UQSay #78

The seventy-eighth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, November 14, 2024.

2–3 PM — Botond Szabo (Bocconi University, Department of Decision Sciences - BIDSA) — [slides]

Uncertainty quantification and estimation with variational Gaussian Processes

We study the theoretical properties of a variational Bayes method in the Gaussian Process regression model. We consider the inducing variables method introduced by Titsias (2009b) and derive sufficient conditions for obtaining contraction rates for the corresponding variational Bayes (VB) posterior. We also derive guarantees and limitations for the associated credible sets. The derived general results are then applied for several specific examples. Finally, we also consider approximation methods from probabilistic numerics (e.g. Lanczos iteration and conjugate gradient descent). We demonstrate their close relationship with the variational Bayes approach and provide guidelines for these approaches as well to obtain optimal (minimax) statistical inference.

References:

• Contraction rates for sparse variational approximations in Gaussian process regression, 2022
• Uncertainty quantification for sparse spectral variational approximations in Gaussian process regression, 2023
• Contraction rates for conjugate gradient and Lanczos approximate posteriors in Gaussian process regression, 2024

Joint work with H. van Zanten (Vrije Universiteit Amsterdam) & D. Nieman (Vrije Universiteit Amsterdam) & B. Stankewitz (University of Potsdam).

Organizing committee: Pierre Barbillon (MIA-Paris), Julien Bect (L2S), Nicolas Bousquet (EDF R&D), Vincent Chabridon (EDF R&D), Amélie Fau (LMPS), Filippo Gatti (LMPS), Clément Gauchy (CEA), Bertrand Iooss (EDF R&D), Alexandre Janon (LMO), Sidonie Lefebvre (ONERA), Didier Lucor (LISN), Sébastien Petit (LNE), Emmanuel Vazquez (L2S), Xujia Zhu (L2S).

Coordinators: Sidonie Lefebvre (ONERA) & Xujia Zhu (L2S)

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 #77

The seventy-seventh UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, October 31, 2024.

2–3 PM — Nadège Polette (Mines Paris PSL - CEA DAM DIF) — [slides]

Mitigating Overconfidence in Bayesian Field Inversion thanks to Hyperparameters Sampling

The objective of Bayesian field inversion is to approximate the posterior distribution of a field thanks to indirect observations. Such problems have to face two issues, the infinite dimensionality of the field and the high number of forward model evaluations required to achieve MCMC convergence. In this study, the field to infer is assumed to be a particular realization of a random field and is modeled by its truncated Karhunen-Loève (KL) decomposition leading to a finite dimensional parametrization. The KL representation relies on an autocovariance function that depends on poorly known hyperparameters. The added value of our work is to introduce a new method, called change of measure, designed to deal with uncertain hyperparameters instead of deterministic ones. In practice, the hyperparameters are jointly sampled with the KL coordinates and the posterior distribution has a hierarchical Bayesian structure. In addition, to reduce the computational cost of the MCMC procedure, the likelihood is estimated with polynomial chaos surrogates of the forward model outputs. Applications on transient diffusion and seismic traveltime tomography problems highlight the interest of not fixing the hyperparameters to deterministic values. Exploring the hyperparameters space is highly valuable because it provides a better estimation of the field uncertainties.

References:

• Change of Measure for Bayesian Field Inversion with Hierarchical Hyperparameters Sampling, 2024

Joint work with O. Le Maître (CNRS, CMAP) & P. Sochala (CEA DAM DIF) & A. Gesret (Mines Paris PSL).

Organizing committee: Pierre Barbillon (MIA-Paris), Julien Bect (L2S), Nicolas Bousquet (EDF R&D), Vincent Chabridon (EDF R&D), Amélie Fau (LMPS), Filippo Gatti (LMPS), Clément Gauchy (CEA), Bertrand Iooss (EDF R&D), Alexandre Janon (LMO), Sidonie Lefebvre (ONERA), Didier Lucor (LISN), Sébastien Petit (LNE), Emmanuel Vazquez (L2S), Xujia Zhu (L2S).

Coordinators: Sidonie Lefebvre (ONERA) & Xujia Zhu (L2S)

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 #76

The seventy-sixth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, October 17, 2024.

4–5 PM — Habib Najm (Sandia National Laboratories) — [slides]

Uncertainty Quantification in Computational Combustion Models

Uncertainty quantification (UQ) in large scale computational combustion models faces key challenges of high dimensionality and computational cost. These models, particularly when using detailed chemical mechanisms, typically involve large numbers of uncertain parameters. Exploring these high-dimensional spaces necessitates the use of large numbers of computational samples, which, given high computational costs, is prohibitively expensive. I will discuss a UQ workflow, and underlying methods, to address this challenge in practice. These methods include global sensitivity analysis with polynomial chaos (PC) sparse regression, coupled with multilevel multifidelity methods. The combination of these tools is often useful to reliably cut-down dimensionality with acceptable computational costs, identifying a lower-dimensional subspace where the construction of PC surrogates of requisite accuracy is feasible. These surrogates are in turn necessary for both Bayesian inference and forward uncertainty propagation purposes. I will discuss this UQ workflow for problems of practical relevance in combustion. I will also touch on the issue of data availability, and the challenge of Bayesian estimation of uncertain model parameters given sparse or missing data.

Organizing committee: Pierre Barbillon (MIA-Paris), Julien Bect (L2S), Nicolas Bousquet (EDF R&D), Vincent Chabridon (EDF R&D), Amélie Fau (LMPS), Filippo Gatti (LMPS), Clément Gauchy (CEA), Bertrand Iooss (EDF R&D), Alexandre Janon (LMO), Sidonie Lefebvre (ONERA), Didier Lucor (LISN), Sébastien Petit (LNE), Emmanuel Vazquez (L2S), Xujia Zhu (L2S).

Coordinators: Sidonie Lefebvre (ONERA) & Xujia Zhu (L2S)

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 #75

The seventy-fifth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, October 3, 2024.

2–3 PM — Olivier Laurent (SATIE, Université Paris-Saclay - U2IS, ENSTA) — [slides]

A Symmetry-Aware Exploration of Bayesian Neural Network Posteriors

The distribution of modern deep neural networks (DNNs) weights -- crucial for uncertainty quantification and robustness -- is an eminently complex object due to its extremely high dimensionality. This paper presents one of the first large-scale explorations of the posterior distribution of deep Bayesian Neural Networks (BNNs), expanding its study to real-world vision tasks and architectures. Specifically, we investigate the optimal approach for approximating the posterior, analyze the connection between posterior quality and uncertainty quantification, delve into the impact of modes on the posterior, and explore methods for visualizing the posterior. Moreover, we uncover weight-space symmetries as a critical aspect for understanding the posterior. To this extent, we develop an in-depth assessment of the impact of both permutation and scaling symmetries that tend to obfuscate the Bayesian posterior. While the first type of transformation is known for duplicating modes, we explore the relationship between the latter and L2 regularization, challenging previous misconceptions. Finally, to help the community improve our understanding of the Bayesian posterior, we release a large-scale dataset of model checkpoints, including thousands of real-world models, along with our code.

References:

• A Symmetry-Aware Exploration of Bayesian Neural Network Posteriors, ICLR, 2024 - [webpage]- [code]
• TorchUncertainty package - [associated bdd]

Joint work with Emanuel Aldea (SATIE) & Gianni Franchi (U2IS, ENSTA).

Organizing committee: Pierre Barbillon (MIA-Paris), Julien Bect (L2S), Nicolas Bousquet (EDF R&D), Vincent Chabridon (EDF R&D), Amélie Fau (LMPS), Filippo Gatti (LMPS), Clément Gauchy (CEA), Bertrand Iooss (EDF R&D), Alexandre Janon (LMO), Sidonie Lefebvre (ONERA), Didier Lucor (LISN), Sébastien Petit (LNE), Emmanuel Vazquez (L2S), Xujia Zhu (L2S).

Coordinators: Sidonie Lefebvre (ONERA) & Xujia Zhu (L2S)

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 #74

The seventy-fourth UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, September 19, 2024.

3–4 PM — Alexander Terenin (Cornell University) — [slides]

Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent

The ability to deploy Gaussian-process-based decision-making systems such as Bayesian optimization at scale has traditionally been limited by computational costs arising from the need to solve large linear systems. The de facto standard for solving linear systems at scale is via the conjugate gradient algorithm - in particular, stochastic gradient descent is known to converge near-arbitrarily-slowly on quadratic objectives that correspond to Gaussian process models’ linear systems. In spite of this, we show that it produces solutions which have low test error, and quantify uncertainty in a manner that mirrors the true posterior. We develop a spectral characterization of the error caused by finite-time non-convergence, which we prove is small both near the data, and sufficiently far from the data. Stochastic gradient descent therefore only differs from the true posterior between these regions, demonstrating a form of implicit bias caused by benign non-convergence. We conclude by showing, empirically, that stochastic gradient descent achieves state-of-the-art performance on sufficiently large-scale regression tasks, and produces uncertainty estimates which match the performance of significantly more expensive baselines on large-scale Bayesian optimization.

References:

• Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent, NeurIPS, 2023 - [github sgd-gp]
• Stochastic Gradient Descent for Gaussian Processes Done Right, ICLR, 2024
• Numerically Stable Sparse Gaussian Processes via Minimum Separation using Cover Trees, JMLR, 2024 - [github conjugate-gradient-sparse-gp]

Organizing committee: Pierre Barbillon (MIA-Paris), Julien Bect (L2S), Nicolas Bousquet (EDF R&D), Vincent Chabridon (EDF R&D), Amélie Fau (LMPS), Filippo Gatti (LMPS), Clément Gauchy (CEA), Bertrand Iooss (EDF R&D), Alexandre Janon (LMO), Sidonie Lefebvre (ONERA), Didier Lucor (LISN), Sébastien Petit (LNE), Emmanuel Vazquez (L2S), Xujia Zhu (L2S).

Coordinators: Sidonie Lefebvre (ONERA) & Xujia Zhu (L2S)

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 #73

The seventy-third UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, May 16, 2024.

2–3 PM — Gaël Poette (CEA DAM, CESTA - ENSEIRB-MATMECA) — [slides]

Building and solving reduced models for the uncertain linear Boltzmann equation (sometimes intrusiveness, a.k.a. physics-informedness, is worth it)

In this talk, I will present one way to build reduced models which aim at being able to efficiently propagate uncertainties through the linear Boltzmann equation. Only one run of a Monte-Carlo code will be enough to solve the system and accurately estimate statistical observables (mean, variance, tolerance intervals, sensitivity indices) on your physical observables of interest. From the Machine Learning community point of view, the presented reduced model can be cast in the category of 'Physics Informed' ones in the sense that the structure of the partial differential equation is intensively used in order to build the reduced model. In this context, we will show that such a physics-informed strategy (from the literature from another generation, it is also called an 'intrusive' strategy) combined with the relevant numerical scheme can be very competitive with respect to the best of the non-intrusive ones.

From the transport community point of view, the reduced model I will present can be understood, in a sense, as an attempt to make the best of two worlds: Pn reduced models and Monte-Carlo resolution schemes.The Pn model, well-known in neutronics and photonics, is built with respect to the uncertain variables (in the literature relative to uncertainty quantification, it is commonly called Polynomial Chaos) rather than with respect to the angular one. And the resulting reduced model is solved with a Monte-Carlo scheme (due to the high dimensional context we are in). The reasons for such a combination will be detailed and numerical results will be presented (for the instationary linear Boltzmann but also for eigenvalue computations). Performance comparisons will be presented on neutronics (keff computations) and photonics applications. If time allows it, discussions on the asymptotical errors (noise) will be tackled.

Some parts of this talk come from a joint work with Emeric Brun (CEA, DES).

References:

• G. Poëtte (2019) A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation. J. Comp. Phys. DOI:10.1016/j.jcp.2019.01.052,
• G. Poëtte (2020) Spectral convergence of the generalized Polynomial Chaos reduced model obtained from the uncertain linear Boltzmann equation. Mathematics and Computers in Simulation. 10.1016/j.matcom.2020.04.009.
• G. Poëtte (2021) Efficient uncertainty propagation for photonics: Combining implicit semi-analog monte carlo (ISMC) and monte carlo generalised polynomial chaos (MC-gPC). J. Comp. Phys. DOI:10.1016/j.jcp.2021.110807,
• G. Poëtte (2022) Numerical analysis of the Monte-Carlo noise for the resolution of the deterministic and uncertain linear Boltzmann equation (comparison of non-intrusive gPC and MC-gPC). J. of Comp. and Theor. Transport. DOI:10.1080/23324309.2022.2063900.
• G. Poëtte, E. Brun (2022) Efficient uncertain keff computations with the Monte Carlo resolution of generalised Polynomial Chaos based reduced models. J. Comp. Phys. DOI:10.1016/j.jcp.2022.111007.

Organizing committee: Pierre Barbillon (MIA-Paris), Julien Bect (L2S), Nicolas Bousquet (EDF R&D), Vincent Chabridon (EDF R&D), Amélie Fau (LMPS), Filippo Gatti (LMPS), Clément Gauchy (CEA), Bertrand Iooss (EDF R&D), Alexandre Janon (LMO), Sidonie Lefebvre (ONERA), Didier Lucor (LISN), Sébastien Petit (LNE), Emmanuel Vazquez (L2S), Xujia Zhu (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 #72

The seventy-second UQSay seminar on UQ, DACE and related topics will take place online on Thursday afternoon, April 25, 2024.

3–4 PM — John Jakeman (Sandia National Laboratories) — [slides]

Surrogate Modeling for Efficiently, Accurately and Conservatively Estimating Measures of Risk

This talk will present a surrogate-based framework for conservatively estimating risk from limited evaluations of an expensive physical experiment or simulation. Focus will be given to the computation of risk measures that quantify tail statistics of the loss, such as Average Value at Risk (AVaR). Monte Carlo (MC) sampling can be used to approximate such risk measures, however MC requires a large number of model simulations, which can make accurately estimating risk intractable for computationally expensive models. Given a set of samples surrogates are constructed such that the estimate of risk, obtained from the surrogate, is always greater than the empirical estimate obtained from the training data. These surrogates not only limit over-confidence in model reliability but produce estimates of risk that converge much faster to the true risk, than purely sampled based estimates.

The first part of the talk will discuss how to use the risk quadrangle, which rigorously connects stochastic optimization and statistical estimation, to construct conservative surrogates that can be tailored to the specific risk preferences of the model stakeholder. Surrogates constructed using least squares and quantile regression are specific cases of this framework. The second part of the talk will then present an approach, based upon stochastic orders, for constructing surrogates that are conservative with respect to families of risk measures, which is useful when risk preferences are difficult to elicit. This approach uses first and second order stochastic dominance to respectively enforce that the surrogate over-estimates probability of failure and AVaR for a finite set of thresholds. The conservative surrogates constructed introduce a bias that allows them to conservatively estimate risk. Theoretical results will be provided that show that for orthonormal models such as polynomial chaos expansions, this bias decays at the same rate as the root mean squared error in the surrogate. Numerous numerical examples will be used to confirm that that risk-aware surrogates do indeed overestimate risk while converging at the expected rate.

Reference: Surrogate modeling for efficiently, accurately and conservatively estimating measures of risk, RESS, 2022 - [github PyApprox].

Joint work with Drew Kouri (Sandia).

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.