Source-linked AI summary
An active-learning algorithm that combines sparse polynomial chaos expansions and bootstrap for structural reliability analysis
S. Marelli, B. Sudret
TL;DR
Structural reliability analysis needs efficient estimates of small failure probabilities, but PCEs can be inaccurate in response tails and lack local metamodel error estimates. The paper combines sparse PCE with bootstrap-based local error estimation and active experimental-design enrichment, demonstrating the approach on analytical and structural problems.
Problem
PCE use in structural reliability is limited by poor tail accuracy and the absence of a natural local metamodel error estimate.
Method
The method bootstraps regression-based sparse PCEs to estimate local prediction error and iteratively enriches an experimental design using an active-learning strategy near the limit-state surface.
Results
The approach is demonstrated on a series-system analytical benchmark, a truss structure, and a complex realistic frame structure problem.
Takeaways & Limitations
Bootstrap-PCE supplies local error information that supports active reliability analysis with sparse polynomial-chaos surrogates.
Takeaways & Limitations
FORM and SORM tend to fail for highly nonlinear structural models, while the order of magnitude of PF may be unknown in general applications.
Abstract
from arXiv · showhide
Polynomial chaos expansions (PCE) have seen widespread use in the context of uncertainty quantification. However, their application to structural reliability problems has been hindered by the limited performance of PCE in the tails of the model response and due to the lack of local metamodel error estimates. We propose a new method to provide local metamodel error estimates based on bootstrap resampling and sparse PCE. An initial experimental design is iteratively updated based on the current estimation of the limit-state surface in an active learning algorithm. The greedy algorithm uses the bootstrap-based local error estimates for the polynomial chaos predictor to identify the best candidate set of points to enrich the experimental design. We demonstrate the effectiveness of this approach on a well-known analytical benchmark representing a series system, on a truss structure and on a complex realistic frame structure problem.
1 Introduction
Structural reliability analysis requires estimating failure probabilities for complex engineering models, but conventional simulation and surrogate approaches face efficiency, nonlinearity, tail-accuracy, or local-error challenges. The paper addresses these gaps with bootstrap-based local error estimation for sparse PCE and an active-learning strategy evaluated on analytical and structural examples.
- Motivation: Monte Carlo simulation can require very large sample sets to estimate small failure probabilities in complex, computationally expensive engineering models.The cited passage indicates sample requirements typically scale on the order of ∼10k+2.
- Motivation: FORM and SORM can be more efficient than simulation but usually rely on linearisation and tend to fail for highly nonlinear structural models.
- Motivation: PCEs are efficient surrogate models with global convergence behaviour, yet their limited tail accuracy has restricted their use in structural reliability.
- Research gap: Active-learning surrogate methods require local error estimates to enrich model evaluations near the limit-state surface, whereas PCEs lack a natural equivalent to Kriging variance.
- Contribution: The proposed method derives a bootstrap-based local error estimator for regression-based sparse PCE and uses it in an active-learning strategy.
- Evaluation: The approach is showcased on a series-system benchmark and a realistic structural frame engineering example.
2 Methodology
The methodology combines sparse polynomial chaos regression with bootstrap resampling to quantify local prediction variability and an active reliability algorithm to refine the experimental design. A fast variant reuses the sparse basis, while the adaptive procedure targets limit-state regions and seeks Monte Carlo-comparable failure-probability estimates with fewer model evaluations.
- 2.1 Polynomial Chaos Expansions: PCE represents a model response as a generalized polynomial expansion in random inputs, with coefficients estimated from an experimental design by least-squares regression.The expansion uses orthogonal polynomial families associated with the input marginals.
- 2.1 Polynomial Chaos Expansions: Sparse regression is used when the coefficient count is high or the experimental design is limited, helping avoid overfitting and maintain feasible sample sizes.The implementation adopts a full degree-adaptive sparse PCE based on hybrid-LARS.
- 2.2 Bootstrap-PCE: Bootstrap resampling repeatedly samples the finite experimental design with replacement, producing multiple coefficient estimates and surrogate responses.The resulting estimators characterize variability caused by the finite experimental-design size.
- 2.2.2 Bootstrap-PCE: Bootstrap-PCE constructs B surrogate trajectories whose pointwise response samples support empirical local error bounds and bounds on derived quantities.The trajectories can be interpreted as bootstrap-resampled surrogate models.
- 2.2.2 Bootstrap-PCE: The fast bPCE approach identifies the sparse polynomial basis once and bootstraps only the final ordinary least-squares regression on that basis.This reduces computational cost when full sparse-basis bootstrapping is expensive, especially in high dimensions or with large designs.
- 2.3 Active bPCE-based reliability analysis: A-bPCE begins with an initial design, estimates failure probabilities on an MCS sample, evaluates a learning function, and enriches the design iteratively near the limit state.The algorithm adapts a prior active PC-Kriging MCS strategy and uses bPCE surrogates.
- 2.3 Active bPCE-based reliability analysis: The adaptive procedure aims to obtain a failure-probability estimate comparable to direct large-sample MCS using a much smaller experimental design.
4. Update the bPCE surrogate on the new ED and return to Step 1
The algorithm iteratively enriches an experimental design using bootstrap-based misclassification uncertainty, then returns a failure-probability estimate with uncertainty bounds. A fixed Monte Carlo sample can stabilize convergence, while clustered enrichment supports simultaneous evaluations.
- The algorithm terminates by returning the bPF from the current PCE and error bounds based on bootstrap replicates.
- The initial experimental design can use space-filling samples such as Latin hypercube sampling, pseudo-random sequences, or uniform ball sampling.
- A single sufficiently large XMCS sample is used throughout to reduce Monte Carlo noise and can produce stabler convergence.This common-random-numbers strategy simplifies notation but is not required by the algorithm.
- When the failure-probability scale is unknown, the recommended alternative is to target a desired coefficient of variation at each iteration.
- The bootstrap learning function ranks candidate points by misclassification probability, selecting points where failed and safe replicates are most balanced.UF BR = 1 corresponds to consistent classification, whereas UF BR = 0 represents maximum epistemic classification uncertainty.
- The single-point enrichment criterion can be extended to K points by clustering the limit-state margin and minimizing UF BR within each region.The K expensive model evaluations can be performed simultaneously when parallel computing is available.
3 Results on benchmark applications
The benchmark applications include the four-branch function, an analytical series-system reliability problem with multiple failure regions and a composite limit-state surface. Its simple two-dimensional structure provides a common test for the proposed reliability-analysis procedure.
- The four-branch function is a common benchmark for reliability-analysis functions.
- It represents a series system comprising four components with different failure criteria.
- Despite being analytical and simple, the benchmark contains multiple failure regions and a composite limit-state surface.Its limit state is two-dimensional.
a. Initial experimental design
The active-learning procedure starts from an initial experimental design, enriches it near the adaptively learned limit-state surface, and calibrates a sparse PCE with bootstrap-based variability estimates. Across analytical and structural examples, it achieves reliable failure-probability estimates with substantially reduced model-evaluation costs in the reported comparisons.
- Four-branch benchmark: The initial design contains Nini = 20 points, and three points are added at each enrichment iteration.The design is updated iteratively using bootstrap samples to identify regions with relatively large prediction variability.
- Four-branch benchmark: Convergence requires 49 iterations and Ntot = 185 model evaluations, with added points concentrating near the adaptively learned limit-state surface.The convergence plot reports bootstrap-based 95% confidence bounds for the estimated failure probability.
- Four-branch benchmark: The final sparse PCE for the four-branch example contains P = 12 basis elements with degree up to p = 5.Degree-adaptive sparse PCE models based on LARS are recalibrated at each iteration.
- Two-dimensional truss structure: For the truss example, A-bPCE converges with 129 model evaluations versus 300 for AK-MCS, while its estimate includes the reference value within the confidence interval.The reported comparison also states that A-bPCE provides a stable failure-probability estimate and confidence intervals at lower cost than FORM for this example.
- Top-floor displacement of a structural frame: For the structural frame, A-bPCE converges at approximately 200 model evaluations with a degree-2 sparse PCE containing 30 non-zero coefficients, and the reference solution lies within its confidence bounds.The reliability-index confidence bounds indicate stability within 2% of the calculated values.
4 Conclusions and outlook
The paper combines bootstrap resampling with sparse polynomial chaos expansions to provide local error estimates and drive active enrichment of a small experimental design. The approach shows performance comparable to AK-MCS across an analytical benchmark and two increasingly complex engineering applications, while motivating extensions for rare-event estimation and broader surrogate models.
- Conclusions: Bootstrap resampling and sparse regression enable local error estimation for the standard polynomial chaos predictor.This feature supports active-learning applications that were hindered by the lack of local PCE error estimates.
- Conclusions: The method uses local error estimates to greedily enrich a relatively small initial experimental design for efficiently estimating failure probabilities of complex systems.Its active-learning construction is analogous to AK-MCS.
- Results: Comparable performance to AK-MCS was obtained on a simple analytical benchmark and two high-dimensional engineering applications of increasing complexity.The applications comprise the paper’s benchmark and engineering demonstrations.
- Extensions: Reliability analysis can be extended beyond simple Monte Carlo simulation using importance sampling, line sampling, or subset simulation for better failure-probability estimates.Such extensions are particularly relevant at very low probabilities of failure.
- Outlook: Future directions include combining the approach with other regression-based surrogate techniques and using parallel computing when enriching designs with multiple points.The paper identifies these as extensions for computationally expensive models and broader surrogate modeling.
- Extensions: The bPCE approach can also be applied outside a pure reliability-analysis context because it provides an effective local error estimate for PCE.The paper points to reliability-based design optimization as an example of an external application context.