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Our work aims to quantify the benefit of storage flexibilities such as a battery on several short term electricity markets. We especially focus on two different markets, the intraday market (ID) and the activation market of the automatic Frequency Restoration Reserve (aFRR), also known as the secondary reserve. We propose algorithms to optimize the management of a small battery (<= 5 MWh) on these markets. In this talk, we first present the modeling of the problem, then we show some theoretical results and numerical simulations. We conclude by listing some limitations of the method.

(joint work with Aurélien Alfonsi, Rafaël Coyaud, Damiano Lombardi and Luca Nenna)

Optimal Transport (OT) problems arise in a wide range of applications, from physics to economics. Getting numerical approximate solution of these problems is a challenging issue of practical importance. In this work, we investigate the relaxation of the OT problem when the marginal constraints are replaced by some moment constraints. Using Tchakaloff's theorem, we show that the Moment Constrained Optimal Transport problem (MCOT) is achieved by a finite discrete measure. Interestingly, for multimarginal OT problems, the number of points weighted by this measure scales linearly with the number of marginal laws, which is encouraging to bypass the curse of dimension. Moreover, the same type of sparsity results also hold for quantum optimal transport problems stemming from electronic structure calculations. These theoretical results guided the design of new numerical schemes for the resolution of these problems which yielded promising numerical results in high-dimensional contexts. The end of the talk will be devoted to the remaining open problems related to the mathematical analysis of these schemes.

The existence of Kaehler-Einstein metrics on Fano 3-folds can be determined by studying lower bounds of stability thresholds. An effective way to verify such bounds is to construct flags of point-curve-surface inside the Fano 3-folds. This approach was initiated by Abban-Zhuang, and allows us to restrict the computation of bounds for stability thresholds only on flags. We employ this machinery to prove K-stability of terminal quasi-smooth Fano 3-fold hypersurfaces. This is deeply intertwined with the geometry of the hypersurfaces: in fact, birational rigidity and superrigidity play a crucial role. The superrigid case had been attacked by Kim-Okada-Won. In this talk, I will discuss the K-stability of strictly rigid Fano hypersurfaces via Abban-Zhuang Theory. This is a joint work with Takuzo Okada.

The Benjamin-Ono (BO) equation is a nonlocal asymptotic model for the unidirectional propagation of weakly nonlinear, long internal waves in a two-layer fluid. The equation was introduced formally by Benjamin in the '60s and has been a source of active research since. For instance, the study of the long-time behavior of solutions, stability of traveling waves, and the low regularity well-posedness of the initial value problem. However, despite the rich theory for the BO equation, it is still an open question whether its solutions are close to the ones of the original physical system.

In this talk, I will explain the main steps involved in the rigorous derivation of the BO equation.

Networks of hyperbolic PDEs arise in different applications, e.g. modeling water- or gas-networks or road traffic. In the first part of this talk we discuss modeling aspects of coupling conditions for hyperbolic PDEs.

Starting from an kinetic description we derive coupling conditions for the associated macroscopic equations. For this process a detailed description of the boundary layer is important. In the second part appropriate numerical methods are considered.

Different high order approaches are compared and applications to district heating or water networks are discussed.

It is known that the partition function $p(n)$ obeys Benford's law in any integer base $b\ge 2$. In a recent paper, Douglass and Ono asked for an explicit version of this result. In my talk, I will show that for any string of digits of length $f$ in base $b$, there is $n\le N(b,f)$, where 

$$N(b,f):=\exp\left(10^{32} (f+11)^2(\log b)^3\right)$$

such that $p(n)$ starts with the given string of digits in base $b$. The proof uses a lower bound for a nonzero linear form in logarithms of algebraic numbers with algebraic coefficients due to  Philippon and Waldschmidt. A similar result holds for the plane partition function. 

Présentation des membres de l'équipe

Freiman's $3k-4$ Theorem states that if a subset $A$ of $k$ integers has a Minkowski sum $A+A$ of size at most $3k-4$, then it must be contained in a short arithmetic progression. We prove a function field analogue that is also a generalisation: it states that if $K$ is a perfect field and if $S\supset K$ is a vector space of dimension $k$ inside an extension $F/K$ in which $K$ is algebraically closed, and if the $K$-vector space generated by all products of pairs of elements of $S$ has dimension at most $3k-4$, then $K(S)$ is a function field of small genus, and $S$ is of small codimension inside a Riemann-Roch space of $K(S)$. Joint work with Alain Couvreur. 

Je discuterai un travail récent avec Yann Chaubet et Daniel Han-Kwan (Nantes). Nous nous sommes intéressés à la dynamique en temps long de l'équation de Vlasov non-linéaire sur une variété à courbure négative lorsque le noyau d'interaction est lisse. J'expliquerai que, pour des petites données initiales lisses et supportées loin de la section nulle, les solutions de cette équation convergent à vitesse exponentielle vers un état d'équilibre du problème linéaire. Pour obtenir un tel résultat, on fait appel à des outils d'analyse microlocale développés initialement dans le contexte de l'étude des systèmes dynamiques chaotiques (Baladi, Dyatlov, Faure, Sjöstrand, Tsujii, Zworski).

We study non-conservative hyperbolic systems of balance laws and are interested in development of well-balanced (WB) numerical methods for such systems. One of the ways to enforce the balance between the flux terms and source and non-conservative product terms is to rewrite the studied system in a quasi-conservative form by incorporating the latter terms into the modified global flux. The resulting system can be quite easily solved by Riemann-problem-solver-free central-upwind (CU) schemes. This approach, however, does not allow to accurately treat non-conservative products. We therefore apply a path-conservative (PC) integration technique and develop a very robust and accurate path-conservative central-upwind schemes (PCCU) based on flux globalization. I will demonstrate the performance of the WB PCCU schemes on a wide variety of examples.

Séminaire IOP banalisé

https://www.math.u-bordeaux.fr/~skupin/conf-pthomas-2024.html

Une variété est dite PSC si elle admet une métrique riemannienne complète à courbure scalaire positive. Vers la fin des années 1970, des résultats de Schoen et Yau reposant sur la théorie des surfaces minimales et, en parallèle, des méthodes basées sur la théorie de l’indice développées par Gromov et Lawson, ont permis de classifier les 3-variétés fermées PSC : ce sont exactement celles qui se décomposent en sommes connexes de variétés sphériques et de produits S2xS1. Dans cet exposé, nous présenterons un résultat de décomposition des 3-variétés PSC non compactes : si sa courbure scalaire décroît assez lentement, alors la variété se décompose en somme connexe (possiblement infinie) de variétés sphériques et S2xS1. Ce résultat fait suite à des travaux récents de Gromov et de Wang.

Il s'agit d'un travail en collaboration avec F. Balacheff et S. Sabourau.

During this talk I will present a work in progress, joint with Félix Baril-Boudreau and Alexandre Benoist on the conjecture by Lang and Trotter that generalizes to elliptic curves Artin's conjecture on primitive roots.

Seminaire joint avec Optimal

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In the Kidney Exchange Problem (KEP), we consider a pool of altruistic donors and incompatible patient-donor pairs. 

Kidney exchanges can be modelled in a directed weighted graph as circuits starting and ending in an incompatible pair or as paths starting at an altruistic donor.

The weights on the arcs represent the medical benefit which measures the quality of the associated transplantation.

For medical reasons, circuits and paths are of limited length and are associated with a medical benefit to perform the transplants.

The aim of the KEP is to determine a set of disjoint kidney exchanges of maximal medical benefit or maximal cardinality (all weights equal to one).

In this work, we consider two types of uncertainty in the KEP which stem from the estimation of the medical benefit (weights of the arcs) and from the failure of a transplantation (existence of the arcs).

Both uncertainty are modelled via uncertainty sets with constant budget.

The robust approach entails finding the best KEP solution in the worst-case scenario within the uncertainty set.

We modelled the robust counter-part by means of a max-min formulation which is defined on exponentially-many variables associated with the circuits and paths.

We propose different exact approaches to solve it: either based on the result of Bertsimas and Sim or on a reformulation to a single-level problem.

In both cases, the core algorithm is based on a Branch-Price-and-Cut approach where the exponentially-many variables are dynamically generated.

The computational experiments prove the efficiency of our approach.  

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À préciser.

Invitée de Pascal Vallet.

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