This work addresses the interpolation of probability measures within a spatial statistics framework. We develop a Kriging approach in the Wasserstein space, leveraging the quantile function representation of the one-dimensional Wasserstein distance. To mitigate the inaccuracies in semivariogram estimation that arise from sparse datasets, we combine this formulation with cross-validation techniques. In particular, we introduce a variant of the virtual cross-validation formulas tailored to quantile functions. The effectiveness of the proposed method is demonstrated on a controlled toy problem as well as on a real-world application from nuclear safety.

Classical many-body potentials that have a rather low number of parameters and are based on physically-inspired functional forms are expected to display a reasonable transferability, though their flexibility is limited. Ways to improve them when their transferability is unsatisfactory and when new interesting structures/targeted properties emerge are not clearly established. Here, model screening and sensitivity analysis techniques are combined to get insights on the intrinsic capabilities of classical potentials for typical sets of targeted properties, and we propose to use the outcomes of the sensitivity analysis to improve potential performances. Usefulness is illustrated on existing simple second moment potentials (SMA), and we refit one such model for the study of small irradiation defects in α-zirconium. Application of this approach is recommended for more complex interaction models with up to tens of parameters, such as other classical many-body potentials or tight-binding electronic structure models, that are possible to re-optimize while gaining a better understanding of the role of their parameters.

In the Best-Estimate Plus Uncertainty (BEPU) framework, the use of best-estimate code requires to go through a Verification, Validation and Uncertainty Quantification process (VVUQ). The relevance of the experimental data in relation to the physical phenomena of interest in the VVUQ process is crucial. Adequacy analysis of selected experimental databases addresses this problem. The outcomes of the analysis can be used to select a subset of relevant experimental data, to encourage designing new experiments or to drop some experiments from a database because of their substantial lack of adequacy. The development of a specific transparent and reproducible approach to analyze the relevance of experimental data for VVUQ still remains open and is the topic of this contribution.In this paper, the concept of adequacy initially introduced in the OECD/NEA SAPIUM (Systematic APproach for model Input Uncertainty quantification Methodology) activity is formalized. It is defined through two key properties, called representativeness and completeness, that allows considering the multifactorial dimension of the adequacy problem. A new systematic approach is then proposed to analyze the adequacy of a set of experimental databases. It relies on the introduction of two sets of criteria to characterize representativeness and completeness and on the use of multi-criteria decision analysis method to perform the analysis. Finally, the approach is applied in the framework of the new OECD/NEA ATRIUM activity which includes a set of practical IUQ exercises in thermal-hydraulics to test the SAPIUM guideline in determining input uncertainties and forward propagating them on an application case. It allows evaluating the adequacy of eight experimental databases coming from the Super Moby-dick, Sozzi-Sutherland and Marviken experiments and identifying the most adequate ones.

This paper deals with the approximation of discrete probability measure-valued data by a new subdivision scheme. Its construction relies on a coupling between linear subdivision and optimal transport. A mathematical analysis is performed to study its convergence. Two test cases are finally described to emphasize its capability: the first one is related to point cloud interpolation while the second one is a first attempt in the framework of image approximation.

This paper is devoted to the construction of a new fast-to-evaluate model for the prediction of 2D crack paths in concrete-like microstructures. The model generates piecewise linear cracks paths with segmentation points selected using a Markov chain model. The Markov chain kernel involves local indicators of mechanical interest and its parameters are learnt from numerical full-field 2D simulations of cracking using a cohesive-volumetric finite element solver called XPER. This model does not include any mechanical elements. It is the database, derived from the XPER crack, that contains the mechanical information and optimizes the probabilistic model. The resulting model exhibits a drastic improvement of CPU time in comparison to simulations from XPER.

Subdivision schemes are widely used in numerical mathematics such as signal/image approximation, analysis and control of data or numerical analysis. However, to develop their full power, subdivision schemes should be incorporated into a multiresolution analysis that, mimicking wavelet analyses, provides a multi-scale decomposition of a function, a curve, or a surface. The ingredients needed to define a multiresolution analysis associated to a subdivision scheme are a decimation scheme and detail operators. Their construction is not straightforward as soon as the subdivision scheme is non-interpolatory. This paper is devoted to the construction of decimation schemes and detail operators compatible with general subdivision schemes, including non-linear ones. Analysis of the performances of the constructed analyses is carried out. Some numerical applications are presented in the framework of image approximation.

Artificial Intelligence (AI) algorithms have shown their capability to complement human analysis in the understanding of complex phenomena. Their advantages essentially rely on the flexibility of their construction that allows capturing complex relationships between inputs and outputs of a problem and also on their ability to adapt themselves via a learning phase to the available information. As a result, AI is widely used in many scientific fields and especially those related to the nuclear industry. This work deals an application of AI based on several classical machine learning approaches for the study of Reactivity Initiated Accidents (RIA) in the CABRI experimental pulse reactor located at the Cadarache research center, southern France. We focus on the interpretation of the fuel rod behaviour during the power pulse using the online fuel motion monitoring system called the hodoscope. The objective of this paper is to investigate how AI algorithms can be used for the automatic detection of a generic fuel delocalization from the signals recorded by the hodoscope.

Uncertainty analysis is a key element in nuclear power plant deterministic safety analysis using best-estimate thermal-hydraulic codes and best-estimate-plus-uncertainty methodologies. If forward uncertainty propagation methods have now become mature for industrial applications, the input uncertainty quantification (IUQ) on the physical models still requires further investigations. The Organisation for Economic Co-operation and Development/Nuclear Energy Agency PREMIUM project attempted to benchmark the available IUQ methods, but observed a strong user effect due to the lack of best practices guidance. The SAPIUM project has been proposed toward the construction of a clear and shared systematic approach for IUQ. The main outcome of the project is a first "good-practices" document that can be exploited for safety study in order to reach consensus among experts on recommended practices as well as to identify remaining open issues for further developments. This paper describes the systematic approach that consists of five elements in a step-by-step approach to perform a meaningful model IUQ and validation as well as some good-practice guideline recommendations for each step.

•BEPU using value and uncertainty as arguments needs in an evidence-based background.•It could be addressed via Data Assimilation and adequately selected IEs.•The “validation paradigms” of V&UQ are not similar in different fields of expertise.•A new generation of multi-physics allows involving in V&UQ wide range of IEs including PMO.•Merging of “paradigms” is a new challenge for multi-physics tools V&UQ.

La construction d'un materiau numerique repose sur des descripteurs morphologiques qui caracterisent un materiau reel a l'aide d'informations statistiques et geometriques. La microstructure numerique generee est qualifie statistiquement semblable par rapport a la realite. Or deux microstructures statistiquement semblables peuvent presenter des comportements de fissuration differents. Ces differences peuvent avoir une influence sur la permeabilite des enceintes. Pour contourner cette limitation, nous proposons dans ce travail un nouvelle approche qui combine descripteurs morphologiques et modele de prediction de fissure pour evaluer la similarite entre 2 microstructures.

PREMIUM (Post BEMUSE Reflood Models Input Uncertainty Methods) was an activity launched with the aim of pushing forward the methods of quantification of physical model uncertainties in thermal-hydraulic codes. The benchmark PREMIUM was addressed to all who apply uncertainty evaluation methods based on input uncertainties quantification and propagation. The benchmark was based on a selected case of uncertainty analysis application to the simulation of quench front propagation in an experimental test facility. Applied to an experiment, enabled evaluation and confirmation of the quantified probability distribution functions on the basis of experimental data. The scope of the benchmark comprised a review of the existing methods, selection of potentially important uncertain input parameters, quantification of the ranges and distributions of the identified parameters using experimental results of tests performed on the FEBA test facility, verification of the performed quantification on the basis of tests performed at the FEBA test facility and validation on the basis of blind calculations of the Reflood 2-D PERICLES experiment. The benchmark has shown dependency of the results on the applied methodology and a strong user effect. The conclusion was that a systematic approach for the quantification of model uncertainties is necessary.

Subdivision schemes have been extensively developed since the eighties with very powerful applications for surface generation. To be implemented for compression, subdivision schemes have to be coupled with decimation operators sharing some consistency relation and with detail operators. The flexibility of subdivision schemes (they can be non-stationary, position or zone dependent, non-linear,…) makes that the construction of consistent decimation operators is a difficult task. In this paper, following the first results introduced in Kui et al. (On the coupling of decimation operator with subdivision schemes for multi-scale analysis. In: Lecture notes in computer science, vol. 10521. Springer, Berlin, pp. 162–185, 2016), we present the construction of multiresolution analyses connected to general subdivision schemes with detailed application to a non-interpolatory linear scheme called shifted Lagrange (Dyn et al., A C2 four-point subdivision scheme with fourth order accuracy and its extensions. In: Mathematical methods for curves and surfaces: Tromsø 2004. Citeseer, 2005) and its non-linear version called shifted PPH (Amat et al., Math. Comput. 80:959–959, 2011).

Taking into account uncertainties is a key issue in nuclear power plant safety analysis using best estimate plus uncertainty methodologies. It involves two main types of treatment depending on the variables of interest input parameters or system response quantity. The OECD/NEA PREMIUM project devoted to the first type of variables has shown that inverse methods for input uncertainty quantification can exhibit strong user-effect. One of the main reasons was the lack of a clear guidance to perform a reliable analysis. This work is precisely devoted to the development of a first good practice guidance document for quantification of thermal-hydraulic code model input uncertainty. The developments have been done in the framework of the OECD/NEA SAPIUM project (January 2017-September 2019). This paper provides a summary of the main project outcome. Recommendations and open issues for future developments are also given.

Gaussian process (GP) models have become popular for approximating and exploring nonlinear systems using scarce input/output samples and prior hypotheses done through mean and covariance functions. While it is common to make stationarity assumptions and use variance-based criteria for exploration, in realistic cases it is not rare that systems under study exhibit a heterogeneous behavior depending on regions of the parameter space. We consider a class of problems where high variations occur along unknown noncanonical directions and we tackle the problem of accommodating nonstationarity from two angles. First we define a novel class of covariances (WaMI-GP) that simultaneously generalizes kernels of multiple index and of tensorized warped GPs, and second, we introduce derivative-based sampling criteria dedicated to the exploration of high-variation regions. The novel GP class is investigated through both mathematical analysis and numerical experiments, and it is shown that it allows encoding much expressiveness while keeping the number of parameters to be inferred moderate. Criteria and models are compared on a mechanical test case from safety studies conducted by IRSN. On this application some of the proposed criteria outperform usual variance-based criteria in the case of a stationary GP model; however, variance-based criteria with WaMI-GP perform even better. Our method is also compared with the treed Gaussian processes (TGP) on this application and on a NASA test case. In the IRSN application, WaMI-GP dominates TGP in static and sequential settings. In the NASA application, while TGP clearly dominates in the static case, for small designs it is outperformed by WaMI-GP in the sequential setup.

The analysis of the behavior of complex computer codes is a key topic in many industrial applications. When taking into account uncertainties, it is for example crucial to ensure that a system cannot deviate from its reference state and lead to unsafe conditions. From a methodological point of view, a computer code can be interpreted as an objective function f : X ∈ D ⊂ IR 7→ y ∈ IR where X stands for the vector of input parameters and y is the response of interest. The relevance of the mathematical treatment essentially relies on the choice of the set of input values called design of experiments (DoEs) where the objective function is evaluated. There exists many approaches to construct such designs. The first one is based on the exploration of the whole input set [1]. However, when the computational budget is limited, this strategy is not satisfactory in the case of objective functions exhibiting heterogeneous behaviors for example. It is then more efficient to refine the design in regions of interest (such as high variation zones) where extra information is required to specify the code response behavior. It leads to adaptive DoEs. Their construction first relies on a modeling of the code response from a DoE with few evaluations then on the optimization of an infill criterion that provides the new evaluations to perform according to the type of regions of interest. This process is applied sequentially in order to update and improve the model at each iteration. In this work, we describe several developments related to the construction of adaptive DoEs. They are all based on a modelling in the frame of Gaussian processes which is very popular for the analysis of computer experiments [2]. After an overview on existing approaches to handle the construction of adaptive DoEs, we introduce two recent contributions. The first one is related to the treatment of optimization problems involving complex computer codes [3]. The second one deals with the analysis of computer code responses exhibiting heterogeneous behaviors [4]. All these developments are illustrated on several test cases coming from IRSN nuclear studies.

Uncertainty assessment is a key step in nuclear applications to ensure that a system cannot move towards unsafe conditions. This topic has already been addressed by several OECD/NEA projects such as UMS or BEMUSE. However, if uncertainty propagation methods have now become mature for industrial applications, the input uncertainties quantification on the physical models still requires further investigations. It is precisely in this context that the SAPIUM project has been proposed in order to reduce as much as possible (or at least better understand) the user-effect observed in the previous PREMIUM activity that was a first attempt to analyze available methods to handle this issue. The underlying idea of the proposed work is not to focus on method benchmarking but on the construction of a clear and shared systematic approach for input uncertainty quantification as it is already addressed in industries and RandD for related topics. The main outcome of the project is a first good practices document that can be exploited for safety study in order to increase the agreement among experts on recommended practices as well as on remaining open issues for further developments. End users are therefore the developers and the users of BEPU methodologies, as well as the organizations in charge of evaluating them. Since it is an on-going activity, this paper describes the general content of the SAPIUM activity. All the details of the contributions will be available in the final document that will be issued in 2019.