Sébastien Riffaud

Sébastien Riffaud

Postdoctoral researcher at École Polytechnique Fédérale de Lausanne

Posts by Collection

portfolio

publications

talks

MOREPAS 2018

Published:

We present a reduced-order approximation of the BGK equation leading to fast and accurate computations. The BGK model describes the dynamics of a gas flow in both hydrodynamic and rarefied regimes. The particles of the gas are represented by a density distribution function depending on physical space, velocity space and time. In this work, the density distribution function is approximated in the velocity space by a small number of basis functions computed offline. In the offline phase, the BGK equation is sampled in order to collect information on the density distribution function. To complete this sampling, optimal transport is used to add new information by interpolating the samples of the density distribution function. Finally, the basis functions are built by Proper Orthogonal Decomposition. During the online phase, the offline knowledge is used to compute approximations of the density distribution function at low cost. To do so, the BGK equation is projected onto the basis functions, leading to a system of partial differential equations. The system obtained is hyperbolic by construction and is solved by an IMEX Runge-Kutta scheme in time and a finite-volume scheme in space. To improve the accuracy, the reduced-oder model is modified to conserve mass, momentum and energy of the gas. Numerical illustrations of 1D and 2D flows are given. In particular, we investigate the reconstruction and the prediction of shock waves, boundary layers and vortices. The results demonstrate the accuracy of the reduced-order model and the significant reduction of the computational cost.

ECCOMAS 2018

Published:

A reduced-order model of the BGK equation leading to fast and accurate computations is presented. The BGK model describes the dynamics of a gas flow in both hydrodynamic and rarefied regimes. In the reduced-order model, the distribution functions are approximated in velocity space by a small number of basis functions computed offline. In the offline phase, the BGK equation is sampled in order to collect information on the distribution functions. Then, optimal transportation provides additional samples by interpolating the distribution functions to complete the sampling. Finally, the basis functions are built by Proper Orthogonal Decomposition. During the online phase, the offline knowledge is used to compute approximations of the density distribution function at low cost. The BGK equation is projected onto the basis functions, leading to an hyperbolic system of partial differential equations. Moreover, the projection is modified to conserve mass, momentum and energy of the gas. The resulting system is then solved by an IMEX Runge-Kutta scheme in time and a finite-volume scheme in space. The results show the significant reduction of the computational cost and the accuracy of the reduced-order model.

SIAM CSE 2019

Published:

A reduced-order model of the BGK equation leading to fast and accurate computations is presented. The BGK model describes the dynamics of a gas flow in both hydrodynamic and rarefied regimes. In the reduced-order model, the distribution functions are approximated in velocity space by a small number of basis functions computed offline. In the offline phase, the BGK equation is sampled in order to collect information on the distribution functions. Then, optimal transportation provides additional samples by interpolating the distribution functions to complete the sampling. Finally, the basis functions are built by Proper Orthogonal Decomposition. During the online phase, the offline knowledge is used to compute approximations of the density distribution function at low cost. The BGK equation is projected onto the basis functions, leading to an hyperbolic system of partial differential equations. Moreover, the projection is modified to conserve mass, momentum and energy of the gas. The resulting system is then solved by an IMEX Runge-Kutta scheme in time and a finite-volume scheme in space. The results show the significant reduction of the computational cost and the accuracy of the reduced-order model.

CMBE 2022

Published:

This talk will present a low-rank tensor method for parameterized linear fluid-structure interaction problems. In this work, the fluid is governed by the incompressible Stokes equations, and we assume a thick-walled solid model. In order to reduce the computational cost associated with the coupled problem, we develop a low-rank tensor method based on the low-rank Generalized Minimal Residual method. This one is then employed in an uncertainty quantification task to perform a large number of simulations. The preliminary results show the significant reduction of the computational cost provided by the proposed method.

CFC 2023

Published:

This work has been motivated by data assimilation in blood flow models. In many realistic applications, the knowledge of the model parameters as well as the boundary conditions is not perfect. We focus here on the estimation of the input parameters from observations of the output solution. The resulting estimate is then exploited to perform uncertainty quantification tasks. However, there are two major difficulties when trying to solve parameter estimation problems. First, some parameters may not be identifiable. For instance, if only partial noisy measurements of the solution are available, several different parameter values may be associated with the same observation. Second, if the parameters depend nonlinearly on the solution, the parameter estimation problem results in a nonlinear non-convex optimization problem that can be difficult to solve. For these reasons, we employ a sequential Markov Chain Monte Carlo (MCMC) method using the Metropolis-Hasting algorithm to estimate the parameters. This bayesian approach consists in sampling the posterior probability distribution of the parameters and allows to identify eventual correlations between the parameter values. However, updating this sampling is computationally expensive since the solution must be evaluated repeatedly for each queried parameter. For this reason, we developed a low-rank solver to significantly reduce the time and storage requirements associated with the MCMC procedure. The performance of the resulting method is demonstrated on three different applications.

M2P 2023

Published:

This work has been motivated by data assimilation in blood flow models. In many realistic applications, the knowledge of the model parameters as well as the boundary conditions is not perfect. We focus here on the estimation of the input parameters from observations of the output solution. The resulting estimate is then exploited to perform uncertainty quantification tasks. However, there are two major difficulties when trying to solve parameter estimation problems. First, some parameters may not be identifiable. For instance, if only partial noisy measurements of the solution are available, several different parameter values may be associated with the same observation. Second, if the parameters depend nonlinearly on the solution, the parameter estimation problem results in a nonlinear non-convex optimization problem that can be difficult to solve. For these reasons, we employ a sequential Markov Chain Monte Carlo (MCMC) method using the Metropolis-Hasting algorithm to estimate the parameters. This bayesian approach consists in sampling the posterior probability distribution of the parameters and allows to identify eventual correlations between the parameter values. However, updating this sampling is computationally expensive since the solution must be evaluated repeatedly for each queried parameter. For this reason, we developed a low-rank solver to significantly reduce the time and storage requirements associated with the MCMC procedure. The performance of the resulting method is demonstrated on three different applications.

UNCECOMP 2023

Published:

This work has been motivated by data assimilation in blood flow models. In many realistic applications, the knowledge of the model parameters as well as the boundary conditions is not perfect. We focus here on the estimation of the input parameters from observations of the output solution. The resulting estimate is then exploited to perform uncertainty quantification tasks. However, there are two major difficulties when trying to solve parameter estimation problems. First, some parameters may not be identifiable. For instance, if only partial noisy measurements of the solution are available, several different parameter values may be associated with the same observation. Second, if the parameters depend nonlinearly on the solution, the parameter estimation problem results in a nonlinear non-convex optimization problem that can be difficult to solve. For these reasons, we employ a sequential Markov Chain Monte Carlo (MCMC) method using the Metropolis-Hasting algorithm to estimate the parameters. This bayesian approach consists in sampling the posterior probability distribution of the parameters and allows to identify eventual correlations between the parameter values. However, updating this sampling is computationally expensive since the solution must be evaluated repeatedly for each queried parameter. For this reason, we developed a low-rank solver to significantly reduce the time and storage requirements associated with the MCMC procedure. The performance of the resulting method is demonstrated on three different applications.

CSQI internal seminar

Published:

This work has been motivated by data assimilation in blood flow models. In many realistic applications, the knowledge of the model parameters as well as the boundary conditions is not perfect. We focus here on the estimation of the input parameters from observations of the output solution. The resulting estimate is then exploited to perform uncertainty quantification tasks. However, there are two major difficulties when trying to solve parameter estimation problems. First, some parameters may not be identifiable. For instance, if only partial noisy measurements of the solution are available, several different parameter values may be associated with the same observation. Second, if the parameters depend nonlinearly on the solution, the parameter estimation problem results in a nonlinear non-convex optimization problem that can be difficult to solve. For these reasons, we employ a sequential Markov Chain Monte Carlo (MCMC) method using the Metropolis-Hasting algorithm to estimate the parameters. This bayesian approach consists in sampling the posterior probability distribution of the parameters and allows to identify eventual correlations between the parameter values. However, updating this sampling is computationally expensive since the solution must be evaluated repeatedly for each queried parameter. For this reason, we developed a low-rank solver to significantly reduce the time and storage requirements associated with the MCMC procedure. The performance of the resulting method is demonstrated on three different applications.

Simulation of data assimilation under a PDE constraint

Published:

This work has been motivated by data assimilation in blood flow models. In many realistic applications, the knowledge of the model parameters as well as the boundary conditions is not perfect. We focus here on the estimation of the input parameters from observations of the output solution. The resulting estimate is then exploited to perform uncertainty quantification tasks. However, there are two major difficulties when trying to solve parameter estimation problems. First, some parameters may not be identifiable. For instance, if only partial noisy measurements of the solution are available, several different parameter values may be associated with the same observation. Second, if the parameters depend nonlinearly on the solution, the parameter estimation problem results in a nonlinear non-convex optimization problem that can be difficult to solve. For these reasons, we employ a sequential Markov Chain Monte Carlo (MCMC) method using the Metropolis-Hasting algorithm to estimate the parameters. This bayesian approach consists in sampling the posterior probability distribution of the parameters and allows to identify eventual correlations between the parameter values. However, updating this sampling is computationally expensive since the solution must be evaluated repeatedly for each queried parameter. For this reason, we developed a low-rank solver to significantly reduce the time and storage requirements associated with the MCMC procedure. The performance of the resulting method is demonstrated on three different applications.

ECCOMAS 2024

Published:

We adopt a hybrid approach that alternates between a high-fidelity model and a reduced-order model to speedup numerical simulations while maintaining accurate approximations. In particular, we develop an error indicator to determine when the reduced-order model is not sufficiently accurate and the high-fidelity model needs to be solved. Then, we propose an adaptive version of the hybrid approach to update the reduced-order model with the high-fidelity snapshots generated when the reduced-order model was not sufficiently accurate. The performance of the method is evaluated on parametrized, time-dependent, nonlinear problems governed by the 1D Burgers’ equation and 2D compressible Euler equations. The results demonstrate the accuracy and computational efficiency of the adaptive hybrid approach.

UNCECOMP 2025

Published:

In recent years, dynamical low-rank (DLR) methods have been employed in various filtering algorithms (e.g., Kalman-like filters, particle filters) to significantly reduce the computational cost associated with the forecast step. The main idea is to approximate the solution by a low-rank matrix (i.e., a SVD-like decomposition) to speedup computations. Compared to projection-based reduced-order models, the difference is that both the reduced basis and the coefficients (i.e., all the factors of the SVD-like decomposition) evolve over time. Several methods have been proposed in the literature to integrate in time the factors of the SVD-like decomposition. In particular, we focus here on the Basis Update & Galerkin (BUG) integrator, which is based on the explicit Euler method and which has recently been extended to second order using the midpoint rule. In this work, we further extend the BUG integrator to high-order Runge-Kutta methods. Notably, any Runge-Kutta method can be used. The performance of the resulting integrators is assessed on two-dimensional, time-dependent, nonlinear problems.

teaching

Numerical analysis

Graduate course, Institut d'Optique Graduate School (Bordeaux), 2020

The goal of this course is to provide fundamental knowledge to be able to formalize a problem into a numerical one, taking into account the existing abilities and limitations in terms of accuracy, stability and efficiency. The course alternates between theory and practice in MatLab to have a first approache of numerical computing software.

Numerical analysis

Undergraduate course, École Polytechnique Fédérale de Lausanne, 2024

This course presents numerical methods for the solution of mathematical problems such as systems of linear and non-linear equations, functions approximation, integration and differentiation, and differential equations.

Analysis

Undergraduate course, École Polytechnique Fédérale de Lausanne, 2025

The course studies fundamental concepts of analysis and the calculus of functions of several variables.