Mean-field game (MFG) systems of partial differential equations (PDE) arise in modeling huge populations of indistinguishable rational agents that play non-cooperative differential games. Mathematically, an MFG system comprises of a Hamilton-Jacobi-Bellman PDE coupled with a Kolmogorov-Fokker-Planck PDE in a highly nonlinear fashion. Hence, theoretical and numerical treatments of MFG systems are highly challenging problems.

Many central problems in geometry, mathematical physics and biology reduce to questions regarding the behavior of solutions of nonlinear evolution equations. The global dynamical behavior of bounded solutions for large times is of significant interest. However, in many real situations, solutions develop singularities in finite time. The singularities have to be analyzed in detail before attempting to extend solutions beyond their singularities or to understand their geometry in conjunction with globally bounded solutions. In this context we have been particularly interested in qualitative descriptions of blowup.

We will present some new methods for source and parameters estimation for partial and fractional differential equations and illustrate the results with some simulations and real applications.

Mean-field game (MFG) systems of partial differential equations (PDE) arise in modeling huge populations of indistinguishable rational agents that play non-cooperative differential games. Mathematically, an MFG system comprises of a Hamilton-Jacobi-Bellman PDE coupled with a Kolmogorov-Fokker-Planck PDE in a highly nonlinear fashion. Hence, theoretical and numerical treatments of MFG systems are highly challenging problems. Day 1: I will show how to transform suitable mean-field game (MFG) systems into infinite-dimensional convex optimization problems. Furthermore, I will present Uzawa’s algorithm and augmented Lagrangian approach for solving convex optimization problems. Finally, I will demonstrate how to apply these methods to approximate solutions of corresponding MFG systems.

Runge-Kutta methods are classical and widespread techniques in the numerical solution of ordinary differential equations (ODEs). Considering partial differential equations, spatial semidiscretisations can be used to obtain systems of ODEs that are solved subsequently, resulting in fully discrete schemes. However, certain stability investigations of high-order methods for hyperbolic conservation laws are often conducted only for the semidiscrete versions.

Space-time conservation element and solution element (CESE) method is a unique finite-volume-type method for computational fluid dynamics (CFD). This approach has several attractive properties, including: (i) unified treatment of the space and time such that only one step is required to construct high-order schemes; (ii) a highly compact stencil regardless of the order of the accuracy; (iii) easiness of extension to any arbitrary shape of polygonal elements. Since its inception, the CESE method has achieved great success in different areas.

The idea behind this presentation is to give to the potential contributors/authors a general view about the Physical Review Family of Journals, and more specifically about Physical Review Applied journal.

In this course we consider multi-phase flows, i.e., flows of one substance which is present as liquid as well as vapor. We focus on models that resolve individual bubbles/droplets and that treat both phases as compressible. We will also discuss incompressible/low Mach limits, since in most applications the liquid is nearly incompressible. Understanding and simulating such small-scale models is important in order to obtain information which can be used in larger scale models for e.g. sprays which play important roles in processes of practical interest as diverse as combustion, chemical engineering, and cloud formation

I will provide an overview on recent researches on acoustic and elastic metamaterials and metasurfaces for controllable wave manipulation, we are developing in my group in the University of Lorraine. I first will present some advances related to low-frequency acoustic and vibration shielding/absorption making use of metamaterials1,2 and metasurfaces3,4, and describe the added value that such artificial engineered materials can bring to consider some innovative applications.

In this course we consider multi-phase flows, i.e., flows of one substance which is present as liquid as well as vapor. We focus on models that resolve individual bubbles/droplets and that treat both phases as compressible. We will also discuss incompressible/low Mach limits, since in most applications the liquid is nearly incompressible. Understanding and simulating such small-scale models is important in order to obtain information which can be used in larger scale models for e.g. sprays which play important roles in processes of practical interest as diverse as combustion, chemical engineering, and cloud formation

The general idea of discrete differential geometry is to find and investigate discrete models that exhibit properties and structures characterisitic of the corresponding smooth geometric objects. This is a challenging problem, since equivalent points of view in the smooth setting may lead to a number of inequivalent treatments in the discrete setting. We will illustrate the paradigm of structure-preserving discretizations on the example of conformal maps by showing how simple definitions lead to surprisingly rich theories.

We analyze the photonic topological phases in dispersive metamaterials which satisfy the degenerate condition at a reference frequency, where the hybrid modes are decoupled and determined by two subsystems with degenerate eigenvalues. By introducing the pseudospin states as the eigenfield basis, the Hamiltonians of the hybrid modes represent the pesudospin-orbit interaction with spin 1, which result in nonzero spin Chern numbers that characterize the topological phases.

With the end of the Moore’s law, it became imperative to find new principles for computing that can avoid the current physical limitations. Among the promising approaches is Quantum Computing which has resulted in substantial national investments in research and development in this area by many nations. This first tutorial is designed to target scientists, engineers and mathematicians who are interested in this rapidly growing field but have not yet invested enough time in learning about it.

In this talk we propose a Flux Corrected Transport (FCT)-like stabilization suitable for high-order Discontinuous Galerkin (DG) finite element discretizations. As a model problem we consider the multi-dimensional transport equation. The method guarantees that the solution satisfies a local Discrete Maximum Principle (DMP). A solution is said to satisfy a DMP locally if any degree of freedom is bounded with respect to the solution at the previous time step in some defined neighborhood.

In this talk, I will start with an overview of my research to date. Then, I will address in more detail the study of the asymptotic behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. More precisely, we consider materials arranged into periodically alternated thin horizontal strips of an elastically rigid component and a softer one with one active slip system. The energies arising from our modeling assumptions are of integral form, featuring linear growth and non-standard differential constraints.

The Cahn-Hilliard equation is a fourth order parabolic partial differential equation (PDE) that is widely used as a phenomenological model to describe the evolution of interfaces in many practical problems, such as, the microstructure formation in materials, fluid flow, etc. It has been observed in the engineering literature that the stochastic version of the Cahn-Hilliard equation provides a better description of the experimentally observed evolution of complex microstructure.