arXiv subjects

M. Moustapha

Publications and source records attributed to M. Moustapha.

8 recordsLinked to original sources

arXiv2026

High-dimensional reliability-based design optimization using stochastic emulators

Reliability-based design optimization (RBDO) is traditionally formulated as a nested optimization and reliability problem. Although surrogate models are generally employed to improve efficiency, the approach remains computationally prohibitive in high-dimensional settings. This paper proposes a novel RBDO framework based on a stochastic simulator viewpoint, in which the deterministic limit-state function and the uncertainty in the model inputs are combined into a unified stochastic representation. Under this formulation, the system response conditioned on a given design is modeled directly through its output distribution, rather than through an explicit limit-state function. Stochastic emulators are constructed in the design space to approximate the conditional response distribution, enabling the semi-analytical evaluation of failure probabilities or associated quantiles without resorting to Monte Carlo simulation. Two classes of stochastic emulators are investigated, namely generalized lambda models and stochastic polynomial chaos expansions. Both approaches provide a deterministic mapping between design variables and reliability constraints, which breaks the classical double-loop structure of RBDO and allows the use of standard deterministic optimization algorithms. The performance of the proposed approach is evaluated on a set of benchmark problems with dimensionality ranging from low to very high, including a case with stochastic excitation. The results are compared against a Kriging-based approach formulated in the full input space. The proposed method yields substantial computational gains, particularly in high-dimensional settings. While its efficiency is comparable to Kriging for low-dimensional problems, it significantly outperforms Kriging as the dimensionality increases.

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arXiv2025

AL-SPCE - Reliability analysis for nondeterministic models using stochastic polynomial chaos expansions and active learning

Reliability analysis traditionally relies on deterministic simulators, where identical inputs yield identical outputs. However, many real-world systems exhibit stochastic behavior, producing non-repeatable outcomes even under identical conditions. Stochastic simulators account for this behavior by representing the response as a random variable, whose intrinsic variability must be considered in reliability analysis. While Monte Carlo simulation can address this problem, its computational cost is often prohibitive. Stochastic emulators have therefore been introduced as surrogate models capable of reproducing the random simulator response at reduced cost. Recent studies have shown their potential for reliability analysis, but accurate estimates may still require relatively large training sets, which can be impractical for expensive models. In this work, we propose an active learning framework to further reduce the computational effort. Focusing on stochastic polynomial chaos expansions (SPCE), we introduce a learning function that identifies regions relevant to reliability estimation where the emulator exhibits high predictive uncertainty. We further exploit the asymptotic normality of the maximum likelihood estimator to quantify local prediction uncertainty. The resulting methodology, termed active learning stochastic polynomial chaos expansions (AL-SPCE), is validated on three types of problems. In all cases, AL-SPCE significantly improves computational efficiency compared with previous surrogate-based approaches and direct Monte Carlo simulation, while maintaining accurate reliability estimates.

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arXiv2022

Multi-objective robust optimization using adaptive surrogate models for problems with mixed continuous-categorical parameters

Explicitly accounting for uncertainties is paramount to the safety of engineering structures. Optimization which is often carried out at the early stage of the structural design offers an ideal framework for this task. When the uncertainties are mainly affecting the objective function, robust design optimization is traditionally considered. This work further assumes the existence of multiple and competing objective functions that need to be dealt with simultaneously. The optimization problem is formulated by considering quantiles of the objective functions which allows for the combination of both optimality and robustness in a single metric. By introducing the concept of common random numbers, the resulting nested optimization problem may be solved using a general-purpose solver, herein the non-dominated sorting genetic algorithm (NSGA-II). The computational cost of such an approach is however a serious hurdle to its application in real-world problems. We therefore propose a surrogate-assisted approach using Kriging as an inexpensive approximation of the associated computational model. The proposed approach consists of sequentially carrying out NSGA-II while using an adaptively built Kriging model to estimate the quantiles. Finally, the methodology is adapted to account for mixed categorical-continuous parameters as the applications involve the selection of qualitative design parameters as well. The methodology is first applied to two analytical examples showing its efficiency. The third application relates to the selection of optimal renovation scenarios of a building considering both its life cycle cost and environmental impact. It shows that when it comes to renovation, the heating system replacement should be the priority.

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arXiv2022

Learning non-stationary and discontinuous functions using clustering, classification and Gaussian process modelling

Surrogate models have shown to be an extremely efficient aid in solving engineering problems that require repeated evaluations of an expensive computational model. They are built by sparsely evaluating the costly original model and have provided a way to solve otherwise intractable problems. A crucial aspect in surrogate modelling is the assumption of smoothness and regularity of the model to approximate. This assumption is however not always met in reality. For instance in civil or mechanical engineering, some models may present discontinuities or non-smoothness, e.g., in case of instability patterns such as buckling or snap-through. Building a single surrogate model capable of accounting for these fundamentally different behaviors or discontinuities is not an easy task. In this paper, we propose a three-stage approach for the approximation of non-smooth functions which combines clustering, classification and regression. The idea is to split the space following the localized behaviors or regimes of the system and build local surrogates that are eventually assembled. A sequence of well-known machine learning techniques are used: Dirichlet process mixtures models (DPMM), support vector machines and Gaussian process modelling. The approach is tested and validated on two analytical functions and a finite element model of a tensile membrane structure.

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arXiv2022

Active learning for structural reliability: survey, general framework and benchmark

Active learning methods have recently surged in the literature due to their ability to solve complex structural reliability problems within an affordable computational cost. These methods are designed by adaptively building an inexpensive surrogate of the original limit-state function. Examples of such surrogates include Gaussian process models which have been adopted in many contributions, the most popular ones being the efficient global reliability analysis (EGRA) and the active Kriging Monte Carlo simulation (AK-MCS), two milestone contributions in the field. In this paper, we first conduct a survey of the recent literature, showing that most of the proposed methods actually span from modifying one or more aspects of the two aforementioned methods. We then propose a generalized modular framework to build on-the-fly efficient active learning strategies by combining the following four ingredients or modules: surrogate model, reliability estimation algorithm, learning function and stopping criterion. Using this framework, we devise 39 strategies for the solution of $20$ reliability benchmark problems. The results of this extensive benchmark (more than $12,000$ reliability problems solved) are analyzed under various criteria leading to a synthesized set of recommendations for practitioners. These may be refined with a priori knowledge about the feature of the problem to solve, i.e. dimensionality and magnitude of the failure probability. This benchmark has eventually highlighted the importance of using surrogates in conjunction with sophisticated reliability estimation algorithms as a way to enhance the efficiency of the latter.

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arXiv2020

A two-level Kriging-based approach with active learning for solving time-variant risk optimization problems

Several methods have been proposed in the literature to solve reliability-based optimization problems, where failure probabilities are design constraints. However, few methods address the problem of life-cycle cost or risk optimization, where failure probabilities are part of the objective function. Moreover, few papers in the literature address time-variant reliability problems in life-cycle cost or risk optimization formulations; in particular, because most often computationally expensive Monte Carlo simulation is required. This paper proposes a numerical framework for solving general risk optimization problems involving time-variant reliability analysis. To alleviate the computational burden of Monte Carlo simulation, two adaptive coupled surrogate models are used: the first one to approximate the objective function, and the second one to approximate the quasi-static limit state function. An iterative procedure is implemented for choosing additional support points to increase the accuracy of the surrogate models. Three application problems are used to illustrate the proposed approach. Two examples involve random load and random resistance degradation processes. The third problem is related to load-path dependent failures. This subject had not yet been addressed in the context of risk-based optimization. It is shown herein that accurate solutions are obtained, with extremely limited numbers of objective function and limit state functions calls.

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arXiv2019

Surrogate-assisted reliability-based design optimization: a survey and a new general framework

Reliability-based design optimization (RBDO) is an active field of research with an ever increasing number of contributions. Numerous methods have been proposed for the solution of RBDO, a complex problem that combines optimization and reliability analysis. Classical approaches are based on approximation methods and have been classified in review papers. In this paper, we first review classical approaches based on approximation methods such as FORM, and also more recent methods that rely upon surrogate modelling and Monte Carlo simulation. We then propose a general framework for the solution of RBDO problems that includes three independent blocks, namely adaptive surrogate modelling, reliability analysis and optimization. These blocks are non-intrusive with respect to each other and can be plugged independently in the framework. After a discussion on numerical considerations that require attention for the framework to yield robust solutions to various types of problems, the latter is applied to three examples (using two analytical functions and a finite element model). Kriging and support vector machines together with their own active learning schemes are considered in the surrogate model block. In terms of reliability analysis, the proposed framework is illustrated using both crude Monte Carlo and subset simulation. Finally, the covariance-matrix adaptation - evolution scheme (CMA-ES), a global search algorithm, or sequential quadratic programming (SQP), a local gradient-based method, are used in the optimization block. The comparison of the results to benchmark studies show the effectiveness and efficiency of the proposed framework.

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arXiv2016

Quantile-based optimization under uncertainties using adaptive Kriging surrogate models

Uncertainties are inherent to real-world systems. Taking them into account is crucial in industrial design problems and this might be achieved through reliability-based design optimization (RBDO) techniques. In this paper, we propose a quantile-based approach to solve RBDO problems. We first transform the safety constraints usually formulated as admissible probabilities of failure into constraints on quantiles of the performance criteria. In this formulation, the quantile level controls the degree of conservatism of the design. Starting with the premise that industrial applications often involve high-fidelity and time-consuming computational models, the proposed approach makes use of Kriging surrogate models (a.k.a. Gaussian process modeling). Thanks to the Kriging variance (a measure of the local accuracy of the surrogate), we derive a procedure with two stages of enrichment of the design of computer experiments (DoE) used to construct the surrogate model. The first stage globally reduces the Kriging epistemic uncertainty and adds points in the vicinity of the limit-state surfaces describing the system performance to be attained. The second stage locally checks, and if necessary, improves the accuracy of the quantiles estimated along the optimization iterations. Applications to three analytical examples and to the optimal design of a car body subsystem (minimal mass under mechanical safety constraints) show the accuracy and the remarkable efficiency brought by the proposed procedure.

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