Abstract

Aiming at optimizing the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points of the Möbius energy. The gradients are computed with respect to certain fractional-order Sobolev scalar products that are adapted to the Möbius energy. In contrast to $L^2$-gradient flows, the resulting flows are ordinary differential equations on an infinite-dimensional manifold of embedded curves. In the fully discrete setting, this allows us to completely decouple the time step size from the spatial discretization, resulting in a very robust optimization algorithm that is orders of magnitude faster than following the discrete $L^2$-gradient flow.

Abstract

Similar to the nonlinear Boltzmann equation of gas dynamics, the linear Boltzmann equation satisfies an H Theorem. This means that there exist entropy functionals of the solution that do not increase in time. Among other things, this implies time-irreversibility of the solution. Contrary to the nonlinear Boltzmann equation, however, a rigorous derivation of the linear Boltzmann equation (more precisely the equation describing a Lorentz gas) is available. This derivation, due to Gallavotti (1972), is interesting for several reasons: (i) It starts from a reversible microscopic system. More precisely, the derivation makes active use of reversibility. (ii) Inspection of the derivation shows where entropy dissipation comes in. (iii) It is a strange derivation in that we derive a solution formula and subsequently identify an equation for that solution formula.

In the lecture, we prove the H Theorem in the linear case and sketch the derivation of the Lorentz gas equation, pointing out where entropy decay emerges.

Abstract

The mechanical modeling of living tissues has attracted much attention in the last decade. Applications include tissue repair and growth models of solid tumors. In this latter case, these models are calibrated on medical images and help to predict the evolution of the disease, to decide of treatment scheduling and of the optimal therapy. They are also used to understand the biological effects that permit tumor growth.

These models contain several levels of complexity, both in terms of the biological and mechanical effects, and therefore in their mathematical description. The number of scales, from molecules to the organ and entire body, explains partly this complexity. Here, the analysis of the incompressible limit, and the associated free boundary problem, is mathematically challenging.

In this talk I shall give a general presentation of the field. Departing from the simplest (and irrealistic) model of cell division moving by pressure forces, I will include several additional biological effects and explain the consequences in terms of qualitative behavior of solutions.

Abstract

Physical systems such as gas, water and electricity networks are usually operated in a state of equilibrium and one is interested in stable systems, where small perturbations are damped over time. We will consider gas flow on a network with feedback boundary conditions. We focus on linearized isothermal Euler equations that are diagonalizable with Riemann invariants and analyze the stability of a steady state. Explicit conditions are presented yielding an exponential Lyapunov stability. We will focus both on a Lyapunov function with respect to the $L^2$- and $H^2$-norm. Furthermore, not only the convergence to a steady state of the analytical solution, but also of the numerical approximation is guaranteed. Numerical results illustrate our analysis.

References

[1] G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-d Hyperbolic Systems, Birkhäuser (2016)
[2] S. Gerster, M. Herty, Discretized Feedback Control for Systems of Linearized Hyperbolic Balance Laws, MCRF, Vol. 9, No. 3 (2019)

Abstract

This talk is concerned with the asymptotic analysis via $\Gamma$-convergence of energy-driven discrete systems when the lattice spacing vanishes. We focus on a class of energy functionals for which the energy densities contain both a bulk and a surface scaling. In the discrete-to-continuum limit such energy functionals typically converge to free-discontinuity functionals, which have natural applications in image segmentation and fracture mechanics.

After presenting some prototypical examples I will discuss a recent result obtained in collaboration with Andrea Braides and Marco Cicalese, where we consider a very general class of discrete energies giving rise to a free-discontinuity functional in the $\Gamma$-limit.

Abstract

The heat equation can be interpreted as the gradient flow of the Boltzmann entropy with respect to the metric of optimal transportation for volume forms, as well known since the work of Otto. We make a similar attempt for other type of differential forms, which leads to vector-valued parabolic equations.

Abstract

We show how several parabolic equations and systems can be derived from hyperbolic systems of conservation laws just by performing a suitable quadratic change of time. The simplest example is the heat equation which follows from the isothermal Euler equations. This idea allows us to transfer well known techniques from the hyperbolic world to the parabolic setting, leading to weak-strong uniqueness results for some degenerate parabolic systems, for instance the Muskat equations for fluids and some mean-curvature flows.

Abstract

Optimal transport (OT) has become a fundamental mathematical tool at the interface between optimization, partial differential equations and probability. It has recently emerged as an important approach to tackle a surprisingly wide range of applications, such as shape registration in medical imaging, structured prediction in supervised learning and the training of deep generative networks. In this talk, I will review an emerging class of numerical approaches for the approximate resolution of OT-based optimization problems. This offers a new perspective to scale OT for high dimensional problems in machine learning. More information and references can be found on the website of our book “Computational Optimal Transport.”

Abstract

Stochastic conservation laws (SCL) with quasilinear multiplicative “rough” path dependence in the flux arise in modeling of mean field games. An impressive collection of theoretical results has been developed for SCL in recent years by Gess, Lions, Perthame, and Souganidis. We present the first fully computable numerical methods for pathwise solutions of scalar SCL with, for instance, “rough” paths in the form of Wiener processes. Convergence rates are derived for the numerical methods and we show that for strictly convex flux functions, “rough” path oscillations lead to cancellations in the solution flow map; a property we take advantage of to develop more efficient numerical methods.

Abstract

A small number of models for transportation networks (modelling street, river, or vessel networks, for instance) has been studied intensely during the past decade, in particular the so-called branched transport and the so-called urban planning. They assign to each network the total cost for transporting material from a given initial to a prescribed final distribution and seek the cost-optimal network. Typically, the considered transportation cost per mass is smaller the more mass is transported together, which leads to highly patterned and ramified optimal networks. I will present novel formulations of these models which allow a better interpretation as an optimal design problem.

Abstract

We are interested in the problem of floating bodies on a large temporal and spatial scale. At this scale, the hydrodynamics is usually solved using reduced model such as the Shallow-Water model. However, the floating body introduces a constraint on the maximum value of the water depth. Models with a constraint on the maximum value also appear for traffic flow or chemotaxis modeling and the literature generally refers to congested models. We propose a strategy based on a relaxation of the congestion constraint leading to an hyperbolic model with stiff pressure source term and low-Mach numerical scheme. Numerical test cases and comparisons with analytical solutions are carried out to illustrate the efficiency of the strategy.

Abstract

Small particles moving in a fluid are encountered in various situations in nature and technology. In many cases, gravitation is the driving force for the dynamics of the particles. If the particles are not too small, the system can be microscopically modeled by the Navier-Stokes equations coupled with a system of ODEs for the particles according to Newton’s laws. Although the force acting on each particle due to the gravity is directly proportional to its mass, and we do not include direct (e.g. electromagnetic) interaction between the particles themselves, the motion of the particles will be quite complex in many situations. The complexity arises from the interaction of the particles through the fluid since the presence of each particle induces a disturbance in the fluid flow which again influences all the other particles.

Assuming that the fluid inertia is negligible, we will present different microscopic models for spherical and non-spherical particles with and without inertia. We will then discuss corresponding macroscopic models which consist of systems that couple a Vlasov equation to Stokes equations. In the case of inertialess spherical particles, the macroscopic system can be rigorously derived from the microscopic dynamics in the limit of many small particles.

Abstract

In this course, I will present a hierarchy of models to describe free surface flow. The main part of the course focuses on the derivation of the hierarchy of models from the incompressible free surface Euler equations, also called water waves equations, which serve as a reference model. The complexity of the reduced models increases, from the well-known Shallow Water model to the layerwise Green-Nahgdi equations. The main known properties of the models as well as the open questions will be presented.

Abstract

In this course, I will present a hierarchy of models to describe free surface flow. The main part of the course focuses on the derivation of the hierarchy of models from the incompressible free surface Euler equations, also called water waves equations, which serve as a reference model. The complexity of the reduced models increases, from the well-known Shallow Water model to the layerwise Green-Nahgdi equations. The main known properties of the models as well as the open questions will be presented.

Abstract

In this course, I will present a hierarchy of models to describe free surface flow. The main part of the course focuses on the derivation of the hierarchy of models from the incompressible free surface Euler equations, also called water waves equations, which serve as a reference model. The complexity of the reduced models increases, from the well-known Shallow Water model to the layerwise Green-Nahgdi equations. The main known properties of the models as well as the open questions will be presented.

Abstract

We propose a general framework to analyze control problems for conservation law models on a network where we regard as controls the boundary data on the incoming edges of a junction and the distributional parameters at the junction. Such controls yield, in general, different solutions from those obtained by the standard well-posedness theory based on the definition of suitable junction-Riemann solvers. Compactness results for classes of junction distribution and inflow controls are established. We then show the existence of optimal solutions, within classes of such controls, for several optimization problems in traffic flow models related to different traffic performance indexes considered in the literature. A variational formulation of such optimization problems is also provided and some characterization of optimal solutions is discussed.

This is a joint work with A. Cesaroni, G. M. Coclite and M. Garavello.

Abstract

We examine a class of conservation laws with flux discontinuous in the conserved quantity that emerges in a model of industrial conveyor belt and in supply chains. We first introduce an appropriate notion of pair of entropic solution-flux, we provide existence of entropic solution-fluxes by front-tracking and we show a Kruzkhov’s type stability of such solutions. Next, we analyze the associated Hamilton-Jacobi equation, we derive an Hopf-Lax type representation formula of the solutions and we show how to recover the pair of entropic solution-flux of the conservation law from the gradient of the solution of the Hamilton-Jacobi equation. Finally, we consider the problem of a junction with a buffer (to store processed products) and with incoming and outgoing belts modeled by the class of conservation laws analyzed beforehand. Existence and uniqueness of the solution to the junction problem is established.

This is a joint work with Prof. Fabio Ancona (University of Padova)

Abstract

We will consider the role of convexity of the energy in determining relaxation rates of the corresponding gradient flow. We will then ask whether similar rates can also hold for nonconvex energies. I will explain a method to obtain optimal algebraic rates of relaxation for a gradient flow via algebraic and differential relationships among distance, energy, and dissipation. I will focus on the example of the 1-d Cahn Hilliard equation.

Abstract

In this joint works with Andrea Marchese and coauthors, we propose an Eulerian formulation for the multicommodity flow problem, i.e., a general branched transportation problem in which m different goods are moved simultaneously. Moreover, in the atomic case, we prove that the problem can be relaxed in a mass minimization problem in a class of rectifiable currents with coefficients in a group, allowing to introduce a notion of calibration. Finally, I will show how these tools allow the modelling of the oriented mailing problem.

Abstract

To model the dynamics of interest rate and energy forward markets, linear hyperbolic stochastic partial differential equations (SPDEs) may be utilized. The forward rate is then given as the solution to a transport equation with a space-time stochastic process as driving noise. In order to capture temporal discontinuities and allow for heavy-tailed distributions, we consider Hilbert space valued-Lévy processes (or Lévy fields) as driving noise terms. The numerical discretization of the corresponding SPDE involves several difficulties: Low spatial and temporal regularity of the solution to the problem entails slow convergence rates and instabilities for space/time-discretization schemes. Furthermore, the Lévy noise process admits values in a possibly infinite-dimensional Hilbert space H, hence projections into a finite-dimensional subspace HN ⊂ H for each discrete point in time are necessary. Finally, unbiased sampling from the resulting HN-valued Lévy field may not be possible. We introduce a fully discrete approximation scheme that addresses the above issues. A discontinuous Galerkin approach for the spatial approximation is coupled with a suitable time stepping scheme to avoid numerical oscillations and increase the temporal order of convergence. Moreover, we approximate the driving noise process by truncated Karhunen-Loève expansions. The latter essentially yields a sum of scaled and uncorrelated one-dimensional Lévy processes, which may be simulated with controlled bias by Fourier inversion techniques. This is joint work with Andrea Barth (SimTech, University of Stuttgart)

Abstract

I will give an overview on two seminal papers of Klainerman and Majda, “Singular limits of quasi-linear hyperbolic systems with large parameters and the incompressible limit of compressible fluids”. Comm. Pure Appl. Math. 34 (1981) and “Compressible and incompressible fluids”. Comm. Pure Appl. Math. 35 (1982).

In the first paper, the authors treat a class of systems of conservation laws which depend on a possibly singular parameter. They establish conditions on the flux functions and the initial data under which a uniform a-priori estimate holds for the solutions. Using this, they prove convergence for several systems as the parameter becomes singular. Note that this includes a change of type in the pde. In the second paper, the authors focus on isentropic gas dynamics, and rigorously prove the existence of an asymptotic expansions. Such expansions are the basis of several recent low-Mach number schemes, which are relevant to our RTG.

Abstract

We will consider the role of convexity of the energy in determining relaxation rates of the corresponding gradient flow. We will then ask whether similar rates can also hold for nonconvex energies. I will explain a method to obtain optimal algebraic rates of relaxation for a gradient flow via algebraic and differential relationships among distance, energy, and dissipation. I will focus on the example of the 1-d Cahn Hilliard equation.

Abstract

The goal of this lecture is to explain what are holomorphic curves and how they can be used to study entropy in energy levels of Hamiltonian systems with few degrees of freedom. The exposition will be elementary, in colloquium style. We will describe some open problems that can be attacked with these techniques.

Abstract

When weakly coupling the classical one-dimensional Allen-Cahn interface model to large scale fields, the dynamics drastically changes. For linear coupling this can be rigorously analysed in a rather explicit way. Combining various methods from spatial dynamics and dynamical systems allows to detect and unfold degenerate Takens-Bodganov points for the interface dynamics. This features various periodic, homoclinic and heteroclinic solutions.

This is joint work with Martina Chirilus-Bruckner, Peter van Heijster and Hideo Ikeda.

Abstract

We introduce the concept of dissipative solutions to the compressible Euler system and discuss its basic properties: Weak-weak uniqueness, stability, maximal energy dissipation - entropy production. Then we show that dissipative solutions form a perfect target object for energy dissipating numerical schemes. Introducing the concept of K-convergence (Komlós convergence) we show that Cesàro averages of numerical solutions approach strongly (a.e. pointwise) to a dissipative solution of the Euler system.

Abstract

Coarsening, usually associated with the perpetuated growth of a system’s geometric length scale, is an ubiquitous phenomenon in nature. In material science it can be observed in the phase separation of alloys, formation of droplets in thin films or the growth of polycristalline structures, while also occurring in stochastic particle systems such as the zero-range process. The typical problem is to characterize generic dynamical properties, such as coarsening rates or self-similarity, for a large class of initial configurations. Heuristic considerations, simulations and experiments often give a rough picture, whereas the rigorous mathematical description remains a challenge. In this talk, I will consider a class of toy models for coarsening on a one dimensional infinite lattice, where sites transfer mass via nonlinear backward diffusion and empty sites vanish from the system. While simple, this system already exhibits interesting dynamical properties. I will discuss the existence of initial configurations for which the system coarsens at the expected rate, but in a very organised way. The idea is to analyse the time-reversed evolution and make use of the parabolic theory of De Giorgi, Nash and Moser in the spatially discrete setting.

Abstract

A celebrated result by Jordan, Kinderlehrer and Otto shows that the heat flow on Euclidean space can be viewed as a gradient flow of the entropy in Wasserstein space. In recent years, analog gradient flow characterizations have been obtained in a series of settings, including the heat flow generated by discrete Laplace operators and the time evolution of finite-dimensional open quantum systems. The distinctive common feature of all these evolution equations is Markovianity, i.e., positivity and total mass of the initial value are preserved. For a given (quantum) Markov semigroup, a transport metric on the space of probability measures is constructed, using a Benamou-Brenier-type formula. If the semigroup satisfies a certain gradient estimate, it is shown that the entropy is convex along geodesics for this transport metric and the orbits of the semigroup are curves of steepest descent for the entropy satisfying an evolution variational inequality.

Abstract

We construct new examples on Lavrentiev phenomenon using fractal contact sets. Comparing to the classical examples of Zhikov it is not important that at the saddle point the variable exponent crosses the threshold dimension. As a consequence we give the negative answer to the well-known conjecture that the dimension plays a critical role for the Lavrentiev gap to appear. As an application we present new counterexamples to the density of smooth functions in variable exponent Sobolev spaces and to the regularity of the functional with double-phase potential. The talk is based on joint work with Lars Diening and Mikhail Surnachev.

Abstract

We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. The argument is based on a compactness framework for finite energy solutions, which utilizes compensated compactness.

Abstract

We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. The argument is based on a compactness framework for finite energy solutions, which utilizes compensated compactness.

Abstract

The development of reduced order models for complex applications promises rapid and accurate evaluation of the output of complex models under parameterized variation with applications to problems which require many evaluations, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed. However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large scale applications, in particular for nonlinear and/or time-dependent problems. After a brief introduction to reduced order models, we discuss the development of methods which seek to conserve chosen invariants for nonlinear time-dependent problems. We develop structure-preserving reduced basis methods for a broad class of Hamiltonian dynamical systems, including canonical problems and port-Hamiltonian problems, before considering the more complex situation of Hamiltonian problems endowed with a general Poisson manifold structure which encodes the physical properties, symmetries and conservation laws of the dynamics. Time permitting, we subsequently discuss reduced order modeling of more general hyperbolic problems, discuss the importance of the skew-symmetric form of the governing equations, and the benefits of using the skew-symmetric form for the reduced order model. We demonstrate the methods through the numerical simulation of various fluid flows.

Abstract

The field of model reduction encompasses a broad range of methods that seek efficient low-dimensional representations of an underlying high-fidelity model. A large class of model reduction methods are projection-based; that is, they derive the low-dimensional approximation by projection of the original large-scale model onto a low-dimensional subspace. Model reduction has clear connections to machine learning. The difference in fields is perhaps largely one of history and perspective: model reduction methods have grown from the scientific computing community, with a focus on reducing high-dimensional models that arise from physics-based modeling, whereas machine learning has grown from the computer science community, with a focus on creating low-dimensional models from black-box data streams. This talk will describe two methods that blend the two perspectives and provide advances towards achieving the goals of Scientific Machine Learning. The first method combines lifting–the introduction of auxiliary variables to transform a general nonlinear model to a model with polynomial nonlinearities–with proper orthogonal decomposition (POD). The result is a data-driven formulation to learn the low-dimensional model directly from data, but a key aspect of the approach is that the lifted state-space in which the learning is achieved is derived using the problem physics. The second method combines a low-dimensional POD parametrization of quantities of interest with machine learning methods to learn the map between the input parameters and the POD expansion coefficients. The use of particular solutions in the POD expansion provides a way to embed physical constraints, such as boundary conditions. Case studies demonstrate the importance of embedding physical constraints within learned models, and also highlight the important point that the amount of model training data available in an engineering setting is often much less than it is in other machine learning applications, making it essential to incorporate knowledge from physical models.

Abstract

Positive semi-definite symmetric tensors whose divergence is a bounded measure enjoy a property of enhance integrability. This tool can be used to find new estimates in fluid dynamics, at every level of description, from microscopic to macroscopic. I shall explain how this theory is related to Monge-Ampère equation and transport theory. Then I shall discuss Euler and Boltzmann equations, and molecular dynamics.

Abstract

Data Science aims to develop models that extract knowledge from complex data and represent it to aid Data Driven Decision Making. Mathematical Optimization has played a crucial role across the three main pillars of Data Science, namely Supervised Learning, Unsupervised Learning and Information Visualization. For instance, Quadratic Programming is used in Support Vector Machines, a Supervised Learning tool. Mixed-Integer Programming is used in Clustering, an Unsupervised Learning task. Global Optimization is used in MultiDimensional Scaling, an Information Visualization tool.

Data Science models should strike a balance between accuracy and interpretability. Interpretability is desirable, for instance, in medical diagnosis; it is required by regulators for models aiding, for instance, credit scoring; and since 2018 the EU extends this requirement by imposing the so-called right-to-explanation. In this presentation, we discuss recent Mathematical Optimization models that enhance the interpretability of state-of-art supervised learning tools, such as nearest neighbors, classification trees and support vector machines, while preserving their good learning performance.

Abstract

The classical Minty lemma relies on monotonicity and some notion of continuity of the operator involved. For discontinuous relations a suitable extension has been proved in Bulicek et al. (2012, 2016), which was applied to show existence of weak solutions to fluid equations for non-Newtonian fluids with discontinuous constitutive relation, cf. Bulicek et al. (2012). I will present a splitting and regularising strategy to show convergence (up to subsequences) of finite element approximations using this Minty type convergence lemma. Furthermore, I shall introduce approximations for discontinuous relations satisfying a variant of the Minty type convergence result. This is useful to establish convergence of numerical approximations without splitting the limits. If time allows I will introduce a class of non-monotone relations and sketch issues around the identification of the relation in the existence proof. Parts of this talk are based on joint work with Endre Süli.

Abstract

We reproduce the capacity drop phenomenon at a road merge via point constraints in a first order traffic model. We first construct an enhanced version of the locally constrained model introduced by Haut, Bastin and Chitour in [1], then we propose its counterpart featuring a non-local constraint and finally we compare numerically the two models by constructing an adapted finite volumes scheme. These results are part of an ongoing collaboration with Edda Dal Santo (Univ. dell’Aquila), Carlotta Donadello (Univ. de Franche-Comté) and Massimiliano Daniele Rosini (Univ. di Ferrara).

References

[1] Haut, B., Bastin, G., and Chitour, Y. A macroscopic traffic model for road networks with a representation of the capacity drop phenomenon at the junctions. Proceedings 16th IFAC World Congress, Prague, Czech Republic (2005).

Abstract

Consider two masses resting in front of a wall in a frictionless one-dimensional world. Now set the mass furthest from the wall into motion (constant speed). Count the collisions mass-mass and mass-wall until the state of the system becomes stationary. Increase the mass ratio by powers of 100, repeat the experiment and be amazed! To celebrate Pi Day 2019 I am going to present a surprising fact of trivia about the number $\pi$. This is not a research talk and requires only some knowledge of basic geometric and physical facts. I was made aware of this by a wonderful youtube video by 3blue1brown, who posed this as a riddle in January. To avoid spoilers, maybe refrain from watching the solution video before the talk.

Abstract

An appealing deterministic method of solving kinetic equations is a Galerkin type approach which involves approximating the solution in some finite dimensional space. In this talk, for initial boundary value problems, we develop a Galerkin method which preserve L2-stability of kinetic equations. We mainly focus on the linearised Boltzmann equation of rarefied gas dynamics and we use Hermite polynomials to approximation the solution along the velocity space. Through numerical examples we demonstrate the importance of L2-stability.

Abstract

I will talk about some geometric questions that arise in the study of soft/thin objects with negative curvature. After reviewing basic ideas from discrete differential geometry (DDG), I will motivate the need for new “geometric” methods, based on DDG, for studying the mechanics of leaves, flowers, and sea-slugs. I will present some of our results in this direction. This is joint work with Toby Shearman and Ken Yamamoto.

Abstract

We present several integral-differential equations that describe the evolution of a population structured with respect to a continuous trait. Under some assumptions, solutions are shown to converge toward the evolutionary stable distribution, using the tool of relative entropy. Efficient numerical methods for capturing selection dynamics are highly desired for model prediction, and we have developed entropy satisfying finite volume methods with provable nice properties. The role of nonlinear competition will also be discussed.

Abstract

We present the convergence of a class of entropy stable finite volume schemes for the barotropic and complete compressible Euler equations in the multidimensional case. We study the classical schemes based on the Lax-Friedrichs numerical flux [1], and moreover propose a new finite volume scheme motivated by the Brenner model [2]. Numerical viscosity imposed through upwinding in the latter scheme acts on the velocity field rather than on the convected quantities. For both methods we establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solution of the Euler system. In particular, using the recently established weak-strong uniqueness principle, we show that the numerical solutions converge pointwise to the regular solution of the limit systems at least on the lifespan of the latter. Finally, we discuss convergence of a finite volume scheme for the compressible (barotropic) Navier-Stokes equations without the a priori hypothesis of the existence of smooth solution [3]. Numerical experiments for benchmark tests support our theoretical results.

These results are based on a joint work with Eduard Feireisl (Czech Academy of Sciences in Prague), Mária Lukáčová-Medvid’ová (Johannes Gutenberg University in Mainz) and Bangwei She (Czech Academy of Sciences in Prague).

References

[1] E. Feireisl, M. Mária Lukáčová-Medvid’ová, and H. Mizerová. Convergence of finite volume schemes for the Euler equations via dissipative measure–valued solutions.
arXiv:1803.08401, 2018.

[2] E. Feireisl, M. Mária Lukáčová-Medvid’ová, and H. Mizerová. A finite volume scheme for the Euler system inspired by the two velocities approach.
arXiv:1805.05072, 2018.

[3] E. Feireisl, M. Mária Lukáčová-Medvid’ová, H. Mizerová, and B. She. Convergence of a finite volume scheme for the compressible Navier-Stokes system.
arXiv:1811.02866, 2018.

Abstract

The problem under consideration is relevant to production processes associated with the longitudinal movement of materials, for example, for producing paper webs. For these processes’ transverse disturbances, which in the vertical section are described by the hyperbolic equation of a longitudinally moving string, are extremely undesirable. That gives the problem of damping these oscillations within a finite time.
To solve the problem of damping the oscillations, we suggest reducing it to the trigonometric problem of the moments at an arbitrary time interval. When considering moving materials the construction of the basis systems forming the moment problem is a special challenge, since the hyperbolic equation contains a mixed derivative due to Coriolis acceleration. Therefore, the classical method of separating variables is not applicable in this case. Instead, a new method is used to find self-similar solutions of non-stationary equations, which makes it possible to find the basis systems explicitly.
In the case of paper web, it is necessary to find a minimal in the whole class of admissible perturbations time interval, within which the trigonometric system forming the problem of moments is the Riesz basis, that make it possible through using the system conjugate with it to find the optimal control way in the form of a series and, therefore, to build a so-called optimal damper.
As a result of the study, a generalized solution of the problem of transverse oscillations is constructed. For the problem of damping oscillations, the exact damping time is obtained, namely, a time T0 at which the total energy of the system is zero. Optimum control is found in the form of a Fourier series.
The results of the talk are published in L.A. Muravey, V.M. Petrov, A.M. Romanenkov, The Problem of Damping the Transverse Oscillations on a Longitudinally Moving String, Mordovia University Bulletin [Vestnik Mordovskogo universiteta], 2018, 28(4):472–485, DOI: https://doi.org/ 10.15507/0236-2910.028.201804.472-485.

Abstract

Thermodynamics of gases wants to compute the evolution of the fields of density, velocity and temperature. It relies on the conservation laws of mass, momentum and energy, but these equations alone do not yield field equations. Instead they contain further unknown fields, like stress tensor and heat flux which have to be closed by material laws. The balance law of entropy, in particular the entropy production, can be used to restrict the possible closures and even derive simple models which fit to well known empirical laws.

Abstract

A symplectic variety comes along with a symplectic form and the Gromov width of such a variety is the size of the largest ball that can be embedded respecting this form. In this talk we will compute the Gromov width of a class of symplectic varieties.

Along the way, we will pass through representation theory of Lie algebras, standard monomial theory, canonical bases, Newton-Okounkov bodies, toric degenerations, to finally reduce this geometric question to a purely combinatorial problem of embedding a standard simplex into a convex polytope.

Abstract

We will consider three different classes of results regarding the structure of curves of measures in metric spaces, all frequently referred to as “superposition principles”, in particular: (A) the results relating the family of measures solving the first order PDE called continuity equation, with the measures on the space of solutions to the associated characteristic ODE; (B) the results giving the structure of curves of positive Borel measures absolutely continuous with respect to some Kantorovich-Wasserstein distance, in terms of measures over some space of rectifiable curves, and, finally, (C) the results on decomposition (or, more appropriately, “disintegration”) of general normal currents (in the sense of De Rham or in the sense of De Giorgi-Ambrosio-Kirchheim) in simpler ones associated with rectifiable curves.

We will discuss the modern general results relating (A), (B) and (C) as well as their applications, in particular showing that all those principles descend from a single superposition principle for normal metric currents.

Abstract

Solving complex models with high accuracy and within a reasonable computing time has motivated the search for numerical schemes that exploit efficiently parallel computing architectures. For a given Partial Differential Equation (PDE), one of the main ideas to parallelize a simulation is to break the problem into subproblems defined over subdomains of a partition of the original domain. The domain can potentially have high dimensionality and be composed of different variables like space, time, velocity or even more specific variables for some problems. While there exist algorithms with very good scalability properties for the decomposition of the spatial variable in elliptic and saddle-point problems, the same cannot be said for the decomposition of time of even simple systems of ODEs. This is despite the fact that research on time domain decomposition is currently very active and has by now a history of at least 50 years (back to at least to work of J. Nievergelt in 1964) during which several algorithms have been explored. As a consequence, time domain decomposition is to date only a secondary option when it comes to deciding what algorithm/method distributes the tasks in a parallel cluster… and this is certainly a pity.

In 2001, with J.L. Lions and G. Turinici, we have introduced the parareal algorithm that has put back some attention on this issue. Our algorithm has achieved accelerations in wall clock restitution thanks to the use of decomposition of the time interval into sub-intervals each of them being treated on independent processors. Since then, the number of contributions has increased and a community is meeting regularly to treat the remaining difficulties, with a success that leads groups in industries to consider parallelism in time as a realistic avenue.

We shall present in details the parareal in time method, with the current achievements, in particular we shall present a recent version that allow to expect full efficiency of the algorithms in terms of parallel speed-up. We shall also present some open problems that remain to be solved.