The main challenges in designing numerical methods for approximating nonlinear cross-diffusion systems is that the diffusion matrix may not be symmetric or positive semidefinite, and that a maximum principle may be not available. In this talk, we present a Local Discontinuous Galerkin method for discretizing nonlinear cross-diffusion systems, which is based on the boundedness-by-entropy...
Nowadays, many engineering problems require computing some quantities of interest, which are usually linear functionals applied to the solution of a partial differential equation. Error estimations of such functionals are called "goal-oriented" error estimations. Such estimations are based on the resolution of an adjoint problem, whose solution is used in the estimator definition, and on the...
Polynomial interpolation is a key aspect in numerical analysis, used in very classical settings as for reconstructing a field from measures, computing integrals by quadratures formulas or selecting basis functions in finite element methods. We will review the roles of the three main characters featured in this action, namely, the representation of the domain by a mesh, the polynomial basis,...
MHD simulations including small viscous and resistive effects are fundamental for simulations related to magnetic fusion.
However, due to the needed long time simulations and the very different wave speeds, implicit or semi-implicit methods are unavoidable.
On the other hand, div B = 0 as well as other symmetries and invariants need to be preserved by the numerical algorithm.
To this aim,...
In its standard presentation, the de Rham complex organises the gradient, curl and divergence operator into a sequence that embeds the well-known calculus relations: the image of one operator (e.g. gradient) is included in the kernel of the following one (e.g. curl). The de Rham theorem states that the gaps between these images and kernels, embedded into the cohomology of the complex, is...
Discretization methods based on differential complexes have many advantageous properties in terms of stability, framework for analysing the discrete formulation, and preservation of important quantities such as the mass, helicity, or the pressure robustness in fluid dynamics.
The premise of this kind of approach appeared early on with elements based on the compatibility between the geometry...
Model order reduction for parameterized partial differential equations is a very active research area that has seen tremendous development in recent years from both theoretical and application perspectives. A particular promising approach is the reduced basis method that relies on the approximation of the solution manifold of a parameterized system by tailored low dimensional approximation...
The aim of this talk is to present novel global space-time methods for the approximation of the time-dependent Schrödinger equation, using Kato theory. The latter can be used in conjunction with low-rank tensor formats (such as Tensor Trains for instance) to derive new variational principles to compute dynamical low-rank approximations of the solution, which are different from the...
The numerical solution of PDEs often ends up with a large system of ODEs, and a canonical choice for the solution of such systems of “method of lines” is the class of Runge-Kutta (RK) methods. Indeed, RK methods are used routinely for integration of large systems of ODEs encountered in various applications. But the standard stability arguments of RK method fail to cover arbitrarily large...
This talk will review structural properties of the equations used to model porous flows involving multiple components undergoing phase transitions. These equations only model the gross properties of these problems since a precise description of the physical system is neither available nor computationally tractable. The saddle point structure resulting from the interaction between dissipation...
The numerical simulation of systems involving fluid-structure-contact interaction raises many modeling, mathematical and numerical difficulties. It is in particular crucial for numerous biomedical applications such as the simulation of cardiac valve dynamics (native or artificial) for instance. Fluid-structure interaction without contact is already challenging due to the moving geometries and...
Electronic structure calculations are widely used to predict the physical properties of molecules and materials. They require to solve nonlinear partial differential and eigenvalue equations. These equations are generally numerically very demanding, especially since they are parameterized by the positions of the nuclei in the molecule and must be solved a large number of times when these...
We propose a new approach to discretize the von Neumann equation, which is efficient in the semi-classical limit. This method is first based on the so called Weyl’s variables to address the stiffness associated with the equation. Then, by applying a truncated Hermite expansion of the density operator, we successfully handle this stiffness. Additionally, we develop a finite volume approximation...
First, we'll introduce the neural methods used to solve PDEs, such as PINNs or the Deep Ritz method. It will be shown that these approaches can fit within the framework of classical Galerkin methods, where only the approximation space changes. The advantages and shortcomings of these approaches will be discussed. Next, a "prediction-correction" approach will be proposed, in which neural...
The radiative transfer equation is a kinetic PDE modelling the specific radiation intensity carried by a population of photons described by a statistical description, i.e. a transport equation on the fraction of photons travelling in a given direction. It is well known that as the Knudsen number (which is the ratio of the mean free path length to a representative physical length scale) goes to...
We present a divergence-free semi-implicit finite volume scheme for the simulation of the ideal magnetohydrodynamics (MHD) equations which is stable for large time steps controlled by the local transport speed at all Mach and Alfvén numbers. An operator splitting technique allows to treat the convective terms explicitly while the hydrodynamic pressure and the magnetic field contributions are...
Numerical methods for the simulation of transient systems with structure-preserving properties are known to exhibit greater accuracy and physical reliability, in particular over long durations. These schemes are often built on powerful geometric ideas for broad classes of problems, such as Hamiltonian or reversible systems. However, there remain difficulties in devising higher-order- in-time...
This talk discusses a stability result for the Monge-Ampère operator in a (potentially regularized) Hamilton-Jacobi-Bellman format as a consequence of Alexandrov's classical maximum principle. The main application is guaranteed a posteriori error control in the $L^\infty$ norm for the difference of the Monge-Ampère solution and the convex hull of a fairly arbitrary $C^1$-conforming finite...
Joint work with Yohance A. P. Osborne
Mean field games (MFG) are models for differential games involving large numbers of players, where each player is solving a dynamic optimal control problem that may depend on the overall distribution of players across the state space of the game. In a standard formulation, the Nash equilibria of the game are characterized by the solutions of a coupled...
