Global sensitivity analysis of complex numerical simulators is often limited by the small number of model evaluations that can be afforded. In such settings, surrogate models built from a limited set of simulations can substantially reduce the computational burden, provided that the design of computer experiments is enriched efficiently. In this context, we propose an active learning approach that, for a fixed evaluation budget, targets the most informative regions of the input space to improve sensitivity analysis accuracy. More specifically, our method builds on recent advances in active learning for sensitivity analysis (Sobol' indices and derivative-based global sensitivity measures, DGSM) that exploit derivatives obtained from a Gaussian process (GP) surrogate. By leveraging the joint posterior distribution of the GP gradient, we develop acquisition functions that better account for correlations between partial derivatives and their impact on the response surface, leading to a more comprehensive and robust methodology than existing DGSM-oriented criteria. The proposed approach is first compared to state-of-the-art methods on standard benchmark functions, and is then applied to a real environmental model of pesticide transfers.

Many industrial problems require the use of black-box numerical simulators, for which it is essential to determine the set of so-called feasible input parameters. A set of parameters is feasible if the output of the code on these parameters satisfies given constraints, for example, by remaining below a certain threshold. Active learning is an effective approach to solve this type of problem, by sequentially enriching a design of experiments using a well-chosen acquisition criterion, based here on a Gaussian process surrogate model. In this work, we focus specifically on simulators with vector outputs. We propose several enrichment strategies to simultaneously explore the entire collection of feasible sets associated with each output component. These enrichment strategies are first tested and compared on analytical test functions, before being applied to the pre-calibration of a simulator dedicated to wind turbine design. The aim is to identify input parameter configurations that respect the vibration constraints imposed on the simulator outputs. Numerical results demonstrate the efficiency of the three proposed strategies, which are compared against two baseline strategies (random sampling and Sobol’ sequence).

In this work, we propose a novel method to tackle the problem of multiobjective optimization under parameteric uncertainties, by considering the Conditional Pareto Sets and Conditional Pareto Fronts. Based on those quantities we can define the probability of coverage of the Conditional Pareto Set which can be interpreted as the probability for a design to be optimal in the Pareto sense. Due to the computational cost of such an approach, we introduce an Active Learning method based on Gaussian Process Regression in order to improve the estimation of this probability, which relies on a reformulation of the EHVI. We illustrate those methods on a few toy problems of moderate dimension, and on the problem of designing a cabin to highlight the differences in solutions brought by different formulations of the problem.

In the context of air quality control, our objective is to quantify the impact of uncertain inputs such as meteorological conditions and traffic parameters on pollutant dispersion maps. It is worth noting that the majority of sensitivity analysis methods are designed to deal with scalar or vector outputs and are ill suited to a map-valued output space. To address this, we propose two classes of methods. The first technique focuses on pointwise indices. Sobol indices are calculated for each position on the map to obtain Sobol index maps. Additionally, aggregated Sobol indices are calculated. Another approach treats the maps as sets and proposes a sensitivity analysis of a set-valued output with three different types of sensitivity indices. The first ones are inspired by Sobol indices but are adapted to sets based on the theory of random sets. The second ones adapt universal indices defined for a general metric output space. The last set indices use kernel-based sensitivity indices adapted to sets. The proposed methodologies are implemented to carry out an uncertainty analysis for time-averaged concentration maps of pollutants in an urban environment in the Greater Paris area. This entails taking into account uncertain meteorological aspects, such as incoming wind speed and direction, and uncertain traffic factors, such as injected traffic volume, percentage of diesel vehicles, and speed limits on the road network.

Numerical models are essential for comprehending intricate physical phenomena in different domains. To handle their complexity, sensitivity analysis, particularly screening is crucial for identifying influential input parameters. Kernel-based methods, such as the Hilbert-Schmidt Independence Criterion (HSIC), are valuable for analyzing dependencies between inputs and outputs. Implementing HSIC requires data from the original model, which leads to the need of efficient sampling strategies to limit the number of costly numerical simulations. While, for independent input variables, existing sampling methods like Latin Hypercube Sampling (LHS) are effective in estimating HSIC with reduced variance, incorporating dependence is challenging. This article introduces a novel LHS variant, quantization-based LHS (QLHS), which leverages Voronoi vector quantization to address dependent inputs. The method provides good coverage of the range of variations in the input variables. The article outlines expectation estimators based on QLHS in various dependency settings, demonstrating their unbiasedness. The method is applied to several models of growing complexities, first on simple examples to illustrate the theory, then on more complex environmental hydrological models, when the dependence is known or not, and with more and more interactive processes and factors. The last application is on the digital twin of a French vineyard catchment (Beaujolais region) to design a vegetative filter strip and reduce water, sediment, and pesticide transfers from the fields to the river. QLHS is used to compute HSIC measures and independence tests, demonstrating its usefulness, especially in the context of complex models.

This work is motivated by goal-oriented sensitivity analysis of inputs/outputs of complex simulators. More precisely we are interested in the ranking of the uncertain input variables that impact the most a feasible design domain. Most sensitivity analysis methods deal with scalar outputs. In this paper, we propose a way to perform sensitivity analysis when dealing with set-valued outputs. Our new methodology is driven by sensitivity analysis on excursion sets but can also be applied to setvalued simulators as in viability field, or when dealing with maps such as pollutant concentration maps or flooding zone maps. We propose a method based on the Hilbert Schmidt Independence Criterion (HSIC) with a kernel tailored to sets as outputs. A first contribution is the proof that this kernel is characteristic (i.e injectivity of the embedding in the associated Reproducing Kernel Hilbert Space), a required property for the HSIC interpretation in a sensitivity analysis context. We propose then to compute the HSIC-ANOVA indices which allow a decomposition of the input contributions. Using these indices, we can identify which inputs should be neglected (screening) and we can rank the others by influence (ranking). The estimation of these indices is also adapted to the set-valued outputs. Finally we test the proposed method on two test cases of excursion sets.

Numerical modeling is essential for comprehending intricate physical phenomena in different domains. To handle complexity, sensitivity analysis, particularly screening, is crucial for identifying influential input parameters. Kernel-based methods, such as the Hilbert Schmidt Independence Criterion (HSIC), are valuable for analyzing dependencies between inputs and outputs. Moreover, due to the computational expense of such models, metamodels (or surrogate models) are often unavoidable. Implementing metamodels and HSIC requires data from the original model, which leads to the need for space-filling designs. While existing methods like Latin Hypercube Sampling (LHS) are effective for independent variables, incorporating dependence is challenging. This paper introduces a novel LHS variant, Quantization-based LHS, which leverages Voronoi vector quantization to address correlated inputs. The method ensures comprehensive coverage of stratified variables, enhancing distribution across marginals. The paper outlines expectation estimators based on Quantization-based LHS in various dependency settings, demonstrating their unbiasedness. The method is applied on several models of growing complexities, first on simple examples to illustrate the theory, then on more complex environmental hydrological models, when the dependence is known or not, and with more and more interactive processes and factors. The last application is on the digital twin of a French vineyard catchment (Beaujolais region) to design a vegetative filter strip and reduce water, sediment and pesticide transfers from the fields to the river. Quantization-based LHS is used to compute HSIC measures and independence tests, demonstrating its usefulness, especially in the context of complex models.

Many model inversion problems occur in industry. These problems consist in finding the set of parameter values such that a certain quantity of interest respects a constraint, for example remains below a threshold. In general, the quantity of interest is the output of a simulator, costly in computation time. An effective way to solve this problem is to replace the simulator by a Gaussian process regression, with an experimental design enriched sequentially by a well chosen acquisition criterion. Different inversion-adapted criteria exist such as the Bichon criterion (also known as expected feasibility function) and deviation number . There also exist a class of enrichment strategies (stepwise uncertainty reduction—SUR) which select the next point by measuring the expected uncertainty reduction induced by its selection. In this paper we propose a SUR version of the Bichon criterion. An explicit formulation of the criterion is given and test comparisons show good performances on classical test functions.

We consider Reliability-based Robust Design Optimization (RRDO) where it is sought to optimize the mean of an objective function while satisfying constraints in probability. The high computational cost of the simulations underlying the objective and constraints strongly limits the number of evaluations and makes this type of problems particularly challenging. The numerical cost issue and the parametric uncertainties have been addressed with Bayesian optimization algorithms which leverage Gaussian processes of the objective and constraint functions. Current Bayesian optimization algorithms call the objective and constraint functions simultaneously at each iteration. This is often not necessary and overlooks calculation savings opportunity. This article proposes a new efficient RRDO Bayesian optimization algorithm that optimally selects for evaluation, not only the usual design variables, but also one or several constraints along with the uncertain parameters. The algorithm relies on a multi-output Gaussian model of the constraints. The coupling of constraints and their separated selection are gradually implemented in three algorithm variants which are compared to a reference Bayesian approach. The results are promising in terms of convergence speed, accuracy and stability as observed on a two, a four and a 27-dimensional problem.

This paper deals with the optimization of the short-term production planning of a real cascade of run-of-river hydropower plants. Water inflows and electricity prices are subject to data uncertainty and they are modeled by a finite set of joint scenarios. The optimization problem is written with a two-stage stochastic dynamic mixed-integer linear programming formulation. This problem is solved by replacing the value function of the second stage with a surrogate model. We propose to evaluate the feasibility of fitting the surrogate model by supervised learning during a pre-processing step. The learning data set is constructed by Latin hypercube sampling after discretizing the functional inputs. The surrogate model is chosen among linear models and the dimension of the functional inputs is reduced by principal components analysis. Validation results for one simplified case study are encouraging. The methodology could however be improved to reduce the prediction errors and to be compatible with the time limit of the operational process.

As computer codes simulate complex physical phenomena, they involve a very large number of variables. To gain time, industrial experts build metamodels on a restricted set of variables, the most influential ones, while the others are fixed. The set of variables is then enlarged progressively to improve knowledge on the studied output. Several designs of experiment are generated, which belong to subspaces included in each other and of increasing dimension. The goal of this paper is to create a metamodel adapted to this inefficient design process, that exploits the structure of all previous runs. An approach based on Gaussian process regression and called seqGPR (sequential Gaussian process regression) is introduced. At each new step of the study (when new variables are released), the output is supposed to be the realization of the sum of two independent Gaussian processes. The first one models the output at the previous step. The second one is a correction term which must be null on the subspace studied at the previous step, that is to say null on a continuum of points. First, some candidate Gaussian processes for the correction terms are suggested. Then, an EM (expectation-maximization) algorithm is implemented to estimate the parameters of the processes. Finally, the metamodel seqGPR is compared to a standard kriging metamodel on three test cases and gives better results.

In this paper we deal with the estimation of a feasible set defined by an inequality constraint on the output of a time-consuming black-box simulator. We focus on the setting where the black-box simulator takes as inputs both a set of scalar controlled variables and a functional uncontrolled variable. We then place ourselves in a probabilistic framework, modelling the functional uncontrolled variable by a random process. The inequality constraint is formulated as the expectation of the output of the simulator conditional on the values taken by the set of controlled variables. We propose an original method to solve the above feasibility problem with a reduced number of evaluations of the costly simulator. A Gaussian Process model of the simulator is learned in the joint space of controlled and uncontrolled input variables, on the basis of a set of simulations which is enriched through a sequential procedure. This procedure aims to reduce the estimation error of the feasible set by evaluating the simulator on new points chosen sequentially in the joint input space according to specific enrichment criteria. It involves as a preliminary step the reduction of the dimension of the uncontrolled input space. A variation of this strategy is also proposed, which increases adaptively the dimension of the reduced space, leading to an improvement in terms of number of calls to the simulator. The procedure we propose is compared with other sampling procedures and another modelling approach on analytical examples. Finally our methodology is implemented on an automotive industrial application. For this application, the feasible set to be recovered is the set of values of controlled variables of a gas after-treatment device, leading to the respect of pollutant emission standards of a vehicle under driving profile uncertainties.

Significant amounts of pollutant are measured in surface water, their presence due in part to the use of pesticides in agriculture. One solution to limit pesticide transfer by surface runoff is to implement vegetative filter strips (VFS) along rivers. These buffer zones are identified as the best management practices of choice for runoff mitigation to prevent and limit the transfer of pollutants from agricultural fields to water resources. They are mandatory or highly advised depending on the country and conditions. Since their location is part of the farmer’s field, the sizing of these strips is a major issue. However, to be efficient, they need to be properly designed

Significant amounts of pollutant are measured in surface water, their presence due in part to the use of pesticides in agriculture. One solution to limit pesticide transfer by surface runoff is to implement vegetative filter strips (VFS) along rivers. The sizing of these strips is a major issue, with influencing factors that include local conditions (climate, soil, etc.). The BUVARD modeling toolkit was developed to design VFSs throughout France according to these properties. This toolkit includes the numerical model VFSMOD, which quantifies dynamic effects of VFS site-specific pesticide mitigation efficiency. However, the toolkit is quite complex to use with many input uncertain parameters (quantitative - such as the slope, the Curve Number - or qualitative -such as the soil type or the rainfall event), making it not easy to use for risk management. In this study, a metamodeling (or model dimension reduction) approach is proposed to ease the use of BUVARD and to help users design VFSs that are adapted to specific contexts. Different reduced models, or surrogates, are compared, based on Bayesian learning approaches or not: Polynomial Chaos Expansions, Mixed-kriging, and Deep-GP. Mixed-kriging is a kriging method that was implemented with a covariance kernel for a mixture of qualitative and quantitative inputs. Kriging and Deep-GP are built by couple of modalities and PCE and Mixed-kriging are built considering mixed quantitative and qualitative variables. As a last step, Finally, we perform a global sensitivity analysis with the help of the two surrogate models with the best accuracy. The results show that they give the same ranking of the importance of the input parameters. The metamodel is a simple way to provide a relevant first guess to help design the pollution reduction device. In addition, the surrogate model is a relevant uncertainty tool, to visualize the impact that lack of knowledge of some parameters of filter efficiency can have when performing risk analysis and management.

In this paper we propose an efficient sampling strategy to solve an inversion problem subjected to functional uncertainties. More precisely, we aim at characterizing a control variable region defined by exceedance above a prescribed threshold of specific Quantities of Interest (QoT). This study is motivated by an automotive industrial application consisting in the identification of the set of values of control variables of a gas after-treatment device, in line with pollutant emission standards of a vehicle under driving profile uncertainties. In that context, driving profile uncertainties are modelled by a functional random variable and the constrained response in the inversion problem is formulated as the expectation over this functional random variable only known through a set of realizations. As often in industrial applications, this problem involves time-consuming computational models. We thus propose an approach that uses Gaussian Process meta-models built on the joint space of control and uncertain input variables. Specifically, we define a learning criterion based on uncertainty in the excursion of the Gaussian Process and derive tractable expressions for variance reduction in such a framework. Applications to analytical examples, followed by the automotive industrial test case show the accuracy and the efficiency brought by the procedure we propose.

We consider the problem of chance constrained optimization where it is sought to optimize a function and satisfy constraints, both of which are affected by uncertainties. The real world declinations of this problem are particularly challenging because of their inherent computational cost. To tackle such problems, we propose a new Bayesian optimization method. It applies to the situation where the uncertainty comes from some of the inputs, so that it becomes possible to define an acquisition criterion in the joint controlled-uncontrolled input space. The main contribution of this work is an acquisition criterion that accounts for both the average improvement in objective function and the constraint reliability. The criterion is derived following the Stepwise Uncertainty Reduction logic and its maximization provides both optimal controlled and uncontrolled parameters. Analytical expressions are given to efficiently calculate the criterion. Numerical studies on test functions are presented. It is found through experimental comparisons with alternative sampling criteria that the adequation between the sampling criterion and the problem contributes to the efficiency of the overall optimization. As a side result, an expression for the variance of the improvement is given.

In this paper we propose an efficient sampling strategy to solve an inversion problem subjected to functional uncertainties. More precisely, we aim at characterizing a control variable region defined by exceedance above a prescribed threshold of specific Quantities of Interest (QoT). This study is motivated by an automotive industrial application consisting in the identification of the set of values of control variables of a gas after-treatment device, in line with pollutant emission standards of a vehicle under driving profile uncertainties. In that context, driving profile uncertainties are modelled by a functional random variable and the constrained response in the inversion problem is formulated as the expectation over this functional random variable only known through a set of realizations. As often in industrial applications, this problem involves time-consuming computational models. We thus propose an approach that uses Gaussian Process meta-models built on the joint space of control and uncertain input variables. Specifically, we define a learning criterion based on uncertainty in the excursion of the Gaussian Process and derive tractable expressions for variance reduction in such a framework. Applications to analytical examples, followed by the automotive industrial test case show the accuracy and the efficiency brought by the procedure we propose.