Water masses of given temperature and salinity are advected without much blending over long periods of time in the ocean. At the numerical level, the discretization of the equation creates a spurious mixing (or numerical diffusion) that artificially mixes the water masses. It can be of the same order of magnitude as the physical mixing. This is especially true in climate simulations where the grid is coarse and the time of integration is long. This project is concerned with two different aspects of the spurious mixing. First, we will study a numerical procedure to quantify precisely in space and time the spurious mixing. The method is both different from usual entropy inequality in maths that are limited to first order schemes, and from global diagnosis based on water mass transformation in oceanography. It should allow us to better understand the geography and effects of spurious mixing. It can be applied to any transported quantity or to the evolution of total energy. Second, we will develop antidiffusive advection schemes for the salinity and temperature in the European ocean general circulation model NEMO. This approach is different from the usual « higher order, finer grids » strategy and has been successfully employed for atmospheric pollution and biphasic flows. Preliminary results obtained by the principal investigator on a new second order antidiffusive scheme show a gain of accuracy in the smooth regions and a correct behavior in 2 space dimensions. This is a major improvement compared to the existing first order antidiffusive schemes, which should allow us to go beyond its use in the vertical direction only. In conclusion, this project proposes a mathematical approach on the question of spurious mixing in ocean general circulation models. The team gathers experts on numerical analysis and ocean modeling. We will collaborate closely together with the aim of having a direct contribution to NEMO.

The problem of estimating distances between probbaility measures arises in many areas of science (physics, computer science, biology,...), to quantify how accurately some phenomenon is approximated by a given stochastic model. Stein's method is a set of techniques for estimating such distances via well-chosen integration by parts formulas. Since it was first introduced by C. Stein in the 70s, it has found many applications, in areas including quantitative central limit theorems, statistical physics and combinatorics. This project aims at developing new techniques and applications, by leveraging ideas from mathematical analysis (PDE techniques, variational methods, optimal transport, functional analysis). Fields of application we shall explore include statistics, random matrix theory and free probability, stochastic processes, high-dimensional geometry, statistical physics and Markov Chain Monte Carlo algorithms.

Ferromagnetic materials are increasingly being used in microelectronics to design reliable, fast and energy-efficient digital data recording devices. The objective of the project is to improve the modelling and simulation of these devices by taking into account their complex geometries (e. g. nanowire networks, curved nanowires) and the multiphysical nature of phenomena: electromagnetic, mechanical (magnetostriction) and thermal effects. Finally, this project will make it possible to optimize these devices to increase the reliability of storage facilities and to design new devices to control magnetization via an electrical current or mechanical action.

The goal of this project is to address new directions of research in control theory for partial differential equations, triggered by models from ecology and biology. In particular, our projet will deal with the development of new methods which will be applicable in many applications, from the treatment of cancer cells to the analysis of the thermic efficiency of buildings, and from control issues for the biological control of pests to cardiovascular fluid flows. To achieve these objectives, we will have to solve several theoretical issues in order to design efficient control methods. We have thus targeted four main challenges. The first axis of study concerns the control of parabolic systems, in particular when the control does not act on all the components of the system. In this case, the main difficulty comes from the fact that the components of the system in which the control does not act have to be controlled indirectly through the coupling with the other components of the system. Our goal is to better understand the control properties of such systems, and to develop new robust techniques, which can be applied in particular in nonlinear settings. The second axis will focus on the analysis of the control properties of partial differential equations containing nonlocal terms. Such nonlocal terms appear in many applications, as soon as the dynamics of the system depends on global quantities, or locally averaged quantities. This is for instance the case in population dynamics, where the total number of birth depends on the whole population. New tools should be developed in this context, where the classical techniques developed so far in control theory do not apply. The third point of study aims at addressing in depth major applications involving fluid mechanics. We will in particular study the modeling of the thermic efficiency of buildings and its related control issues, the modeling of cardiovascular flows and stabilization issues around periodic trajectories in models of interactions of viscous or viscoelastic flow (modeling the blood) and an elastic structure (modeling the blood vessels), and the design of waves energy converter of optimal efficiency. The fourth axis of study concerns the control of partial differential equations when the trajectories are required to satisfy some constraints. The study of such question has started very recently and should be developed further and extended to many models, in particular in order to take into account positivity constraints on controlled trajectories, or on some components of it, which are required for the feasibility of the control strategies. This question appears very naturally when some quantities are intrinsically positive, which is the case for instance when they model a population density or a concentration of a chemical reactant. To sum up, with this project, our goal is to reduce the gap between theory and applicative areas and make significant advances for the use of more accurate models and more efficient control designs in real life applications.

The GeoPor project proposal aims to gather experienced and promising researchers from different fields of mathematics, more precisely optimal transportation for the analysis of partial differential equations (PDEs) and numerical analysis, for studying the equations governing multi-phase flows in porous media, mixing analysis, differential geometry, and scientific computing. Indeed, the project has been thought as a transversal bridge across the mathematical communities, the common factor being the problems to study, that are the equations governing multiphase flows and their wide range of applications (e.g., oil-engineering, CO2 sequestration, nuclear waste repository management, hydrogeology). In the recent years, new and promising results have been obtained separately in the field of optimal transportation, in the design and the study of advanced numerical methods, and in the development of adaptive strategies for solving numerically some PDEs based on a posteriori error analysis. We aim to take advantage of all these new developments, and to apply this new material to the study, from a theoretical and numerical point of view, of nonlinear degenerated parabolic systems and equations. We claim that the gradient flow approach is relevant for studying multiphase flows in porous media. Indeed, the natural and physically consistent way to write the equations governing multiphase flows in porous media let some degenerate mobilities appear leading to the study of degenerate parabolic equations and making the gradient flow writing completely natural. As a major advantage of this writing, only physically meaningful quantities appear, in contrary to the classical mathematical approaches. Keeping the physical relevance as a constant matter, we propose to design what we call nonlinear methods for approximating the solutions of such (possibly) degenerate parabolic equations and systems. The methods we have in mind are based on schemes that are widely used in oil-engineering despite very partially studied: they are robust and allow enrichments of the physics. Taking advantage of the numerous new tools developed in the recent years we will design robust advanced numerical methods whose convergence will be proven. Bearing in mind that we are concerned with real world applications, we will extend the recent works on a posteriori error analysis to the new methods we will design. We expect to derive fully computable, guaranteed error bounds, so that the maximal error in the calculation can be certified. Moreover, relying on mesh and time stepping adaptivity, but also on adaptive stopping criteria for linear and nonlinear solvers, we hope to propose algorithms that will save an important part of the computational time, so that the methods we will propose will be competitive, especially in the industrial context. Last but not least, this project will create new interactions between different communities as well as with international researchers. These interactions will be consolidated thanks to the organization of several workshops and one international conference, and by the implication in the formation of young researchers in France and abroad.