L2S | UQSay seminar | CNRS - CentraleSupelec

UQSay: UQ, DACE & related topics @ Paris-Saclay

UQSay is a series of seminars on the broad area of Uncertainty Quantification (UQ) and related topics (Read more…), organized by L2S, MSSMAT, LMT and EDF R&D.

Upcoming seminars

See https://www.uqsay.org/upcoming/.

All seminars

UQSay #54

The fifty-fourth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, February 2, 2022.

2–3 PM — Brian Staber (Safran Tech) — [slides]

Quantitative performance evaluation of Bayesian neural networks

Due to the growing adoption of deep neural networks in many fields of science and engineering, modeling and estimating their uncertainties has become of primary importance. Various approaches have been investigated including Bayesian neural networks, ensembles, deterministic approximations, amongst others. Despite the growing litterature about uncertainty quantification in deep learning, the quality of the uncertainty estimates remains an open question. In this work, we attempt to assess the performance of several algorithms on sampling and regression tasks by evaluating the quality of the confidence regions and how well the generated samples are representative of the unknown target distribution. Towards this end, several sampling and regression tasks are considered, and the selected algorithms are compared in terms of coverage probabilities, kernelized Stein discrepancies, and maximum mean discrepancies. Joint work with Sébastien Da Veiga (ENSAI). Ref: arXiv:2206.06779

Organizing committee: Pierre Barbillon (MIA-Paris), Julien Bect (L2S), Nicolas Bousquet (EDF R&D), Amélie Fau (LMPS), Filippo Gatti (LMPS), Bertrand Iooss (EDF R&D), Alexandre Janon (LMO), Sidonie Lefebvre (ONERA), Didier Lucor (LISN), Emmanuel Vazquez (L2S). Coordinators: Julien Bect (L2S) & Sidonie Lefebvre (ONERA)

Practical details: the seminar will be held online using Microsoft Teams. If you want to attend this seminar (or any of the forthcoming online UQSay seminars), and if you do not already have access to the UQSay group on Teams, simply send an email and you will be invited. Please specify which email address the invitation must be sent to (this has to be the address associated with your Teams account). You will find the link to the seminar on the “General” UQSay channel on Teams, approximately 15 minutes before the beginning.

The technical side of things: you can use Teams either directly from your web browser or using the “fat client”, which is available for most platforms (Windows, Linux, Mac, Android & iOS). We strongly recommend the latter option whenever possible. Please give it a try before the seminar to anticipate potential problems.

UQSay #53

The fifty-third UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, January 19, 2022.

2–3 PM — Felipe Tobar (Initiative for Data & AI, Universidad de Chile) — [slides]

Computationally-efficient initialisation of Gaussian processes: The generalised variogram method

We present a computationally-efficient strategy to find the hyperparameters of a Gaussian process (GP) avoiding the computation of the likelihood function. Motivated by the fact that training a GP via ML is equivalent (on average) to minimising the KL-divergence between the true and learnt model, we set to explore different metrics/divergences among GPs that are computationally inexpensive and provide estimates close to those of ML. In particular, we identify the GP hyperparameters by projecting the empirical covariance or (Fourier) power spectrum onto a parametric family, thus proposing and studying various measures of discrepancy operating on the temporal or frequency domains. Our contribution extends the Variogram method developed by the geostatistics literature and, accordingly, it is referred to as the Generalised Variogram method (GVM). In this talk, we will start with a brief introduction to Gaussian processes, then present the proposed GVM and finally provide experimental validation using synthetic and real-world data. Joint work with Elsa Cazelles & Taco de Wolff. Ref: arXiv:2210.05394.

UQSay #52

The fifty-second UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, January 5, 2023.

2–3 PM — Georgios Karagiannis (Durham University)

Bayesian spanning treed co-kriging for high dimensional output emulation

We propose a new Bayesian emulator, called Bayesian spanning treed co-kriging, suitable to analyze computer models with non-stationary massive outputs in the multifidelity setting. Our motivation comes from a real-life application with a storm surge simulator. Given certain assumptions on the Bayesian model, we introduce a suitable stochastic mechanism that facilitates predictions in a principal manner. The good performance of our method is demonstrated in benchmark examples, while our method is implemented for the analysis of a surge simulator given simulations at different fidelity levels.

UQSay #51

The fifty-first UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, November 17, 2022.

2–3 PM — Cécile Mercadier (Institut Camille Jordan) — [slides]

Hoeffding–Sobol and Möbius decompositions for (tail-)dependence analysis

Methods to analyse dependence and tail dependence are well established. Using for instance the copula function or the stable tail dependence function, and their empirical versions, one can construct non parametric statistics, parametric inference, as well as testing or resampling procedures. My talk will reflect upon the use of g sensitivity analysis for extreme value theory and copula modeling. Through my recent publications, I will explain what their links are and the benefit in mixing these domains. Joint work with Christian Genest, Paul Ressel & Olivier Roustant. Refs:

C. Mercadier, O. Roustant & C. Genest (2022). Linking the Hoeffding–Sobol and Möbius formulas through a decomposition of Kuo, Sloan, Wasilkowski, and Wozniakowski. Statistics & Probability Letters, vol. 185 [hal-03220809],
C. Mercadier & P. Ressel (2021). Hoeffding–Sobol decomposition of homogeneous co-survival functions: from Choquet representation to extreme value theory application. Dependence Modeling, 9(1):179–198 [hal-03200817],
C. Mercadier & O. Roustant (2019). The tail dependograph. Extremes, 22:343–372 [hal-01649596].

UQSay #50

The fiftieth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, October 13, 2022.

2–3 PM — Gilles Stoltz (LMO, Université Paris-Saclay – CNRS) — [slides]

Multi-armed bandit problems: a statistical view, focused on lower bounds

Multi-armed bandit problems correspond to facing K unknown probability distributions, having to sequentially pull one of them, and observing a realization thereof at each pull. Two goals will be considered. (1) The realizations are payoffs, and the sum of these payoffs is to be maximized. This goal is achieving by minimizing regret, which is defined as the expected performance of the best arm minus the expected sum of payoffs achieved by a strategy. Two types of bounds may be defined, depending on whether they may depend on the specific bandit problem or only on the model (the class of possible distributions). We will recall classical strategies like UCB and MOSS, as well as a new strategy combining both, called KL-UCB-Switch. We will review upper bounds on the regret and detail which lower bounds may be achieved, and how. We will deal with one interesting extension, the adaptation to the unknown range of the distributions, i.e., when the distribution are supported on a compact interval that is unknown as well. The case of regret minimization is very well understood in the literature, contrary to: (2) A second goal can be to identify the best arm, i.e., control the probability that after T observations (sampled adaptively) the strategy does not identify the arm with the highest expectation. This is called best arm identification with a fixed budget. Limited results are available. We will describe a typical strategy, called successive rejects, that drops one distribution after the other after horse racing them. We will also indicate how we are currently laying the foundations of a non-parametric approach to this problem, based on KL divergences, as opposed to typical approaches based on differences between expectations. Joint work with Antoine Barrier, Aurélien Garivier, Hédi Hadiji & Pierre Ménard. Refs:

KL-UCB-Switch: JMLR, 23(179):1−66, 2022,
Lower bounds for regret minimization: MOOR, 4(2):377-766, 2019 [HAL],
Adaptation to the unknown range: HAL-02794382,
Best-arm identification: HAL-03792668.

UQSay #49

The forty-ninth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, September 29, 2022.

2–3 PM — Jonas Latz (Heriot-Watt University, Edinburgh) — [slides]

Stochastic gradient descent in continuous time: discrete and continuous data

Optimisation problems with discrete and continuous data appear in statistical estimation, machine learning, functional data science, robust optimal control, and variational inference. The “full” target function in such an optimisation problem is given by the integral over a family of parameterised target functions with respect to a discrete or continuous probability measure. Such problems can often be solved by stochastic optimisation methods: performing optimisation steps with respect to the parameterised target function with randomly switched parameter values. In this talk, we discuss a continuous-time variant of the stochastic gradient descent algorithm. This so-called stochastic gradient process couples a gradient flow minimising a parameterised target function and a continuous-time ‘index’ process which determines the parameter. We first briefly introduce the stochastic gradient processes for finite, discrete data which uses pure jump index processes. Then, we move on to continuous data. Here, we allow for very general index processes: reflected diffusions, pure jump processes, as well as other Lévy processes on compact spaces. Thus, we study multiple sampling patterns for the continuous data space. We show that the stochastic gradient process can approximate the gradient flow minimising the full target function at any accuracy. Moreover, we give convexity assumptions under which the stochastic gradient process with constant learning rate is geometrically ergodic. In the same setting, we also obtain ergodicity and convergence to the minimiser of the full target function when the learning rate decreases over time sufficiently slowly. Joint work with Kexin Jin, Chenguang Liu & Carola-Bibiane Schönlieb. Refs: DOI:10.1007/s11222-021-10016-8, arXiv:2112.03754, arXiv:2203.11555.

UQSay #48

The forty-eighth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, June 2, 2022.

2–3 PM — Valentin Resseguier (Scalian Innovation Lab, INRAE) — [slides]

Fast generation of prior for Bayesian estimation problems in fluid mechanics

We are interested in real-time estimation and short-term forecasting of 3D fluid flows, using limited computational resources. This is possible through the coupling between data, numerical simulations and sparse fluid flow measurements. Here, the term data refers to numerical simulation outputs. To achieve these ambitious goals, synthetic (i.e. simulated) data and intrusive surrogate models drastically reduce the problem dimensionality – typically from 10 7 to 10. Unfortunately, even with corrections, the accumulated errors of these surrogate models increase rapidly over time due to the chaotic and intermittent nature of fluid mechanics. Therefore, deterministic predictions are hardly possible outside the learning time interval. Data assimilation can alleviate these problems by (i) providing a set of simulations covering probable futures (without increasing the computational cost) and (ii) constraining these online simulations with measurements. We addressed this Uncertainty Quantification (UQ) problem (i) with a multi-scale physically-based stochastic parameterization called “Location uncertainty models” (LUM) [1-3] and new statistical estimators based on stochastic calculus, signal processing and physics [3]. The deterministic ROM coefficients are obtained by a Galerkin projection whereas the correlations of the noises are estimated from the residual velocity, the physical model structure, and the evolution of the resolved modes. We solved problem (ii) with a particle filter [4]. Whether we consider UQ [3] or DA [4] applications, our method greatly exceeds the state of the art, for ROM degrees of freedom smaller than 10 and moderately turbulent 3D flows (Reynolds number up to 300). Joint work A. M. Picard & M. Ladvig (Scalian), and D. Heitz (INRAE). Refs: [1] hal-01391420, [2] hal-02558016, [3] hal-03169957 & [4] hal-03445455.

UQSay #47

The forty-seventh UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, May 19, 2022.

2–3 PM — Mélanie Rochoux (CECI, Cerfacs, CNRS) — [slides]

Assimilating fire front position and emulating boundary-layer flow simulations for wildland fire behavior ensemble prediction and reanalysis

Monitoring wildfire behavior has recently emerged as a key public policy issue due to the occurrence of extreme events, in particular in the Euro-Mediterranean area that is exposed to more frequent and more severe wildfires under climate change. Key to this modeling is the development of an event-scale numerical simulation capability as a means to understand and predict the interactions between the atmosphere and the wildfire that drive its behavior. In this framework, my research aims at designing and evaluating a wildland fire behavior reanalysis capability to reconstruct as best as possible wildland fire progression at landscape-to-atmospheric scales. This approach combines information coming from a coupled atmosphere/fire model (Costes et al. 2021) and from airborne thermal infrared images (Paugam et al. 2021) through an ensemble-based data assimilation algorithm that infers more realistic environmental factors and estimates the time-evolving fire front position. My talk will provide an overview of the different components required to build this reanalysis capability, with two main focus: i) a front data assimilation methodology to address position errors in the fire front progression (Rochoux et al. 2018; Zhang et al. 2019), and ii) a non-intrusive reduced-order modeling approach combining principal component analysis and adaptive Gaussian processes to accurately and efficiently explore the physical parameter space and predict the atmospheric boundary-layer flow patterns (Nony et al. 2021). In the long-term, these methods will be applied to the Meso-NH/Blaze coupled atmosphere/fire model to design a wildland fire behavior ensemble prediction and reanalysis capability. Joint work with Bastien Nony & Thomas Jaravel (Cerfacs), Didier Lucor (LISN), Annabelle Collin & Philippe Moireau (Inria), Cong Zhang & Arnaud Trouvé (University of Maryland). Refs:

M.C. Rochoux, A. Collin, C. Zhang, A. Trouvé, D. Lucor and P. Moireau (2018). Front shape similarity measure for shape-oriented sensitivity analysis and data assimilation for eikonal equation. ESAIM: Proceedings and Surveys, EDP Sciences, 63:258–279, DOI:10.1051/proc/201863258.
C. Zhang, A. Collin, P. Moireau, A. Trouvé and M.C. Rochoux (2019). State-parameter estimation approach for data-driven wildland fire spread modeling: application to the 2012 RxCADRE S5 field-scale experiment. Fire Safety Journal, 105:286–299, DOI:<10.1016/j.firesaf.2019.03.009.
B.X. Nony, M.C. Rochoux, D. Lucor and T. Jaravel (2021). Compound parametric metamodeling of large-eddy simulations for micro-scale atmospheric dispersion. 20th International Conference on Harmonisation within Atmospheric Dispersion Modelling for Regulatory Purposes, Tartu (Estonia), 14–18 June, 2021.
A. Costes, M.C. Rochoux, C. Lac and V. Masson (2021) Subgrid-scale fire front reconstruction for ensemble coupled atmosphere-fire simulations of the FireFlux I experiment. Fire Safety Journal, 126:103475, DOI:10.1016/j.firesaf.2021.103475.
R. Paugam, M.J. Wooster, W.E. Mell, M.C. Rochoux, J-B. Filippi, G. Rücker, O. Frauenberger, E. Lorenz, W. Schroeder and N. Govendor (2021). Orthorectification of helicopter-borne high resolution experimental burn observation from infra red handheld imagers. Remote Sensing, 13(23):4913, DOI:10.3390/rs13234913.

UQSay #45

The forty-fifth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, March 31, 2022.

2–3 PM — Nathalie Bartoli (ONERA) — [slides]

Bayesian optimization to solve mono- or multi-fidelity constrained black box problem

This work aims at developing new methodologies to optimize computational ostly complex systems (e.g., aeronautical engineering systems). The proposed surrogate-based method (often called Bayesian Optimization) uses adaptive sampling to promote a trade-off between exploration and exploitation. Our in-house implementation, called SEGOMOE, handles a high number of design variables (continuous, discrete or categorical) and nonlinearities by combining mixtures of experts (local surrogate models) for the objective and/or the constraints. An extension to multi-fidelity is also included when a variety of information is available. The performance of the proposed approach has been evaluated on both a benchmark of analytical constrained and unconstrained problems a well as a set of realistic aeronautical applications. Refs:

P. Saves, N. Bartoli, Y. Diouane, T. Lefebvre, J. Morlier, C. David, S. Defoort (2022). Multidisciplinary design optimization with mixed categorical variables for aircraft design. In AIAA SCITECH 2022 Forum (p. 0082).
R. C. Arenzana, A. López-Lopera, S. Mouton, N. Bartoli, T. Lefebvre (2021, July). Multifidelity Gaussian Process model for CFD and Wind Tunnel data fusion. In Proceedings of the International Conference on Multidisciplinary Design Optimization of Aerospace Systems (AEROBEST 2021) (pp. 1-758).
R. Priem, H. Gagnon, I. Chittick, S. Dufresne, Y. Diouane, and N. Bartoli (2020). An efficient application of Bayesian optimization to an industrial MDO framework for aircraft design. In AIAA AVIATION 2020 FORUM (p. 3152).
R. Priem, N. Bartoli, Y. Diouane, A. Sgueglia (2020), Upper trust bound feasibility criterion for mixed constrained Bayesian optimization with application to aircraft design, Aerospace Science and Technology
M. Meliani, N. Bartoli, T. Lefebvre, M.-A. Bouhlel, J. R. R. A. Martins, J. Morlier, Multi-fidelity efficient global optimization: Methodology and application to airfoil shape design, 20th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, June 2019, Dallas, United States
N. Bartoli, T. Lefebvre, S. Dubreuil, R. Olivanti, R. Priem, N. Bons, J. R. R. A. Martins, J. Morlier (2019), Adaptive modeling strategy for constrained global optimization with application to aerodynamic wing design, Aerospace Science and Technology Journal, vol. 90, p. 85-102
M.-A. Bouhlel, J. T. Hwang, N. Bartoli, R. Lafage, J. Morlier, J. R. R. A. Martins (2019), A Python surrogate modeling framework with derivatives, Advances in Engineering Software
M.-A. Bouhlel, N. Bartoli, A. Otsmane and J. Morlier, Improving kriging surrogates of high-dimensional design models by Partial Least Squares dimension reduction, Structural and Multidisciplinary Optimization, vol 53, no5, pp 935-952, 2016

UQSay #44

The forty-fourth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, March 17, 2022.

2–3 PM — Nicola Pedroni (Politecnico di Torino) — [slides]

Quantification of Mixed Aleatory and Epistemic Uncertainties for Robust Design Optimization, in the Presence of Scarce and Functional Data

The quantitative analyses of the phenomena occurring in complex, safety-critical (e.g., civil, nuclear, aerospace and chemical) dynamic engineering systems are based on mathematical models. In practice, not all the characteristics of the system under analysis can be captured in the model: thus, uncertainty is present in the values of the input parameters and in the model hypotheses and structure. This is due to: (i) the intrinsically random nature of several of the phenomena occurring during system operation (aleatory uncertainty, here represented by multivariate probability distributions); (ii) the incomplete knowledge about some phenomena and operating conditions, often due to the scarcity of quantitative data available, which may be either very sparse or prohibitively expensive to collect (epistemic uncertainty, here described by intervals or sets). The characterization and quantification of this mixed uncertainty is of paramount importance for: (i) making robust decisions in safety-critical systems applications; (ii) optimally designing and operating such systems; and (iii) driving resource allocation for uncertainty reduction. This talk addresses the “NASA Langley Uncertainty Quantification Challenge on Optimization Under Uncertainty” with respect to two issues: (i) calibration of the mathematical model of an aerospace system and joint quantification of mixed (probabilistic) aleatory and (set-based) epistemic uncertainties; and (ii) system design optimization, in the presence of scarce and functional (time series) data (i.e., observations coming from the real system). With reference to issue (i), the parametric Sliced Normal (SN) class of distributions is employed, whose flexibility and versatility allow characterizing multivariate data and complex parameter dependencies with minimal effort. The modeling power of SNs is tested within a frequentist (optimization-based) framework and a Bayesian inverse approach. With reference to issue (ii), an iterative framework is developed to robustly optimize the design of the system (e.g., by minimizing the worst-case, epistemic upper bound of its failure probability). The issue is addressed by an efficient combination of: (i) Monte Carlo Simulation (MCS) to propagate the aleatory uncertainty described by probability distributions; (ii) Genetic Algorithms (GAs) to solve the optimization problems related to the propagation of epistemic uncertainty by interval analysis; and (iii) fast-running Artificial Neural Networks (ANNs) to reduce the computational time related to the repeated model evaluations. As a final remark, since the outputs of the system models of interest are functions of time, both issues are addressed in the space defined by the orthonormal bases resulting from a Singular Value Decomposition (SVD) of the real system observations. Ref: DOI:10.1016/j.ymssp.2021.108206.

UQSay #43

The forty-third UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, February 17, 2022.

2–3 PM — D. Austin Cole (GlaxoSmithKline, Inc.) — [slides]

Locally induced Gaussian processes for modelling large-scale simulations

Gaussian processes (GPs) serve as flexible surrogates for complex surfaces, but buckle under the cubic cost of matrix decompositions with big training data sizes. Geospatial and machine learning communities suggest pseudo-inputs, or inducing points, as one strategy to obtain an approximation easing that computational burden. However, we show how placement of inducing points and their multitude can be thwarted by pathologies, especially in large-scale dynamic response surface modeling tasks. As a remedy, we suggest porting the inducing point idea, which is usually applied globally, over to a more local context where selection is both easier and faster. In this way, our proposed methodology (LIGP) hybridizes global inducing point and data subset-based local GP approximation. A cascade of strategies for planning the selection of local inducing points is provided, and comparisons are drawn to related methodology with emphasis on computer surrogate modeling applications. We show that local inducing points extend their global and data-subset component parts on the accuracy—computational efficiency frontier. Next, we show how LIGP also provides benefits for stochastic simulation experiments by separating signal from noise with nugget estimation and replication. Woodbury identities allow local kernel structure to be expressed in terms of unique design locations only, increasing the amount of data (i.e., the neighborhood size) that may be leveraged without additional flops. Illustrative examples are provided on benchmark data and a variety of real-world simulation experiments, including satellite drag and epidemic management. Joint work with Ryan Christianson, Robert B. Gramacy and Mike Ludkovski. Ref: DOI:10.1007/s11222-021-10007-9 and arXiv:2109.05324.

UQSay #41

The forty-first UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, January 20, 2022.

2–3 PM — Nora Lüthen (ETH Zürich) — [slides]

Poincaré chaos expansions for derivative-enhanced surrogate modelling and sensitivity analysis

Variance-based global sensitivity analysis, and in particular Sobol’ analysis, is widely adopted to determine the importance of input variables to a computational model. Sobol’ indices can be computed cheaply based on spectral methods like polynomial chaos expansions (PCE). Another option is given by the recently developed Poincaré chaos expansions (PoinCE), whose orthonormal tensor-product basis is generated from the eigenfunctions of one-dimensional Poincaré differential operators. The Poincaré differential operator is a special case of Sturm-Liouville operator and has recently been revisited for sensitivity analysis (Roustant et al. 2017). Solving the associated eigenproblem yields the Poincaré constant for a large class of one-dimensional measures with bounded support. The associated eigenfunctions form an orthonormal basis with the special (and characterizing) property that derivatives of the basis form again an orthogonal basis with respect to the same measure (Lüthen et al. 2021). The expansion of a model in terms of this basis allows the analytical computation of Sobol’ indices and derivative-based sensitivity indices (DGSM) directly from the expansion coefficients. Furthermore, the special property of the derivatives makes PoinCE particularly well suited to account for derivative information in the computation of sensitivity indices (Roustant et al. 2020). Indeed the expansions involving either model or derivative evaluations are connected, and computations can be reused. Assuming that partial derivative evaluations of the computational model are available, we compute spectral expansions in terms of Poincaré basis functions or basis partial derivatives, respectively, by sparse regression. We show on numerical examples that the derivative-based expansions provide accurate estimates for Sobol’ indices, even outperforming PCE in terms of bias and variance, and explore the performance of PoinCE as a surrogate model. Joint work with Olivier Roustant, Fabrice Gamboa, Bertrand Iooss, Stefano Marelli and Bruno Sudret. Ref: arXiv:2107.00394.

UQSay #40

The fortieth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, January 6, 2022.

2–3 PM — Didier Dubois (CNRS, IRIT, Univ. Paul Sabatier)

New uncertainty theories (The limited expressiveness of single probability measures) — [slides]

The variability of physical phenomena and partial ignorance about them motivated the development of probability theory in the two last centuries. However, the mathematical framework of probability theory, together with the Bayesian credo claiming the inevitability of unique probability measures for representing agents’ beliefs, have blurred the distinction between variability and ignorance. Modern theories of uncertainty, by putting together probabilistic and set-valued representations of information, provide a better account of the various facets of uncertainty.

Organizing committee: Pierre Barbillon (MIA-Paris), Julien Bect (L2S), Nicolas Bousquet (EDF R&D), Amélie Fau (LMPS), Filippo Gatti (LMPS), Bertrand Iooss (EDF R&D), Alexandre Janon (LMO), Sidonie Lefebvre (DOTA), Didier Lucor (LISN), Emmanuel Vazquez (L2S). Coordinator: Julien Bect (L2S).

UQSay #39

The thirty-ninth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, December 16, 2021.

2–3 PM — Gianni Franchi (U2IS, ENSTA Paris) — [slides]

Encoding the latent posterior of Bayesian neural networks for Uncertainty Quantification

Bayesian neural networks (BNNs) have been long considered an ideal, yet unscalable solution for improving the robustness and the predictive uncertainty of deep neural networks. While they could capture more accurately the posterior distribution of the network parameters, most BNN approaches are either limited to small networks or rely on constraining assumptions such as parameter independence. These drawbacks have enabled prominence of simple, but computationally heavy approaches such as Deep Ensembles, whose training and testing costs increase linearly with the number of networks. In this work we aim for efficient deep BNNs amenable to complex computer vision architectures, e.g. ResNet50 DeepLabV3+, and tasks, e.g. semantic segmentation, with fewer assumptions on the parameters. We achieve this by leveraging variational autoencoders (VAEs) to learn the interaction and the latent distribution of the parameters at each network layer. Our approach, Latent-Posterior BNN (LP-BNN), is compatible with the recent BatchEnsemble method, leading to highly efficient ({in terms of computation and} memory during both training and testing) ensembles. LP-BNNs attain competitive results across multiple metrics in several challenging benchmarks for image classification, semantic segmentation and out-of-distribution detection. Joint work with Andrei Bursuc, Emanuel Aldea, Séverine Dubuisson & Isabelle Bloch Ref: arXiv:2012.02818

Organizing committee: Pierre Barbillon (MIA-Paris), Julien Bect (L2S), Nicolas Bousquet (EDF R&D), Didier Clouteau (MSSMAT), Amélie Fau (LMT), Filippo Gatti (MSSMAT), Bertrand Iooss (EDF R&D), Alexandre Janon (LMO), Sidonie Lefebvre (DOTA), Fernando Lopez-Caballero (MSSMAT), Didier Lucor (LISN), Emmanuel Vazquez (L2S). Coordinator: Julien Bect (L2S).

UQSay #38

The thirty-eighth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, December 2, 2021.

2–3 PM — Luc Pronzato (CNRS, Univ. Côte d’Azur) — [slides]

Maximum Mean Discrepancy, Bayesian integration and kernel herding for space-filling design

A standard objective in computer experiments is to predict/interpolate the behaviour of an unknown function f on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, minimax-distance) or a discrepancy criterion measuring distance to uniformity. We focus our attention to sequential constructions where design points are added one at a time. The presentation is based on the survey [4], built on several recent results [2, 5, 6] that show how energy functionals can be used to measure distance to uniformity. We investigate connections between design for integration of f with respect to a measure µ (quadrature design), construction of the (continuous) BLUE for the location model, and minimization of energy (kernel discrepancy) for signed measures. Integrally strictly positive definite kernels define strictly convex energy functionals, with an equivalence between the notions of potential and directional derivative showing the strong relation between discrepancy minimization and more traditional design of optimal experiments, as used for instance in [3]. Kernel herding algorithms, which are special instances of vertex-direction methods used in optimal design [1, 7], can be applied to the construction of point sequences with suitable space-filling properties. Several illustrative examples are presented Refs:

F. Bach, S. Lacoste-Julien, and G. Obozinski. On the equivalence between herding and conditional gradient algorithms. In Proc. 29th Annual International Conference on Machine Learning, pages 1355–1362, 2012.
S.B. Damelin, F.J. Hickernell, D.L. Ragozin, and X. Zeng. On energy, discrepancy and group invariant measures on measurable subsets of Euclidean space. J. Fourier Anal. Appl., 16:813–839, 2010.
S. Mak and V.R. Joseph. Support points. Annals of Statistics, 46(6A):2562–2592, 2018.
L. Pronzato and A.A. Zhigljavsky. Bayesian quadrature, energy minimization and space-filling design. SIAM/ASA J. Uncertainty Quantification, 8(3):959–1011, 2020.
S. Sejdinovic, B. Sriperumbudur, A. Gretton, and K. Fukumizu. Equivalence of distance-based and RKHS-based statistics in hypothesis testing. The Annals of Statistics, 41(5):2263–2291, 2013.
B.K. Sriperumbudur, A. Gretton, K. Fukumizu, B. Schölkopf, and G.R.G. Lanckriet. Hilbert space embeddings and metrics on probability measures. Journal of Machine Learning Research, 11:1517–1561, 2010.
M. Welling. Herding dynamical weights to learn. In Proc. 26th Annual International Conference on Machine Learning, pages 1121–1128, 2009.

UQSay #37

The thirty-seventh UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, November 18, 2021.

2–3 PM — Toni Karvonen (University of Helsinki) — [slides]

Parameter estimation in Gaussian process regression for deterministic functions

In fields such as kriging, modelling of computer experiments, and probabilistic numerical computation, Gaussian process (GP) regression is used to interpolate deterministic functions which are observed without noise on compact sets. This talk reviews recent theoretical work on estimation of parameters (in particular via maximum likelihood) of the covariance kernel of the GP prior in such a setting, as well as the effect parameter estimation has on uncertainty quantification under model misspecification. We also discuss results on sample path properties of GPs that we use to characterise data-generating functions which resemble samples from a GP and to highlight the difference in assuming that the data are generated by some deterministic function or by a stochastic process. The results are based on the theory of reproducing kernel Hilbert spaces and function approximation in Sobolev spaces, which are briefly reviewed. Ref: DOI:10.1137/20M1315968, arXiv:2103.03169, arXiv:2110.02810.

UQSay #36

The thirty-sixth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, November 4, 2021.

2–3 PM — Thomas Santner (Ohio State University) — [slides]

Using Combined Physical and Computer Experiments to Solve Bioengineering Problems

Bioengineering seeks to solve problems at the confluence of Engineering and Biology. Classical Bioengineering applications concerned the engineering design, and analysis of the performance of prosthetic joints, such as hips and knees, in multiple operating environments. More recent Bioengineering applications are concerned with designing replacement tissues, and analyzing different treatments for joint tissue injuries. Finite element methods can be used to numerically approximate the stresses and strains in the human bone when prosthetic joints are implanted or when cushioning tissues such as menisci are damaged. Prediction methodology from the computer experiments literature can be used to approximate the stresses and strains for a wide variety of potential prosthetic designs, to study their performance in multiple environments, and to determine the sensitivity of the prosthetic designs to specific engineering and environmental inputs. This talk will provide an overview of two such projects and describe how computer experiment methodology, including calibration to cadaver data, was used to provide insight into their solution. Ref: DOI:doi.org/10.1007/978-1-4757-3799-8.

UQSay #35

The thirty-fifth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, October 21, 2021.

3–4 PM — Polina Kirichenko (New York University) — [slides]

Scaling Bayesian Deep Learning: Subspace Inference

Bayesian methods can provide full-predictive distributions and well-calibrated uncertainties in modern deep learning. The Bayesian approach is especially relevant in scientific and healthcare applications—where we wish to have reliable predictive distributions for decision making, and the facility to naturally incorporate domain expertise. With a Bayesian approach, we not only want to find a single point that optimizes a loss, but rather to integrate over a loss landscape to form a Bayesian model average. The geometric properties of the loss surface, rather than the specific locations of optima, therefore greatly influence the predictive distribution in a Bayesian procedure. By better understanding loss geometry, we can realize the significant benefits of Bayesian methods in modern deep learning, overcoming challenges of dimensionality. In this talk, I review work on Bayesian inference and loss geometry in modern deep learning, including challenges, new opportunities, and applications. Refs: arxiv.org:1505.05424, arxiv:1706.04599, arxiv:1609.04836, arxiv:1506.02142, stoclangevin_v6.pdf, arxiv:1612.01474, arxiv:1902.02476, arxiv:1907.07504

UQSay #34

The thirty-fourth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, October 7, 2021.

2–3 PM — Elmar Plischke (T.U. Clausthal) — [slides]

Optimal-transport-based sensitivity measures and their computation

The theory of optimal transport and the use of Wasserstein distances are attracting increasing attention in statistics and machine learning. At the same time, the definition of sensitivity measures for multivariate responses is a topical research subject. This work examines the construction of probabilistic sensitivity measures using the theory of optimal transport. We obtain a new family of indicators that can deal with multivariate outputs. We test estimators based on alternative algorithmic approaches for computing optimal transport problems, showing promising results and fast execution times for resonable sample sizes. Joint work with E. Borgonovo & G. Savarè (Bocconi Univ.), A. Figalli (ETH Zürich) Ref: preprint + code snippets.

The technical side of things: you can use Teams either directly from you web browser or using the “fat client”, which is available for most platforms (Windows, Linux, Mac, Android & iOS). We strongly recommend the latter option whenever possible. Please give it a try before the seminar to anticipate potential problems.

UQSay #33

The thirty-third UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, July 1, 2021.

2–3 PM — Iason Papaioannou (T.U. Munich) — [slides]

Reliability sensitivity analysis with FORM

This talk discusses reliability sensitivity analysis with the first-order reliability method (FORM). Classical sensitivity indices, which are often used to assess the influence of the input random variables on the probability of failure, are the FORM $\alpha$-factors. These factors are the directional cosines of the the most likely failure point in an underlying independent standard normal space and are obtained as by-products of the FORM analysis. The talk reviews a set of alternative reliability sensitivity indices and their estimation with FORM. Focus is put on variance-based reliability sensitivities that emerge from the variance decomposition of the indicator function of the failure event. The resulting first-order and total-effect reliability sensitivities can be estimated as a function of the FORM reliability indices and the $\alpha$-factors. The second part of the talk addresses decision-oriented sensitivities based on the concept of value of information. In particular, the indices associated with a decision related to the safety of an existing system are presented and their estimation with FORM is examined. The accuracy of the FORM approximations of the various sensitivities is demonstrated with numerical examples. Joint work with Daniel Straub. Ref: DOI:10.1016/j.ress.2021.107496 (preprint) and arxiv:2104.00986.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #32

The thirty-second UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, June 17, 2021.

2–3 PM — Andreas Fichtner (ETH Zürich) — [slides]

Probabilistic Full-Waveform Inversion

In the course of the past decade, full-waveform inversion has matured from a largely idealistic dream into a commonly applied method to image the internal structure of inaccessible bodies. Despite undeniable success, a major problem remains: The quantification of uncertainties in this often strongly nonlinear inverse problem. In this lecture, I will present a series of computational approaches that brings probabilistic full-waveform inversion with complete uncertainty quantification within reach: 1) Hamiltonian Monte Carlo sampling of the posterior probability density treats model parameters as particles that orbit through model space, obeying Hamilton’s equations from classical mechanics. The scaling properties of Hamiltonian Monte Carlo allow us to consider high-dimensional model spaces that often cannot be considered with more traditional, derivative-free sampling methods. 2) Autotuning based on limited-memory quasi-Newton methods provides nearly optimal mass matrices for Hamiltonian Monte Carlo, thereby largely removing laborious manual tuning. A factorised version of the L-BFGS algorithm, in particular, can increase the effective sample size by more than an order of magnitude. 3) Wavefield-adapted spectral-element meshes exploit prior knowledge on the geometry of wavefields. Such prior knowledge is frequently available for media that are smooth relative to the minimum wavelength. Wavefield-adapted meshes have the potential to drastically reduce the number of elements, leading to a computational forward modelling cost that makes Monte Carlo sampling possible. Joint work with Lars Gebraad & Christian Boehm. Ref: DOI:10.1029/2019JB018428.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #31

The thirty-first UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, June 3, 2021.

2–3 PM — Adrien Touboul (IRT SystemX & CERMICS) — [slides]

Uncertainty Quantification in graphs of functions through sample reweighting

The needs for multidisciplinary simulations in the design of complex industrial systems motivate the development of Uncertainty Quantification and Sensitivity Analysis methods that are compatible with disciplinary autonomy. This presentation focuses on decomposition methods based on sample reweighting. The design process is modeled by a graph, whose nodes are simulation codes and edges are exchanges of variables. The first part of this presentation is dedicated to the study of one particular reweighting method, based on the minimization of a Wasserstein distance. An explicit expression of the weights is exhibited in terms of Nearest Neighbors and some consistency results and rates of convergence are derived. The second part is dedicated to the general propagation of the weights in directed acyclic graphs, inspired from an existing algorithm of Amaral, Allaire & Willcox (2014). A general framework is developed to characterize the consistency of the global algorithm in terms of local weighting condition at each node. We observe that some weighting schemes can be obtained naturally from nonparametric linear regressions and linear smoothers. An interesting equivalence with some already existing tools in the literature permits to simplify the numerical computations part. The final algorithm does not require that the simulation codes have to be run at the same time or in a specific order. Hence, it allows for disciplinary autonomy. Joint work with Julien Reygner. Ref: hal-02968059.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #30

The thirtieth UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, May 20, 2021.

2–3 PM — Clément Gauchy (CEA & École polytechnique) — [slides]

An information geometry approach for robustness analysis in uncertainty quantification of computer codes

Robustness analysis is an emerging field in the uncertainty quantification domain. It involves analyzing the response of a computer model—which has inputs whose exact values are unknown—to the perturbation of one or several of its input distributions. Practical robustness analysis methods therefore require a coherent methodology for perturbing distributions; we present here one such rigorous method, based on the Fisher distance on manifolds of probability distributions. Further, we provide a numerical method to calculate perturbed densities in practice which comes from Lagrangian mechanics and involves solving a system of ordinary differential equations. The method introduced for perturbations is then used to compute quantile-related robustness indices. We illustrate these “perturbed-law based” indices on several numerical models. We also apply our methods to an industrial setting: the simulation of a loss of coolant accident in a nuclear reactor, where several dozen of the model’s physical parameters are not known exactly, and where limited knowledge on their distributions is available. Joint work with Jérôme Stenger, Roman Sueur et Bertrand Iooss. Refs: DOI:10.1080/00401706.2021.1905072.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #30

The thirtieth UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, May 20, 2021.

2–3 PM — Clément Gauchy (CEA & École polytechnique) — [slides]

An information geometry approach for robustness analysis in uncertainty quantification of computer codes

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #29

The twenty-ninth UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, May 6, 2021.

2–3 PM — Stefano Mariani (DICA @ Politecnico di Milano) — [slides]

Online damage detection and model updating via proper orthogonal decomposition and recursive Bayesian filters

An approach based on the synergistic use of proper orthogonal decomposition (POD) and Kalman filtering is proposed for the online health monitoring of damaged structures. The reduced-order model of the structure is obtained during the initial training stage of monitoring; afterward, effective estimations of structural damage are provided online by tracking the evolution in time of stiffness parameters and projection bases handled in the model order reduction procedure. Such tracking is accomplished via two Kalman filters: a first one to deal with the time evolution of a joint state vector, gathering the reduced-order state and the stiffness terms degraded by damage; a second one to deal with the update of the reduced-order model in case of damage evolution. Both filters exploit the information conveyed by measurements of the structural response to the external excitations. Focusing on multi-story shear building, the capability and performance of the proposed approach are assessed in terms of tracked variation of the stiffness terms, identified damage location and speed-up of the whole health monitoring procedure. Joint work with Saeed Eftekhar Azam, Giovanni Capellari, Francesco Caimmi. Refs: 10.1016/j.engstruct.2017.12.031, 10.1007/s11071-017-3530-1, 10.3390/s16010002, 10.1504/IJSMSS.2015.078355, 10.1016/j.engstruct.2013.04.004.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #28

The twenty-eighth UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, April 22, 2021.

2–3 PM — Chris Oates (Newcastle University and Alan Turing Inst.) — [slides]

Optimal Thinning of MCMC Output

There is a recent trend in computational statistics to move away from sampling methods and towards optimisation methods for posterior approximation. These include discrepancy minimisation, gradient flows and control functionals—all of which have the potential to deliver faster convergence than a Monte Carlo method. In this talk we will see how ideas from discrepancy minimisation can be applied to the problem of optimal thinning of MCMC output. Joint work with Marina Riabiz, Wilson Chen, Jon Cockayne, Pawel Swietach, Steve Niederer, Lester Mackey. Ref: arXiv:2005.03952 and http://stein-thinning.org.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #27

The twenty-seventh UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, April 1, 2021.

2–3 PM — Julien Pelamatti (EDF R&D) — [slides]

Bayesian optimization of variable-size design space problems

Within the framework of complex system design, it is often necessary to solve mixed variable optimization problems, in which the objective and constraint functions can depend simultaneously on continuous and discrete variables. Additionally, complex system design problems occasionally present a variable-size design space. This results in an optimization problem for which the search space varies dynamically (with respect to both number and type of variables) along the optimization process as a function of the values of specific discrete decision variables. Similarly, the number and type of constraints can vary as well. In this paper, two alternative Bayesian optimization-based approaches are proposed in order to solve this type of optimization problems. The first one consists of a budget allocation strategy allowing to focus the computational budget on the most promising design sub-spaces. The second approach, instead, is based on the definition of a kernel function allowing to compute the covariance between samples characterized by partially different sets of variables. The results obtained on analytical and engineering related test-cases show a faster and more consistent convergence of both proposed methods with respect to the standard approaches. Joint work with Loic Brevault (ONERA), Mathieu Balesdent (ONERA), El-Ghazali Talbi (Inria Lille), Yannick Guerin (CNES). Ref: arXiv:2003.03300.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #26

The twenty-sixth UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, March 18, 2021.

2–3 PM — Amaya Nogales Gómez (I3S, Sophia Antipolis) — [slides]

Incremental space-filling design based on coverings and spacings: improving upon low discrepancy sequences

The paper addresses the problem of defining families of ordered sequences {x_i} i∈N of elements of a compact subset X of R^d whose prefixes X_n = {x_i} i=1, …, n, for all orders n, have good space-filling properties as measured by the dispersion (covering radius) criterion. Our ultimate aim is the definition of incremental algorithms that generate sequences X_n with small optimality gap, i.e., with a small increase in the maximum distance between points of X and the elements of X_n with respect to the optimal solution X_n. The paper is a first step in this direction, presenting incremental design algorithms with proven optimality bound with respect to one-parameter families of criteria based on coverings and spacings that both converge to dispersion for large values of their parameter. Joint work with Luc Pronzato and Maria-Joao Rendas. Ref: hal-02987983.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #25

The twenty-fifth UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, March 4, 2021.

2–3 PM — Victor Picheny (Secondmind)

Bayesian optimisation: ablation study, global performance assessment and improvements based on trust regions

Bayesian Optimisation algorithms (BO) are global optimisation methods that iterate by constructing and using conditional Gaussian processes (GP). It is a common claim that BO is state-of-the-art for costly functions. However, this claim is weakly supported by experimental evidence, as BO is most often compared to itself, rather than to algorithms of different nature. In this work, we study the performance of BO within the well-known COmparing Continuous Optimizers benchmark (COCO). We first analyse the sensitivity of BO to its own parameters, enabling us to answer general questions regarding the choice of the GP kernel or its trend, the initial GP budget, and the suboptimisation of the acquisition function. Then, we study on which function class and dimension BO is relevant when compared to state-of-the-art optimisers for expensive functions. The second part of this talk describes a new BO algorithm to improve scalability with dimension, called TREGO (trust-region-like efficient global optimisation). TREGO alternates between regular BO steps and local steps within a trust region. By following a classical scheme for the trust region (based on a sufficient decrease condition), we demonstrate that our algorithm enjoys strong global convergence properties, while departing from EGO only for a subset of optimization steps. The COCO benchmark experiments reveal that TREGO consistently outperforms EGO and closes the performance gap with other state-of-the-art algorithms in conditions (high budget and dimension) for which BO was struggling to compete previously. Joint work Youssef Diouane, Rodolphe Le Riche, Alexandre Scotto Di Perrotolo. Ref: arXiv:2101.06808 & DiceOptim.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #24

The twenty-fourth UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, February 18, 2021.

2–3 PM — Amandine Marrel (CEA & IMT)

ICSCREAM methodology for the Identification of penalizing Configurations using SCREening And Metamodel — Application to high-dimensional thermal-hydraulic numerical experiments

In the framework of risk assessment in nuclear accident analysis, best-estimate computer codes are used to estimate safety margins. Several inputs of the code can be uncertain, due to a lack of knowledge but also to the particular choice of accidental scenario being considered. The objective of this work is to identify the most penalizing (or critical) configurations of several input parameters (called “scenario inputs”), independently of the uncertainty of the other inputs. Critical configurations of the scenario inputs correspond to high values of the code output Y, defined here by exceeding the 90%-quantile. However, thermal-hydraulic codes are too CPU-time expensive to be directly used to propagate the input uncertainties and solve the inversion problem. The adopted solution consists in fitting the code output by a metamodel, built from a reduced number of code simulations. When the number of input parameters is very large (e.g., around a hundred here), the metamodel building remains a challenge. To overcome this, we have developed a methodology, called ICSCREAM for Identification of penalizing Configurations using SCREening And Metamodel. Applied from a Monte Carlo sample of code simulations, the ICSCREAM methodology judiciously combines a step of SA to identify and rank the main influential inputs and to reduce the dimension, before building a Gaussian process (GP) metamodel. SA relies on new statistical independence tests that aggregate information of global and target Hilbert-Schmidt independence criteria. The GP is then efficiently built with a sequential process, where the inputs are taken into account in a more or less fine way, according to their supposed influence. Finally, the GP metamodel is intensively used to estimate the conditional probabilities of Y exceeding the critical value, according to each inputs to be penalized. Accurate uncertainty propagation, not feasible with the computational costly model, become therefore accessible with the ICSCREAM methodology. Joint work with Bertrand Iooss (EDF R&D & IMT) and Vincent Chabridon (EDF R&D). Ref: hal-02535146.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #23

The twenty-third UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, February 4, 2021.

2–3 PM — Clémentine Prieur (LJK, Univ. Grenoble Alpes)

Global sensitivity analysis for models described by stochastic differential equations

Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest. One of the statistical tools used to quantify the influence of each input variable on the quantity of interest are the Sobol’ sensitivity indices. In this paper, we consider stochastic models described by stochastic differential equations (SDE). We focus the study on mean quantities, defined as the expectation with respect to the Wiener measure of a quantity of interest related to the solution of the SDE itself. Our approach is based on a Feynman-Kac representation of the quantity of interest, from which we get a parametrized partial differential equation (PDE) representation of our initial problem. We then handle the uncertainty on the parametrized PDE using polynomial chaos expansion and a stochastic Galerkin projection. Joint work with Pierre Étoré, Dang Khoi Pham & Long Li. Ref: hal-01926919.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #22

The twenty-second UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, January 21, 2021.

14h–15h — Cédric Travelletti (University of Bern)

Implicit Update for Large-Scale Inversion under GP prior

We present an almost matrix-free update method for posterior Gaussian process distributions under sequential observations of linear functionals. By introducing a novel implicit representation of the posterior covariance matrix, we are able to extract posterior covariance information on large grids and to provide a framework for sequential data assimilation when covariance matrices cannot fit in memory. This is useful in Bayesian linear inverse problems with Gaussian priors, where the matrices involved grow quadratically in the number of elements in the discretization grid, creating memory bottlenecks when inverting on fine-grained discretizations. We illustrate our method by applying it to an excursion set recovery task arising from a gravimetric inverse problem on Stromboli volcano. In this setting, we demonstrate computation and sequential updating of exact posterior mean and covariance at resolutions finer than what state-of-the-art techniques can handle and showcase how the proposed framework enables implementing large-scale probabilistic excursion set estimation and also deriving efficient experimental design strategies tailored to this goal. Joint work with David Ginsbourger (Univ. Bern) and Niklas Linde (Univ. Lausanne). Ref: Volcapy (github).

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #20

The twentieth UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, December 17, 2020.

14h–15h — Bojana Rosic (University of Twente, Netherlands)

Inverse methods for damage estimation in concrete given small data sets

One of the main issues in material science is estimation of the constitutive laws given experimental data that may come in different forms ranging from the microscopic images to the macroscopic data collected by strain gauges for example. As data are often heterogeneous, of multi-scale/temporal nature, possibly ambiguous and of low quality due to missing values, the process of learning is often requiring the careful application of existing or design of new data fusion algorithms that are bounded to small data sets. In this talk will be presented the computationally efficient Bayesian algorithms for the damage estimation. In particular, the special attention will be paid to damage model estimation by using both classical uncertainty quantification as well as machine/deep learning approaches. Joint work with (alphabetical order) X. Chapeleau, P.-E. Charbonnel, L.-M. Cottineau, L. De Lorenzis, A. Ibrahimbegovic, V. Le Corvec, H.G. Matthies, E. Merliot, M.S. Sarfaraz, D. Siegert, R. Vidal, J. Waeytens and T. Wu. Refs: hal-01379214, arXiv:1909.07209, DOI:10.1007/s00466-020-01942-x, arXiv:1912.03108.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #19

The nineteenth UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, December 3, 2020.

14h–15h — Álvaro Rollón de Pinedo (EDF R&D and Université Grenoble Alpes)

Functional outlier detection applied to nuclear transient simulation analysis

The ever increasing recording and storing capabilities of industrial systems provide a large amount of physical data that can be exploited by engineers. These data may take the form of functions, usually a one-dimensional function of time, but eventually as a multidimensional function of space and time. Finding the subsets of objects that behave abnormally in them is a goal that can prove to be useful in order to avoid spurious results, simulations that do not reproduce certain physical phenomena as expected, or extreme physical events and domains. In the context of nuclear transient simulations, safety reports mostly focus on the study of some scalar parameters (safety criteria), supposed to guarantee the safety of an installation during an accidental transient as long as they do not surpass a previously established threshold. Nevertheless, the state- of-the-art simulations codes (called Best Estimate) provide a much richer and complex information, which can be better taken advantage of through the identification outlying simulations amongst those generated as outputs. The goal of this talk is to introduce the functional outlier detection domain, highlighting its interest in industrial settings, as well as to present our detection technique and the conclusions on the physical analysis of nuclear transients that can be obtained from its use. Joint work with Mathieu Couplet, Bertrand Iooss, Nathalie Marie, Amandine Marrel, Elsa Merle and Roman Sueur. Reference: hal-02965504.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #18

The eighteenth UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, November 19, 2020.

14h–15h — Eyke Hüllermeier (Paderborn University, Germany) — [slides]

Aleatoric and Epistemic Uncertainty in Machine Learning: An Ensemble-based Approach

Due to the steadily increasing relevance of machine learning for practical applications, many of which are coming with safety requirements, the notion of uncertainty has received increasing attention in machine learning research in the last couple of years. This talk will address the question of how to distinguish between two important types of uncertainty, often refereed to as aleatoric and epistemic, in the setting of supervised learning, and how to quantify these uncertainties in terms of suitable numerical measures. Roughly speaking, while aleatoric uncertainty is due to inherent randomness, epistemic uncertainty is caused by a lack of knowledge. As a concrete approach for uncertainty quantification in machine learning, the use of ensemble learning methods will be discussed. Joint work with S. Destercke, V.-L. Nguyen, M. H. Shaker & W. Waegeman. References: arXiv:1910.09457, arXiv:1909.00218, arXiv:2001.00893.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #17

The seventeenth UQSay seminar on UQ, DACE and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, November 5, 2020.

14h–15h — Luc Bonnet (ONERA & MSSMAT) — [slides]

The expected performance of a system can generally differ from its operational performance due to the variability of some parameters. Optimal Uncertainty Quantification is a powerful mathematical tool that can be used to rigorously bound the probability of exceeding a given performance threshold for uncertain operational conditions or system characteristics. Metamodeling is at the heart of this research framework. In this perspective, Kernel Flow, a recent method to obtain a metamodel by kriging developed by Owhadi & Yoo, will be presented. The results obtained will be illustrated by examples in numerical and experimental aerodynamics. Joint work with Eric Savin and Houman Owhadi. References: 10.1016/j.jcp.2019.03.040, 10.1137/10080782X & 10.3390/a13080196.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #16

The sixteenth UQSay seminar on Uncertainty Quantification and related topics, organized by L2S, MSSMAT, LMT and EDF R&D, will take place online on Thursday afternoon, October 22, 2020.

14h–15h — Nicolas Bousquet (EDF R&D)

Well-posed stochastic inversion in uncertainty quantification, with links with sensitivity analysis

Stochastic inversion problems are typically encountered when it is wanted to quantify the uncertainty affecting the inputs of computer models. They consist in estimating input distributions from noisy, observable outputs, and such problems are increasingly examined in Bayesian contexts where the targeted inputs are affected by a mixture of aleatory and epistemic uncertainties. While they are characterized by identifiability conditions, well-posedness constraints of “signal to noise” have to be took into account within the definition of the model, prior to inference. In addition to numeric conditioning notions and regularization techniques used in inverse problems, we propose and investigate an interpretation of well-posedness, in the context of parametric uncertainty quantification and global sensitivity analysis, based on the degradation of Fisher information. It offers an explicitation of such prior constraints considering linear or linearizable operators, this linearization being either local (based on differentiability) or variational. Simulated experiments indicate that, when injected into the modeling process, these constraints can limit the influence of measurement or process noise on the estimation of the input distribution, and let hope for future extensions in a full non-linear framework, for example through the use of linear Gaussian mixtures.

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Amélie Fau (LMT), Bertrand Iooss (EDF R&D).

UQSay #15

The fifteenth UQSay seminar on Uncertainty Quantification and related topics, organized by L2S, MSSMAT, and EDF R&D, will take place online on Thursday afternoon, October 8, 2020.

14h–15h — Sebastian Schöps (TU Darmstadt)

Uncertainty Quantification for Maxwell’s eigenproblem based on isogeometric analysis and mode tracking

Superconducting cavities are used in particle accelerators, e.g. at DESY in Hamburg, Germany. Their resonating electromagnetic field is commonly characterised by eigenmodes and eigenvalues which are very sensitive to small geometry deformations. This presentation proposes an uncertainty quantification workflow based on a Karhunen–Loève expansion of the manufacturing imperfections and eigenvalue tracking based on algebraic and geometric homotopies. Joint work with Niklas Georg, Wolfgang Ackermanna, Jacopo Corno. Reference: DOI:10.1016/j.cma.2019.03.002 (arxiv:1802.02978).

Organizing committee: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Bertrand Iooss (EDF R&D).

UQSay #14

The fourteenth UQSay seminar on Uncertainty Quantification and related topics, organized by L2S, MSSMAT, and EDF R&D, will take place online on Thursday afternoon, September 24, 2020.

14h–15h — Amélie Fau (LMT, ENS Paris-Saclay)

Alternative strategies for adaptive sampling for kriging metamodels

A large variety of strategies have been proposed in the literature to offer optimal dataset for kriging metamodels. Even though adaptive schemes guarantee convergence and improvement of estimation accuracy for instance for Galerkin approaches at least in a goal-oriented sense, using usual adaptive sampling schemes for kriging metamodels might be detrimental, worsing prediction results compared to one-shot sampling techniques. The goal of this seminar is to share our experience on cases leading to this disvantageous behavior. Besides, problems leading to beneficial behavior will be discussed to highlight criteria for deciding about cases of interest for which adaptive sampling strategies are highly promising.

Joint work with Jan Fuhg & Udo Nackenhorst (Leibniz Universität, Hannover).

Reference: DOI:10.1007/s11831-020-09474-6.

Organizers: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Bertrand Iooss (EDF R&D).

Practical details: the seminar will be held online using Microsoft Teams.

If you want to attend this seminar (or any of the forthcoming online UQSay seminars), and if you do not already have access to the UQSay group on Teams, simply send an email and you will be invited. Please specify which email address the invitation must be sent to (this has to be the address associated with your Teams account).

You will find the link to the seminar on the “General” UQSay channel on Teams, approximately 15 minutes before the beginning.

UQSay #13

The thirteenth UQSay seminar on Uncertainty Quantification and related topics, organized by L2S, MSSMAT, and EDF R&D, will take place online on Thursday afternoon, September 10, 2020.

14h–15h — Balázs Kégl (Noah’s Ark Lab, Huawei Paris) — [slides]

DARMDN: Deep autoregressive mixture density nets for dynamical system modelling

Unlike computers, physical engineering systems (such as data center cooling or wireless network control) do not get faster with time. This is arguably one of the main reasons why recent beautiful advances in deep reinforcement learning (RL) stay mostly in the realm of simulated worlds and do not immediately translate to practical success in the real world. In order to make the best use of the small data sets these systems generate, we develop data-driven neural simulators to model the system and apply model-based control to optimize them. In this talk I will present the first step of this research agenda, a new versatile system modelling tool called deep autoregressive mixture density net (DARMDN – pronounced darm-dee-en). We argue that the performance of model-based reinforcement learning is partly limited by the approximation capacity of the currently used conditional density models and show how DARMDN alleviates these limitations. The model, combined with a random shooting controller, establishes a new state of the art on the popular Acrobot benchmark. Our most interesting and counter-intuitive finding is that the “sincos” Acrobot system which requires no multimodal posterior predictives, can be solved with a deterministic model, but only if it is trained as a probabilistic model. A deterministic model that is trained to minimize MSE leads to prediction error accumulation.

Joint work with Gabriel Hurtado and Albert Thomas.

Organizers: Julien Bect (L2S), Emmanuel Vazquez (L2S), Didier Clouteau (MSSMAT), Filippo Gatti (MSSMAT), Fernando Lopez Caballero (MSSMAT), Bertrand Iooss (EDF R&D).

Practical details: the seminar will be held online using Microsoft Teams.

You will find the link to the seminar on the “General” UQSay channel on Teams, approximately 15 minutes before the beginning.

UQSay #07

The seventh UQSay seminar on Uncertainty Quantification and related topics, organized by L2S and MSSMAT, will take place on Thursday afternoon, January 16, 2020, at CentraleSupelec Paris-Saclay (Eiffel building, amphi III).

We will have two talks: 14h — Bertrand Iooss (EDF R&D / PRISME dept.) — [slides]

Iterative estimation in uncertainty and sensitivity analysis

While building and using numerical simulation models, uncertainty and sensitivity analysis are invaluable tools. In engineering studies, numerical model users and modellers have shown high interest in these techniques that require to run many times the simulation model with different values of the model inputs in order to compute statistical quantities of interest (QoI, i.e. mean, variance, quantiles, sensitivity indices…). In this talk we will focus on new issues relative to large scale numerical systems that simulate complex spatial and temporal evolutions. Indeed, the current practice consists in the storage of all the simulation results. Such a storage becoming quickly overwhelming, with the associated long read time that makes cpu time consuming the estimation of the QoI. One solution consists in avoiding this storage and in computing QoI on the fly (also called in-situ). It turns the problem to considering problems of iterative statistical estimation. The general mathematical and computational issues will be posed, and a particular attention will be paid to the estimation of quantiles (via an adaptation of the Robbins-Monro algorithm) and variance-based sensitivity indices (the so-called Sobol’ indices).

Joint work with Yvan Fournier (EDF), Bruno Raffin (INRIA), Alejandro Ribés (EDF), Théophile Terraz (INRIA).

UQSay #03

The third UQSay seminar, organized by L2S and EDF R&D, will take place on Thursday afternoon, June 13, 2019, at CentraleSupelec Paris-Saclay (Eiffel building, amphi V). We will have two talks:

14h — Alexandre Janon (Laboratoire de Mathématique d’Orsay) — [slides]

Part 1: Consistency of Sobol indices with respect to stochastic ordering of input parameters

In the past decade, Sobol’s variance decomposition have been used as a tool – among others – in risk management. We show some links between global sensitivity analysis and stochastic ordering theories. This gives an argument in favor of using Sobol’s indices in uncertainty quantification, as one indicator among others. Reference: https://doi.org/10.1051/ps/2018001 (hal-01026373)

Part 2: Global optimization using Sobol indices

We propose and assess a new global (derivative-free) optimization algorithm, inspired by the LIPO algorithm, which uses variance-based sensitivity analysis (Sobol indices) to reduce the number of calls to the objective function. This method should be efficient to optimize costly functions satisfying the sparsity-of-effects principle. Reference: hal-02154121

15h — Pierre Barbillon (MIA Paris) — [slides]

Sensitivity analysis of spatio-temporal models describing nitrogen transfers, transformations and losses at the landscape scale

Modelling complex systems such as agroecosystems often requires the quantification of a large number of input factors. Sensitivity analyses are useful to determine the appropriate spatial and temporal resolution of models and to reduce the number of factors to be measured or estimated accurately. Comprehensive spatial and temporal sensitivity analyses were applied to the NitroScape model, a deterministic spatially distributed model describing nitrogen transfers and transformations in rural landscapes. Simulations were led on a theoretical landscape that represented five years of intensive farm management and covering an area of 3km2. Cluster analyses were applied to summarize the results of the sensitivity analysis on the ensemble of model outputs.The methodology we applied is useful to synthesize sensitivity analyses of models with multiple space-time input and output variables and could be ported to other models than NitroScape. Reference: https://doi.org/10.1016/j.envsoft.2018.09.010 (arXiv:1709.08608)

Organizers: Julien Bect (L2S) and Bertrand Iooss (EDF R&D). No registration is needed, but an email would be appreciated if you intend to come.

UQSay #01

The first UQSay seminar, organized by L2S, will take place in the afternoon of March 21, 2019, at CentraleSupelec Paris-Saclay (Eiffel building, amphi IV). We will have two talks:

14h – Mickaël Binois (INRIA Sophia-Antipolis) [slides]

Heteroskedastic Gaussian processes for simulation experiments

An increasing number of time-consuming simulators exhibit a complex noise structure that depends on the inputs. To conduct studies with limited budgets of evaluations, new surrogate methods are required to model simultaneously the mean and variance fields. To this end, we present recent advances in Gaussian process modeling with input-dependent noise. First, we describe a simple, yet efficient, joint modeling framework that rely on replication for both speed and accuracy. Then we tackle the issue of leveraging replication and exploration in a sequential manner for various goals, such as obtaining a globally accurate model, for optimization, contour finding, and active subspace estimation. We illustrate these on applications coming from epidemiology and inventory management.

Ref : https://arxiv.org/abs/1710.03206.

15h – François Bachoc (IMT, Toulouse) [slides]

Gaussian process regression model for distribution inputs

Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances. We provide asymptotic results for covariance function estimation and prediction. We also provide numerical comparisons with other forecast methods based on distribution inputs.

Ref : https://arxiv.org/abs/1701.09055.

Organizers : Julien Bect (L2S) and Emmanuel Vazquez (L2S).

No registration is needed, but an email would be appreciated if you intend to come.