Some hope for the Buridan’s donkey… and other stories

• Marc PEIGNE (Laboratoire de Mathématiques et Physique Théorique, Faculté des Sciences et Techniques Université François Rabelais, France.)

We describe the asymptotic behavior of the Buridan’s wandering using technics developed to study iterated Lipschitz functions systems with possibly place dependent probabilities. Under some general conditions on this family of probabilities and using quasi-compact linear operators technics, we obtain a necessary and sufficient condition for the uniqueness of the stationary probability measure for this chain and explore the case when it does not hold

On relaxation Methods for Mathematical Programs with Complementarity Constraints

• Mounir HADDOU (Centre de mathématiques Institut National des Sciences Appliquées de Rennes, France.)

We propose a new family of relaxation schemes for mathematical programs with complementarity constraints that extends the relaxations converging to an M-stationary point. We discuss the properties of the sequence of relaxed non-linear programs as well as stationarity properties of limiting points. We prove under a new and weak constraint qualification, that our relaxation schemes have the desired property of converging to an M-stationary point. Unfortunately, in practice, relaxed problems are only solved up to approximate stationary points and the guarantee of convergence to an M-stationary point is lost. We define a new strong approximate stationarity condition and prove that we can maintain our guarantee of convergence and attain the desired goal of computing an M-stationary point. A comprehensive numerical comparison between existing relaxations methods is performed and shows promising results for our new methods. We also propose di↵erent extensions to tackle MPVC ( vanishing constraints) and MOCC (cardinality constraints) problems.

Numerical solution of a class of delay differential-algebraic equations by half-explicit methods

• Hoang Linh VU (Faculty of Mathematics - Mechanics and Informatics, Vietnam National University, Hanoi, Vietnam.)

Delay differential-algebraic equations (DAEs) can be used for modeling real-life phenomena that involve simultaneously time-delay effect and constraints. It is also known that solving delay DAEs is more complicated than solving non-delay ones because interpolation errors for the solution in the past time may arise in addition to discretisation errors. Recently, we have investigated the efficient use of half-explicit methods for strangeness-free DAEs (without delay). In this talk, we propose and analyse some efficient half-explicit methods for a class of structured strangeness-free DAEs with constant delay. Convergence results for half-explicit linear multistep (HELM) methods and half-explicit Runge-Kutta (HERK) methods are obtained. Numerical experiments are also given for illustration.

Reflections on waves and waves reflections

• Fabrice PLANCHON

We will review recent results on the propagation of waves on domains, with emphasis on two extreme cases: the exterior or the interior of a stricly convex domain. Understanding the wave localization, its amplitude and how it decays is fundamental for several (unrelated) problems and we will describe an interesting interplay between geometrical aspects and degenerate oscillatory integrals that model waves

Optimized wavelet-Gaussian mixed bases for electronic structure calculations in quantum chemistry

• Quang Huy TRAN (IFPEN)

Many applications of quantum chemistry involve ab initio simulations. These are feasible thanks to well-known approximations to the Schrödinger equation, such as Hartree-Fock’s or Density Functional Theory. More than 70 softwares are available to chemists in this field, the most common ones being VASP, Gaussian and ABINIT. A key difference between them lies in the basis functions selected to express the molecular orbitals. One of the newcomers, the massively parallel program BigDFT, uses wavelet bases for performance considerations. To better capture the cusp singularities of the orbitals in the all-electron calculations without increasing the complexity of BigDFT, we suggest enriching the wavelet basis by Gaussian functions centered at each nucleus position. To optimize the construction of additional Gaussian functions, we rely on a combination of a posteriori error estimates and the greedy algorithm. We adapt the ideas from Maday and his co-authors to establish that the dual norm of the residue can serve as an effective estimate for the energy decrease between the pure-wavelet solution and the augmented-basis solution. Furthermore, in a similar spirit with reduced-basis techniques, we recommend the greedy algorithm for building an incremental sequence of additional Gaussian functions. As a proof of concept to this strategy, we investigate a one-dimensional model of Schrödinger type with delta potentials, which represents a system of one electron and several nuclei of known charges and positions. Due to the small number of additional degrees of freedom, wavelet-Gaussian mixed bases exhibit a significant gain in accuracy while having a low computational cost. This testifies to the interest of this approach.

Optimized Schwarz Waveform Relaxation and Applications to Semilinear Equations

• Laurence Halpern

Likelihood in the Symbolic Context

• Richard EMILION (Mathématiques – Analysie, Probabilités, Modélisation, Université d’Orléans, France.)

Our context is the statistic of a random distribution, that is a random variable taking values in a space of probability measures. Such a context is useful when dealing with classes of raw data of a large dataset or with objects having a complex behaviour. We present the theoretical framework, some examples of models, the likelihood for finite dimensional estimators and some recent applications.

Domain decomposition-based nonlinear preconditioning for elliptic PDEs

• Felix KWOK (Department of Mathematics, Hong Kong Baptist University, Hong Kong felix_kwok@hkbu.edu.hk)

One way to accelerate the numerical solution of a nonlinear elliptic problem is to use nonlinear preconditioning, which replaces the original discretized problem by an equivalent but easier one. In this talk, we discuss how a fixed point iteration for a nonlinear system can be used in combination of Newton's method to yield highly efficient, nonlinearly preconditioned methods. Our starting point is the Restricted Additive Schwarz Preconditioned Exact Newton (RASPEN) method by Dolean et al.~(2016), which is derived from the nonlinear Restricted Additive Schwarz method. We then show how to extend this method to include a coarse component, as well as how to incorporate optimized transmission conditions of the Robin type. Finally, we will show some applications demonstrating the effectiveness of our approach.