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6/10/25, 9:00 AM
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6/10/25, 9:25 AM
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6/10/25, 9:30 AM
Machine Learning, Neural Networks and Artificial Intelligence are words that one cannot escape from these times. What are some sound mathematical basis for this activity in view of applications to SciML (Scientific Machine Learning) ? This will be the general topic of the course.
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- course 1: The compositional structure of NN functions will be analysed within a convenient functional framework... -
6/10/25, 10:30 AM
Evolutionary partial differential equations (PDEs) sometimes have multi-symplectic structures.
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For these PDEs, multi-symplectic integrators, numerical methods inheriting the structures, have been widely studied.
In this direction, McLachlan and Wilkins proposed multi-symplectic diamond schemes which use a special mesh called a diamond mesh to fully leverage the local nature of PDEs.
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6/10/25, 11:15 AM
In this talk, we present a d-dimensional extension of a fictitious domain penalization technique we previously proposed for Neumann or Robin boundary conditions. We apply Droniou’s approach for non-coercive linear elliptic problems to prove existence and uniqueness of the penalized problem’s solution. To establish convergence, we develop a boundary layer approach inspired by the Dirichlet...
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6/10/25, 12:00 PM
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6/10/25, 2:00 PM
In this talk, I will present recent results on energy cascades and structure formation in Hamiltonian systems. I will introduce two families of solvable systems that explicitly illustrate the dynamical development of energy cascades and the emergence of large- and small-scale structures. Some solutions represent condensate formation through highly coherent dynamics, while all cascade solutions...
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6/10/25, 2:45 PM
We propose a numerical method for the Vlasov-Poisson system which is asymptotically consistent and stable in the quasi-neutral limit, that is, when the Debye length is small compared to the scale of the domain. The Vlasov-Poisson system is written as an hyperbolic system thanks to a spectral decomposition in the basis of Hermite functions with respect to the velocity variable, then a structure...
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6/10/25, 3:30 PM
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6/10/25, 4:00 PM
In this talk we will be concerned with the Nonlinear Schrödinger Equations with Time-Dependent Potential. We will propose a new numerical method based on splitting in time and spectral method in space. Second order splitting will be obtained via linearisation combined with iterated Duhamel formula. In spectral discretization we will employ the basis of Hermite functions, for which we will...
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6/10/25, 4:45 PM
A numerical study of solutions to fractional nonlinear Schrödinger (fNLS) equations is presented. We discuss efficient numerical algorithms to compute fractional derivatives. For the focusing fNLS equation, solitons are constructed numerically and their stability is explored. The possibility of a blow-up of solutions to fNLS for smooth initial data is discussed.
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6/10/25, 5:30 PM
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6/11/25, 9:00 AM
The aim of this mini-course is to give an overview of various methods to compute dynamical low-complexity approximations of the solution of high-dimensional evolution Partial Differential Equations (PDEs) with a specific focus on the Schro ̈dinger equation. In this particular case, the solution of the PDE of interest is in general defined on a very high-dimensional space in the case when the...
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6/11/25, 10:00 AM
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6/11/25, 10:30 AM
In this talk, we consider an energy-preserving finite difference
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scheme for the KdV equation (Furihata, 1996). The scheme preserves
the cubic energy function, and has been empirically known that
it works better than (L2)norm-preserving schemes. However,
since the cubic energy function by itself is not useful
in the mathematical analysis of numerical schemes,
the convergence estimate of... -
6/11/25, 11:15 AM
Dynamical low-rank algorithms have developed into an efficient way to solve high-dimensional problems ranging from plasma physics to quantum mechanics. Those methods are attractive because they reduce a high-dimensional problem into a set of lower-dimensional equations and can thus overcome the curse of dimensionality. This, for example, enables 6D Vlasov simulation on a desktop computer that...
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6/11/25, 12:00 PM
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6/11/25, 2:00 PM
The numerical solution of implicit and stiff differential equations by implicit numerical integrators has been largely investigated and there exist many excellent efficient codes available in the scientific community, as Radau5 (based on a Runge-Kutta collocation method at Radau points) and Dassl, based on backward differentiation formulas, among the others. When solving fractional ordinary...
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6/11/25, 2:45 PM
We present an explicit particle method for the Vlasov-Fokker-Planck equation that conserves energy at the fully discrete level. The method features two key components: a conservative particle discretization for the nonlinear Fokker-Planck operator (also known as the Lenard-Bernstein or Dougherty operator), and an explicit time integrator that ensures energy conservation through an...
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6/11/25, 3:30 PM
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6/11/25, 4:00 PM
We investigate the long-time behavior of a resonance-based low-regularity integrator for the cubic nonlinear Schrödinger equation (NLS). Specifically, we analyze the cubic NLS with a weak nonlinearity characterized by a dimensionless parameter ε ∈ (0, 1]. Through rescaling, this equation is equivalent to the NLS with small initial data. We provide rigorous error estimates for rough initial...
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6/12/25, 9:00 AM
The aim of this mini-course is to give an overview of various methods to compute dynamical low-complexity approximations of the solution of high-dimensional evolution Partial Differential Equations (PDEs) with a specific focus on the Schro ̈dinger equation. In this particular case, the solution of the PDE of interest is in general defined on a very high-dimensional space in the case when the...
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6/12/25, 10:00 AM
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6/12/25, 10:30 AM
In inverse problems and data assimilation, various sources of uncertainty arise. Among them, discretization errors in evolution equations can be significant and should sometimes be treated as a major source of uncertainty to be quantified.
In this talk, we present a posteriori and statistical approaches to estimating such errors. The key idea is to model the discretization error as a random...
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6/12/25, 11:15 AM
If a partial differential operator commutes with a symmetry group (permutations, rotations, reflections, etc.) then it can be decoupled by discretising with a so-called symmetry-adapted basis built from irreducible representations, the basic building blocks of representation theory. In this talk we explore this phenomena using symmetry-adapted multivariate orthogonal polynomials to discretise...
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6/12/25, 12:00 PM
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6/12/25, 2:00 PM
In this talk, I will present a new class of numerical methods for the time integration of evolution equations. The systematic design of these methods mixes the Runge-Kutta collocation formalism with collocation techniques in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to...
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6/12/25, 2:45 PM
Abstract: We consider the issue of strict, fully discrete local energy conservation for a whole class of fully implicit local-charge- and global-energy-conserving particle-in-cell (PIC) algorithms. Earlier studies [1-3] demonstrated these algorithms feature strict global energy conservation. However, whether a local energy conservation theorem exists (in which the local energy update is...
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6/12/25, 3:30 PM
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6/12/25, 4:00 PM
Many thermodynamic mixture and biological multicomponent models can be described by cross-diffusion systems. Although the diffusion matrices are generally neither symmetric and nor positive definite, the systems often possess an entropy (or free energy) structure. We aim to "translate" this entropy structure to fully discrete finite-volume discretizations. The main difficulty is to adapt the...
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6/13/25, 9:00 AM
Machine Learning, Neural Networks and Artificial Intelligence are words that one cannot escape from these times. What are some sound mathematical basis for this activity in view of applications to SciML (Scientific Machine Learning) ? This will be the general topic of the course.
Go to contribution page
- course 1: The compositional structure of NN functions will be analysed within a convenient functional framework... -
6/13/25, 10:00 AM
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6/13/25, 10:30 AM
Abstract:
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There is ample numerical evidente that splitting methods, when applied to the time integration of the (semi-discretized) Schrödinger equation, exhibit numerical resonances at specific values $h_r$ of the time step-size: for these values $h_r$ the errors in the solution and in the energy show a peak. E. Faou has analyzed in detail this phenomenon using backward error analysis... -
6/13/25, 11:15 AM
In this talk, I construct and analyze the decay to equilibrium of a finite volume scheme for a 1D nonlinear kinetic relaxation model describing a recombination-generation reaction of two species, proposed in [Neumann, Schmeiser, KRM 2016]. The study is based on the adaptation of the L² hypocoercivity method of [Dolbeault, Mouhot, Schmeiser, Trans. Amer. Math. Soc. 2015] for the discretization...
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6/13/25, 12:00 PM
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