Jean Baccou

Numerical accuracy and efficiency in the propagation of epistemic and aleatory uncertainties

The need to differentiate between epistemic and aleatory uncertainties is now well admitted by the risk analysis community. One way to do so is to model aleatory uncertainty by classical probability distributions and epistemic uncertainty by means of possibility distributions, and then propagate them by their respective calculus. The result of this propagation is a random fuzzy variable. When dealing with complex models, the computational cost of such a propagation quickly becomes too high. In this paper, we propose a numerical approach, the Random/Fuzzy (RaFu) method, whose aim is to determine an optimal numerical strategy so that computational costs are reduced to their minimum, using the theoretical frameworks mentioned above. We also give some means to take account of the resulting numerical error. The benefits of the RaFu method are shown by comparing it to previously proposed methods.

On four-point penalized Lagrange subdivision schemes

This paper is devoted to the definition and analysis of new subdivision schemes called penalized Lagrange. Their construction is based on an original reformulation for the construction of the coefficients of the mask associated to the classical 4-points Lagrange interpolatory subdivision scheme: these coefficients can be formally interpreted as the solution of a linear system similar to the one resulting from the constrained minimization problem in Kriging theory which is commonly used for reconstruction in geostatistical studies. In such a framework, the introduction in the formulation of a so-called error variance can be viewed as a penalization of the oscillations of the coefficients. Following this idea, we propose to penalize the 4-points Lagrange system. This penalization transforms the interpolatory schemes into approximating ones with specific properties suitable for the subdivision of locally noisy or strongly oscillating data. According to a so-called penalization vector, a family of schemes can be generated. A full theoretical study is first performed to analyze this new type of non stationary subdivision schemes. Then, in the framework of position dependant penalization vector, several numerical tests are provided to point out the efficiency of these schemes compared to standard approaches.

Bayesian Adaptive Reconstruction of Profile Optima and Optimizers

Given a function depending both on decision parameters and nuisance variables, we consider the issue of estimating and quantifying uncertainty on profile optima and/or optimal points as functions of the nuisance variables. The proposed methods are based on interpolations of the objective function constructed from a finite set of evaluations. Here the functions of interest are reconstructed relying on a kriging model but also using Gaussian random field conditional simulations that allow a quantification of uncertainties in the Bayesian framework. Besides this, we introduce a variant of the expected improvement criterion, which proves efficient for adaptively learning the set of profile optima and optimizers. The results are illustrated with a toy example and through a physics case study on the optimal packing of polydisperse frictionless spheres.

POSITION-DEPENDENT LAGRANGE INTERPOLATING MULTIRESOLUTIONS

This paper is devoted to the construction of interpolating multiresolutions using Lagrange polynomials and incorporating a position dependency. It uses the Hartens framework21 and its connection to subdivision schemes. Convergence is first emphasized. Then, plugging the various ingredients into the wavelet multiresolution analysis machinery, the construction leads to position-dependent interpolating bases and multi-scale decompositions that are useful in many instances where classical translation-invariant frameworks fail. A multivariate generalization is proposed and analyzed. We investigate applications to the reduction of the so-called Gibbs phenomenon for the approximation of locally discontinuous functions and to the improvement of the compression of locally discontinuous 1D signals. Some applications to image decomposition are finally presented.

On coupling wavelets with fictitious domain approaches

We define and analyse a numerical algorithm for the approximation of parabolic equations on a general 2D domain with Dirichlet boundary conditions. It couples wavelet approximations with fictitious domain surface Lagrange multiplier approaches. This algorithm turns out to be precise, fast and numerically efficient.

A practical methodology for information fusion in presence of uncertainty: application to the analysis of a nuclear benchmark

This work is devoted to some methodological developments on information fusion in presence of uncertainty and their application in the frame of the BEMUSE Programme. In nuclear safety studies, different uncertainty analyses using different codes and implying different experts are generally performed. It is then useful to define formal methods to combine all these information sources in order to improve the reliability of the expertise process and especially to detect possible conflicts (if any) between the sources. Starting from the IRSN methodology already introduced in Destercke and Chojnacki (Nucl Eng Des 238(9):2484–2493, 2008), this paper presents a more convenient reformulation of its construction to allow its use by engineers. It is then applied to analyse the results coming from the BEMUSE Programme.

Warped Gaussian Processes and Derivative-Based Sequential Designs for Functions with Heterogeneous Variations

Gaussian process (GP) models have become popular for approximating and exploring non-linear systems based on scarce input/output training samples and on prior hypotheses implicitly done through prior mean and covariance functions. While it is common to make stationarity assumptions and use variance-based criteria for space exploration, in realistic test cases it is not rare that systems under study exhibit an heterogeneous behaviour depending on considered regions of the parameter space. With a class of problems in mind where high variations occur along unknown non-canonical directions, we tackle the problem of uncovering and accommodating non-stationarity in function approximation from two angles: first via a novel class of covariances (called WaMI-GP) that simultaneously generalizes kernels of Multiple Index and of tensorized warped GP models and second, by introducing derivative-based sampling criteria dedicated to the exploration of high variation regions. The novel GP class is investigated both through mathematical analysis and numerical experiments, and it is shown that proposed kernels allow encoding much expressiveness while remaining with a moderate number of parameters to be inferred. On the other hand, and independently of non-stationarity assumptions, we conduct (semi-)analytically derivations for our new variance-based infill sampling criteria relying on a change of focus from the GP to the norm of its associated gradient field. Criteria and GP models are first compared on a mechanical test case taken from nuclear safety studies conducted by IRSN. It is found on this application that some of the proposed sampling criteria including derivatives outperform usual variance-based criteria in the case of a stationary GP model, but that it is even better to use standard variance-based criteria with the proposed novel class of covariances. Comparisons are also done with the Treed Gaussian Processes (TGP) both on this application and on a three-dimensional 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 initial designs it is outperformed by WaMI-GP in the sequential set up.

Construction and Analysis of Zone-dependent Interpolatory/Non-interpolatory Stochastic Subdivision Schemes for Non-regular Data

This work is devoted to the definition of stochastic subdivision schemes adapted to the reconstruction of non-regular data. These schemes are constructed in the framework of the Kriging theory. Thanks to the introduction of a zone-dependent error variance in the Kriging approach, they combine interpolatory and non interpolatory subdivision schemes according to a domain segmentation. Their originality relies on the introduction and coupling of three ingredients: a segmentation of the data, a local prediction according to the characteristics of the different zones and an adaption strategy near segmentation points. The convergence of the corresponding 4-point scheme is analyzed. Its behavior is compared with other subdivision schemes on various numerical experiments.

Original article: Kriging-based subdivision schemes: Application to the reconstruction of non-regular environmental data

Abstract: This work is devoted to the construction of new kriging-based interpolating position-dependent subdivision schemes for data reconstruction. Their originality stands in the coupling of the underlying multi-scale framework associated to subdivision schemes with kriging theory. Thanks to an efficient stencil selection, they allow to cope the problem of non-regular data prediction while keeping the interesting properties of kriging operators for the quantification of prediction errors. The proposed subdivision schemes are fully analyzed and an application to the reconstruction of non-regular environmental data is given as well.

Construction of finite-dimensional multiresolutions based on linear subdivision schemes: Application to the 4-point shifted Lagrange scheme

Abstract This paper is devoted to the construction of multiresolution frameworks related to general but linear subdivision schemes applied on sequences of finite length. Thanks to the flexible properties of subdivision schemes, a subdivision-based multiresolution is a promising powerful tool for compression, control and analysis of data. However, its construction is not straightforward either for non-interpolatory subdivisions, or for sequences of finite length. In this paper, given a subdivision for finite length sequences, we provide a first approach to construct compatible multiresolutions thanks to an extension of some classical wavelet results. As it will become clear that this type of approach can become computationally costly in practice, a new edge-adapted method that combines local consistent decimation is then developed to circumvent this limitation. An illustration in the case of the 4-point shifted Lagrange subdivision scheme, for which there is no available multiresolution up to now, is then provided. Finally, some properties of the new multiresolutions are analyzed and their performances are evaluated in the framework of image approximation.