Sébastien Boyaval

EXISTENCE AND APPROXIMATION OF A (REGULARIZED) OLDROYD-B MODEL

We consider the finite element approximation of the Oldroyd-B system of equations, which models a dilute polymeric fluid, in a bounded domain

A fast Monte–Carlo method with a reduced basis of control variates applied to uncertainty propagation and Bayesian estimation

\mathcal{D} \subset \mathbb{R}^{d}

On the modeling and simulation of non-hydrostatic dam break flows

, d = 2 or 3, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conformation tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics or a reduced version, where the tangential component on each simplicial edge (d = 2) or face (d = 3) is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-element. We show that both of these types of schemes satisfy a free energy bound, which involves the logarithm of the conformation tensor, without any constraint on the time step for the backward Euler-type time discretization. This extends the results of Boyaval et al. (Free-energy-dissipative schemes for the Oldroyd-B model, ESAIM: Mat...

Lid-driven-cavity simulations of Oldroyd-B models using free-energy-dissipative schemes

Abstract The reduced-basis control-variate Monte-Carlo method was introduced recently in [S. Boyaval, T. Lelievre, A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm, Commun. Math. Sci. 8 (2010) 735–762 (Special issue “Mathematical Issues on Complex Fluids”)] as an improved Monte-Carlo method, for the fast estimation of many parametrized expected values at many parameter values. We provide here a more complete analysis of the method including precise error estimates and convergence results. We also numerically demonstrate that it can be useful to some parametric frameworks in Uncertainty Quantification, in particular (i) the case where the parametrized expectation is a scalar output of the solution to a Partial Differential Equation (PDE) with stochastic coefficients (an Uncertainty Propagation problem), and (ii) the case where the parametrized expectation is the Bayesian estimator of a scalar output in a similar PDE context. Moreover, in each case, a PDE has to be solved many times for many values of its coefficients. This is costly and we also use a reduced basis of PDE solutions like in [S. Boyaval, C. Le Bris, Y. Maday, N. Nguyen, A. Patera, A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable robin coefficient, Comput. Methods Appl. Mech. Eng. 198 (2009) 3187–3206]. To our knowledge, this is the first combination of various reduced-basis ideas, with a view to reducing as much as possible the computational cost of a simple versatile Monte-Carlo approach to Uncertainty Quantification.

Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation

The numerical simulation of three-dimensional dam break flows is discussed. A non-hydrostatic numerical model for free-surface flows is considered, which is based on the incompressible Navier–Stokes equations coupled with a volume-of-fluid approach. The numerical results obtained for a variety of benchmark problems show the validity of the numerical approach, in comparison with other numerical models, and allow to investigate numerically the non-hydrostatic three-dimensional effects, in particular for the usual test cases where hydrostatic approximations are known analytically. The numerical experiments on actual topographies, in particular the Malpasset dam break and the (hypothetical) break of the Grande-Dixence dam in Switzerland, also illustrate the capabilities of the method for large-scale simulations and real-life visualization.

Polynomial Surrogates for Open-Channel Flows in Random Steady State

In this work, we report on numerical tests in keeping with the study [Boyaval, Lelievre, and Mangoubi, Free-energy-dissipative schemes for the Oldroyd-B model, ESAIM: Mathematical Modelling and Numerical Analysis (M2AN), 43(3): 523–561, 2009], about Finite-Element discretizations of the Oldroyd-B system (for viscoelastic flows of some non-Newtonian fluids) which are stable in the sense of free-energy dissipation.

A Reduced-Basis Approach to Two-Phase Flow in Porous Media

We propose a certified reduced basis approach for the strong- and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the given observational data to estimate only the unknown initial condition of the model, the weak-constraint 4D-Var formulation additionally provides an estimate for the model error and thus can deal with imperfect models. Since the model error is a distributed function in both space and time, the 4D-Var formulation leads to a large-scale optimization problem for every given parameter instance of the PDE model. To solve the problem efficiently, various reduced order approaches have therefore been proposed in the recent past. Here, we employ the reduced basis method to generate reduced order approximations for the state, adjoint, initial condition, and model error. Our main contribution is the development of efficiently computable a posteriori upper bounds for the error of the reduced basis approximation with respect to the underlying high-dimensional 4D-Var problem. Numerical results are conducted to test the validity of our approach.

A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient

Assessing epistemic uncertainties is considered as a milestone for improving numerical predictions of a dynamical system. In hydrodynamics, uncertainties in input parameters translate into uncertainties in simulated water levels through the shallow water equations. We investigate the ability of generalized polynomial chaos (gPC) surrogate to evaluate the probabilistic features of water level simulated by a 1-D hydraulic model (MASCARET) with the same accuracy as a classical Monte Carlo method but at a reduced computational cost. This study highlights that the water level probability density function and covariance matrix are better estimated with the polynomial surrogate model than with a Monte Carlo approach on the forward model given a limited budget of MASCARET evaluations. The gPC-surrogate performance is first assessed on an idealized channel with uniform geometry and then applied on the more realistic case of the Garonne River (France) for which a global sensitivity analysis using sparse least-angle regression was performed to reduce the size of the stochastic problem. For both cases, Galerkin projection approximation coupled to Gaussian quadrature that involves a limited number of forward model evaluations is compared with least-square regression for computing the coefficients when the surrogate is parameterized with respect to the local friction coefficient and the upstream discharge. The results showed that a gPC-surrogate with total polynomial degree equal to 6 requiring 49 forward model evaluations is sufficient to represent the water level distribution (in the sense of the ℓ2

Free-energy-dissipative schemes for the Oldroyd-B model

\ell _2

Hybridization of mixed high-order methods on general meshes and application to the Stokes equations

norm), the probability density function and the water level covariance matrix for further use in the framework of data assimilation. In locations where the flow dynamics is more complex due to bathymetry, a higher polynomial degree is needed to retrieve the water level distribution. The use of a surrogate is thus a promising strategy for uncertainty quantification studies in open-channel flows and should be extended to unsteady flows. It also paves the way toward cost-effective ensemble-based data assimilation for flood forecasting and water resource management.