivity analyses are performed on the surrogates using subset simulation.
- Augmented reliability space: A single global kriging surrogate is built and refined for all nested reliability analyses, avoiding reinitialization from an empty DOE at each iteration.The DOE must cover extreme design choices and random-variable values within the selected confidence region.
- Augmented reliability space: The augmented reliability space keeps dimension n by treating design variables as uncertainty in the random vector, unlike the tensor-product formulation.The augmented vector V ≡ X(Θ) incorporates instrumental design uncertainty and aleatory uncertainty.
- Augmented reliability space: The augmented probability density is illustrated for a Gaussian random variate with a uniform mean, with the augmented space represented by axis V.The design-parameter density is combined with the conditional density of the random vector.
- Nested RBDO strategy: The outer optimization uses Polak-He optimization with a quasi-SQP direction and Goldstein-Armijo line search.Reliability sensitivities are used in the quasi-SQP sub-optimization problem.
- Nested RBDO strategy: Subset simulation provides the nested reliability and reliability-sensitivity analyses, while sensitivity estimation reuses samples from failure-probability estimation.The sensitivity calculation therefore requires no additional simulation runs beyond postprocessing.
5 Applications
The applications validate the surrogate-based RBDO strategy against analytical and literature benchmarks, showing convergence, reliability estimation, and substantial reductions in performance-function evaluations. Across the examples, kriging with subset simulation preserves reliability information while making nested optimization computationally tractable.
- 5.1 Elastic buckling of a straight column – An analytical reference: The elastic-buckling benchmark minimizes average cross-sectional area by optimizing the means of random width and height while enforcing deterministic and probabilistic constraints.The analytical reference uses a rectangular column model with independent random variables and a reliability-based design formulation.
- 5.1.2 Numerical solution: The algorithm converges from both deterministic-optimum and oversized initial designs to the exact square-section solution, with numerical approximation µb = µh ≈231 mm.All deterministic and reliability-based constraints are satisfied, while cost and design variables stabilize.
- 5.1.2 Numerical solution: 20 performance-function evaluations suffice for the surrogate-based optimum, versus about 4 × 10^6 evaluations for the corresponding direct simulation-based RBDO.The kriging design of experiments is enriched once and then remains accurate across the nested design configurations.
- 5.2 Short column case: In the short-column and bracket examples, FORM-based methods can produce non-conservative or underestimated reliability, whereas subset simulation exposes these differences and kriging reduces simulation demand.For the short column, subset simulation gives βsim ≈3.19 with a 5% coefficient of variation; for the bracket, kriging reduces calls from 10^6 to 10^2 while providing an error measure.
6 Conclusion
The paper presents a kriging–subset simulation strategy for RBDO with expensive performance models, quantifying and reducing surrogate error during optimization. It achieves competitive results with few real-model evaluations, while noting scalability limits as problem dimensionality and DOE size increase.
- The proposed RBDO strategy combines kriging surrogates with subset simulation to quantify and sequentially reduce error in optimal failure-probability estimates.The method targets problems where direct simulation is unaffordable and MPFP-based approaches do not quantify failure-probability estimation error.
- The strategy achieved competitive performance against FORM-based approaches on RBDO examples using only a few dozen evaluations of the real performance functions.
- Building surrogates in the augmented reliability space enables reuse across nested RBDO iterations, saving performance-function evaluations.
- The adaptive refinement strategy can add several DOE observations simultaneously, allowing distributed computing to accelerate convergence.
- The number of experiments increases with the number of performance-function variables, and kriging loses numerical efficiency when the DOE exceeds a few thousand experiments.The authors identify this scalability issue as requiring further investigation and mention ongoing work with a nonlinear finite-element model involving 10 variables.