Abstract

The effective behaviour of (classical) many-particle systems can often be described by PDEs for particle densities on larger scales. A way to justify these PDEs are mean-field limits. We will review classical results for regular interaction kernels and clarify why Wasserstein distances are particularly well-suited to derive mean-field limits in this context. We will then discuss recent developments for singular interaction kernels.

Abstract

The evolution of a network of interfaces by mean curvature flow features the occurrence of topology changes and geometric singularities. As a consequence, classical solution concepts for mean curvature flow are in general limited to a finite time horizon. At the same time, the evolution beyond topology changes can be described only in the framework of weak solution concepts (e.g., Brakke solutions), whose uniqueness may fail.

Following the relative energy approach, we prove a quantitative stability estimate holding up to the singular time at which a circular closed curve shrinks to a point. This implies a weak-strong uniqueness principle for weak BV solutions to planar multiphase mean curvature flow beyond circular topology changes. We expect our method to have further applications to other types of shrinkers.

This talk is based on a joint work with Julian Fischer, Sebastian Hensel and Maximilian Moser.

Abstract

The Stokes equations describe the dynamics of a viscous fluid when inertial forces are negligible compared to viscous forces. In this talk, we study the dynamics of two different incompressible Stokes fluids evolving in a horizontally periodic strip. We consider the fluids to have different densities and to be subject to gravity forces. This induces the evolution of the fluids and hence the evolution of the free interface arising between them. There are two differentiated scenarios depending on the regime of the densities: the stable regime (when the denser fluid is below the lighter fluid) and the unstable regime (vice-versa). In both regimes, we study the dynamics of the interface through a contour dynamics approach and we address fundamental questions such as well-posedness, stability and instability of the interface in the stable and unstable regime of the densities.

Abstract

Central-upwind schemes are Riemann-problem-solver-free Godunov-type finite-volume schemes, which are, in fact, non-oscillatory central schemes with a certain upwind flavor: derivation of the central-upwind numerical fluxes is based on the one-sided local speeds of propagation, which can be estimated using the largest and smallest eigenvalues of the Jacobian.

I will introduce two new classes of central-upwind schemes with reduced numerical dissipation. First, we will use a subcell resolution at the projection step to enhance the resolution of contact waves, which are typically badly affected by excessive numerical dissipation present in numerical methods. The second approach is based on the utilization of the local characteristic decomposition for the modification of the numerical diffusion of the central-upwind schemes. Both approaches help to significantly reduce the amount of numerical dissipation present in central-upwind schemes without risking large spurious oscillation. Applications to several hyperbolic systems of conservation laws will be discussed.

Abstract

Growth is a fundamental process in biological systems, as well as in various technological applications, including epitaxial deposition and additive manufacturing. The interaction between growth and mechanics in deformable bodies gives rise to a wealth of very challenging mathematical questions. I will provide a brief overview of the fundamental concepts of morphoelasticity, namely, the theory of elastic deformations in growing bodies. In contrast to the classical case, the reference state of a growing body evolves over time, also in response to external stimuli and stress. In some situations, this calls for a free-boundary formulation, for the actual shape of the undeformed body is also unknown. I plan to discuss the case of surface accretion, posing specific challenges. The focus will be the development of a variational framework where the existence of three-dimensional, quasistatic morphoelastic evolution can be proved. This is work in collaboration with Elisa Davoli (TU Vienna), Katerina Nik (University of Vienna), and Giuseppe Tomassetti (Roma 3).

Abstract

In this talk we will discuss numerical approximations to nonlinear dispersive equations, suited for non-smooth initial data. We will present on the one hand their construction, and on the other hand rigorous low-regularity convergence results in the case of the Gross-Pitaevskii equation.

We will also put forth a novel time discretization to the nonlinear Schrödinger equation, allowing for a low regularity approximation while maintaining good long-time preservation of the mass and energy on the discrete level.

If time permits, higher order extensions will be presented, following new techniques based on decorated trees series expansions inspired by singular stochastic PDEs.

Abstract

Schrödinger’s classical thought experiment aimed for finding the most likely evolution between two subsequent observations of a cloud of independent particles. We introduce and advocate a generalised Schrödinger problem, defined for a wide class of entropy and Fisher information functionals, as a geometric problem on a metric space. Broadly speaking, in Schrödinger’s original situation the metric space is the Wasserstein space and the entropy is Boltzmann’s one. Under very mild assumptions (in particular, without any curvature restrictions) we prove a generic gamma-convergence result of the generalised Schrödinger problem towards the geodesic problem, as the temperature parameter tends to zero. Our novel technique is based on adaptive perturbations by gradient flows. We then study the dependence of the entropic cost on the temperature parameter. A similar technique allows us to prove the so-called Ishihara property for the harmonic maps valued in metric spaces, which means that the pullback of a convex function defined on a metric space by a harmonic map valued in that space is subharmonic. This abstract result implies some conjectures of [Y. Brenier, “Extended Monge-Kantorovich theory” in Optimal transportation and applications (Martina Franca, 2001), volume 1813 of Lecture Notes in Math., pages 91-121. Springer, Berlin, 2003]. The talk will be based on joint works with H. Lavenant, L. Monsaingeon & L. Tamanini.

Abstract

Recurrent neural networks with randomly generated internal weights, and trained with descent-based methods, have become prevalent for learning discrete-time dynamical systems. Despite extensive empirical investigations in the literature, a concrete theoretical understanding of these methods remains elusive, especially with nonlinear architectures such as ReLU networks. In this work, we address this challenge for the regression problem in the overparameterized regime, and establish non-asymptotic sample-complexity, iteration-complexity, and network-width bounds for reservoir computing with ReLU recurrent networks, trained with gradient-descent. Our results reveal that (i) the memory of the system has a detrimental impact on the generalization properties with a non-vanishing error term due to the nonlinearity it introduces, and (ii) only mild overparameterization, i.e., neural network width with polylogarithmic dependence on the size of the training set, suffices for gradient descent to achieve global near-optimality up to a term that vanishes with the memory of the system.

Abstract

We develop sparse grid discontinuous Galerkin (DG) schemes for the Vlasov-Maxwell (VM) equations. The VM system is a fundamental kinetic model in plasma physics, and its numerical computations are quite demanding, due to its intrinsic high-dimensionality and the need to retain many properties of the physical solutions. To break the curse of dimensionality, we consider the sparse grid DG methods that were recently developed for transport equations. Such methods are based on multiwavelets on tensorized nested grids and can significantly reduce the numbers of degrees of freedom. We formulate two versions of the schemes: sparse grid DG and adaptive sparse grid DG methods for the VM system. Their key properties and implementation details are discussed. Accuracy and robustness are demonstrated by numerical tests, with emphasis on comparison of the performance of the two methods, as well as with their full grid counterparts.

Abstract

Homogenization is a useful analytical approach to rigorously derive macroscopic models from microscopic information.

This talk will primarily be focused on a high-level overview of energy methods. We will introduce Two scale convergence– a useful compactness tool independent of homogenization– and it will be used to simplify the computation of the homogenized limit in the classical elliptic problem.

We will also discuss through examples the context in the hierarchy of physical models and the many open problems in this area.

Abstract

The solutions of hyperbolic systems may contain discontinuities. These weak solutions verify not only the original PDEs, but also an entropy inequality that acts as a selection criterion determining whether a discontinuity is physical or not. Obtaining a discrete version of these entropy inequalities when approximating the solutions numerically is crucial to avoid convergence to unphysical solutions or even unstability. However such a task is difficult in general, if not impossible for schemes of order 2 or more. We introduce an optimization framework that enables us to quantify a posteriori the decrease or increase of entropy of a given scheme, locally in space and time. We use it to obtain maps of numerical diffusion and to prove that some schemes do not have a discrete entropy inequality. A special attention is devoted to the widely used second order MUSCL scheme for which almost no theoretical results are known. This is a joint work with Emmanuel Audusse, Vivien Desveaux and Julien Salomon.

Abstract

We consider the parabolic Signorini problem (or parabolic thin obstacle problem), which is a classical free boundary problem motivated from modelling fluids passing through semipermeable membranes. I will review some techniques used to tackle the regularity properties of the solutions as well as the free boundaries. This is based on joint work with Vedansh Arya.

Abstract

Hyperbolic systems of conservation laws exhibit shock waves. The stability of numerical methods relies on upwinding the flux function, while higher order accuracy is based on piecewise polynomial spatial reconstructions and suitable time integration. In this lecture we focus on piecewise linear spatial reconstructions called MUSCL. Two-stage Runge-Kutta methods are popular for time integration, but there are also classical one-stage schemes such as the Hancock method and the central scheme.

To ensure the accuracy order, stability and fast simulations of these schemes, we should be careful about three building blocks: the slope limiter in initial data reconstruction, the robustness of the numerical flux function, and the CFL condition to control the time step. In this course we will analyze these three building blocks for the above mentioned three types of MUSCL schemes independently. The stability results rely on new convex decompositions of the schemes. This makes it possible to improve the schemes in two ways: first, we can allow sharper, more compressive spatial reconstructions. Second, we can allow larger time steps and hence faster simulations than previously reported in the literature. Numerical experiments show that the estimates provide sharp maximum-minimum bounds for scalar conservation laws. For the two-dimensional laws of gas dynamics, we design a scheme which is positivity preserving, and test it on a flow with extremely high Mach number.

Our course includes the following sections:

1. Stability properties of some typical hyperbolic conservation laws.

2. Stability of numerical approximations

    2.1 Two-stage MUSCL scheme

    2.2 One-stage MUSCL-Hancock scheme

    2.3 Central scheme.

The doctoral researchers of the Research Training Group (EDDy) are pleased to announce the next Math Chat (everyone is invited).

Math Chats are informal gatherings hosted by PhD students for bachelor’s and master’s students who are interested in finding out more about what research is like, how to find a thesis advisor, which seminars & opportunities are available through EDDy at the RWTH, etc.

Abstract

Modelling of tumour growth is one of the challenging frontiers of applied mathematics. In the last years, phase field models for tumour growth have been studied intensively. Alike classical free boundary models they use a continuum approach to describe the growth of tumours. However, an advantage to free boundary models is that phase field models allow for topology changes like break up and coalescence. In addition, phase field methods can be used numerically without an explicit tracking of the interface which is necessary for free boundary models. In my talk I will introduce several macroscopic models for tumour growth in which cell-cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. The resulting evolution equation is a Cahn-Hilliard equation taking source and sink terms into account. In addition, nutrient diffusion is incorporated by a coupling to a reaction-equation diffusion. I will show existence, uniqueness and regularity results. I will also discuss how to couple the system to an internal velocity field which either solves a Darcy-type or a (Navier-)Stokes system or a viscoelastic system. Properties of solutions will be illustrated with the help of numerical simulations.

Abstract

In knot theory, we use self-repulsive energies as a tool to measure entanglement. In the first half of the talk, I will give a crash-course introduction into self-repulsive energies. I will present some properties, which were used in order to construct a Riemannian metric measuring this self-entanglement. Later, I’m going to introduce infinite dimensional Riemannian geometry and characterize the tangent-spaces of fractional Sobolev-spaces.

In the second half of the talk I will present the metric and sketch the proof of geodesic completeness (i.e. that geodesics w.r.t the metric are defined for all time). Furthermore, some of the occurring problems and how we overcame them will be explained.

This is ongoing work of me and my collaborators, Prof. Philipp Reiter and Dr. Henrik Schumacher.

Abstract

Maintaining the topology of objects undergoing deformations is a crucial aspect of elasticity models. Impermeability may be implemented via regularization by a suitable nonlocal functional.

In case of elastic solids whose shape is described by the image of a reference domain under a deformation map, self-interpenetrations can be ruled out by claiming global invertibility. Given a suitable stored energy density, the latter is ensured by the Ciarlet–Nečas condition which, however, is difficult to handle numerically in an efficient way. This motivates approximating the latter by adding a self-repulsive functional which formally corresponds to a suitable Sobolev–Slobodeckiĭ seminorm of the inverse deformation.

This is joint work with Stefan Krömer (Prague).

Abstract

In this talk I will discuss basic notions in the theory of differentiable manifolds, aiming towards differential forms and Stokes theorem. With these tools at hand, I intend to explain various applications in differential geometry/topology (Brouwer degree, Brouwer fixed point theorem, Lefschetz index theorem, Gauss-Bonnet theorem etc).

Abstract

In the first part of the talk, I’ll discuss Floer homology for 3-manifolds. The Seiberg-Witten invariants of a smooth 4-manifold M are defined by counting solutions to a certain PDE on M. The theory of Floer homology for 3-manifolds was developed to understand these invariants using cut and paste topology: we split a given 4-manifold M in half along a 3-manifold Y and express the invariants of M in terms of relative invariants of the two halves. The relative invariants live in a vector space (the Floer homology) associated to Y. There’s a related invariant (knot Floer homology) which assigns a vector space to a knot inside a 3-manifold.

A similar thing happens when we split Y along a surface S, but now the relative invariant is an object in a category. The second half of the talk will focus on what happens when one of the two pieces is a solid torus containing a knot. This is an extension of previous joint work with Hanselman and Watson.

Abstract

The circle enjoys several optimality properties: from being the solution to the isoperimetric inequality, to being the shape with the least first frequency, from having the minimum electrostatic capacity, to having the maximal torsional rigidity. Do polygons enjoy similar optimality properties?

In this seminar, we will investigate such a question by considering two prototypical classes of energies. The first, is a nonlocal functional with a Riesz type potential. The second is a crystalline perimeter perturbed with a functional of the former kind. In particular, for the latter we will investigate the case of general kernels in any dimension, while for the former we will focus on the case of Riesz potentials on triangles and quadrilaterals in dimension two.

This talk is based on joint works with Marco Bonacini (Università di Trento) and Ihsan Topaloglu (Virginia Commonwealth University).

Link to Zoom room:

https://rwth.zoom.us/j/4608079046?pwd=RHB6MVE1SGg3WWwyWVhNMFBiMUpUdz09

Abstract

Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by utilizing the “spreading” property of the collision operator. In this talk, we introduce a new proof based on a careful $L^2$ estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions. This talk is based on some joint works with Tong Yang (PolyU Hong Kong) and Jingwei Hu (University of Washington).

Abstract

In this talk we present a new linear model for the propagation of acoustic waves and gravity waves in a simplified ocean. The use of acoustic waves measurements in the ocean is currently seen as a good candidate for the improvement of tsunami early-warning systems. The new model is obtained in the following way: the compressible Euler equation are written in Lagrangian coordinates and linearized around a state at equilibrium corresponding to the ocean at rest. A wave-like equation containing acoustic terms and gravity terms is then derived. The obtained model is compared with the literature by using some simplifications, namely the barotropic assumption, and the limits in the incompressible and in the acoustic regime.

We present then some numerical results. By using a finite element discretization in space and a finite difference scheme in time, we are able to reproduce the simulations available in the literature for the case without topography and with a constant temperature. Finally we show some aspects of the mathematical analysis for the continuous problem, and focus on the functional space needed to prove existence and uniqueness of the solution.

This is a joint work with Jacques Sainte-Maris (Inria Paris, ANGE) and Sébastien Imperiale (Inria Saclay, M3DISIM).

Abstract

We consider the following question: among all curves lying on a given Riemannian manifold with prescribed length and boundary data, which ones minimise the $L^\infty$ norm of the curvature? This extends a paper of Moser considering the same question in Euclidean space. Using the method of $L^p$ approximation we show that minimisers of our problem and also a wider class of “pseudominimiser” curves must satisfy an ODE system obtained as the limit as $p \rightarrow \infty$ of the $L^p$ Euler-Lagrange equations. This system gives us some geometric information about our (pseudo)minimisers. In particular we find that their curvature takes on at most two values: a positive constant $K$, and possibly zero in some places.

This talk is based on joint work with Roger Moser which can be found at https://arxiv.org/abs/2202.07407.

Abstract

Abstract: The Landau-Lifschitz-Gilbert equation is a model describing the magnetisation of a ferromagnetic material. The stochastic model is studied to observe the role of thermal fluctuations. We interpret the linear multiplicative noise appearing by means of rough paths theory and we study existence and uniqueness of the solution to the equation on a one dimensional domain $D$. We show that the map that to the noise associates the unique solution to the equation is locally Lipschitz continuous in the strong norm $L^{\infty}([0,T];H^1(D))\cap L^2([0,T];H^2(D))$, with initial condition in $H^1(D)$. This implies a Wong-Zakai convergence result, a large deviation principle, a support theorem and the Feller property for the associated semigroup.

The talk is based on a joint work with A. Hocquet https://arxiv.org/abs/2103.00926 and on https://arxiv.org/abs/2208.02136.

Abstract

Since the system of field equations is hyperbolic, the shock-structure solution is not always regular, and discontinuous parts (sub-shocks) can be formed when the shock velocity meets a characteristic velocity [1]. In particular, in the case of a hyperbolic system with a convex entropy (symmetric systems), a theorem by Boillat and Ruggeri [2] proved that a sub-shock surely arises when the shock’s velocity becomes greater than the maximum characteristic velocity in the unperturbed state. The question if there are sub-shocks also for shock velocities less than the maximum characteristic is still an open problem. An interesting case of the existence of sub-shocks for shock velocity smaller than the maximum velocity is offered by a binary mixture of polyatomic Eulerian gases with different degrees of freedom of a molecule based on the multi-temperature model of Rational Extended Thermodynamics [3]. For given values of the mass ratio and the specific heats of the constituents, we identify the possible sub-shocks as the Mach number of the shock wave and the concentration of the constituents change [4]. Namely, the regions with no sub-shocks, a sub-shock for only one component, or sub-shocks for both constituents are comprehensively classified. The most interesting case is that the lighter molecule has more degrees of freedom than the heavier one. In this situation, the topology of the various regions becomes different. We also numerically solve the system of the field equations using the parameters in the various regions and confirm whether the sub-shocks emerge. Finally, the relationship between an acceleration wave in one constituent and the sub-shock in the other is explicitly derived.

References:

[1] T. Ruggeri, “Breakdown of shock-wave-structure solutions,” Phys. Rev. E 47, 4135 (1993).

[2] G. Boillat and T. Ruggeri, “On the shock structure problem for hyperbolic system of balance laws and convex entropy,” Cont. Mech. Thermodyn. 10 285–292 (1998).

[3] T. Ruggeri and M. Sugiyama, Classical and Relativistic Rational Extended Thermodynamics of Gases (Springer, Cham, 2021).

[4] T. Ruggeri and S. Taniguchi, “A Complete Classification of Sub-Shocks in the Shock Structure of a Binary Mixture of Eulerian Gases with Different Degrees of Freedom,” Phys. Fluids 34, 066116 (2022).

https://rwth.zoom.us/j/96830789945?pwd=TFpSRGpSOW1tTkpGTkxnV0dHZ1l1dz09

Meeting ID: 968 3078 9945

Passcode: 951945

Abstract

We are interested in quantifying uncertainties that appear in nonlinear hyperbolic partial differential equations arising in a variety of applications from fluid flow to traffic modeling. A common approach to treat the stochastic components of the solution is by using generalized polynomial chaos expansions. This method was successfully applied in particular for general elliptic and parabolic PDEs as well as linear hyperbolic stochastic equations. More recently, gPC methods have been successfully applied to particular hyperbolic PDEs using the explicit form of nonlinearity or the particularity of the studied system structure as, e.g., in the p-system. While such models arise in many applications, e.g., in atmospheric flows, fluid flows under uncertain gas compositions and shallow water flows, a general gPC theory with corresponding numerical methods are still at large. Typical analytical and numerical challenges that appear for the gPC expanded systems are loss of hyperbolicity and positivity of solutions (like gas density or water depth). Any of those effects might trigger severe instabilities within classical finite-volume or discontinuous Galerkin methods. We will discuss properties and conditions to guarantee stability and present numerical results on selected examples.

Abstract

The software ‘Wolfram Mathematica’ is a very powerful computer-algebra system with the significant downside that it is proprietary and commercial. In fact, access to it is expensive and heavily guarded, such that only few schools provide campus licenses and even run-only players or cloud options remain not really available or hardly useful, giving it a fairly antique flavor. Once you get over this nuisance, Mathematica can be extremely helpful and actually fun to use for a wide range of mathematics and computational science, from symbolic manipulation, visualization, and interactive math, to rapid prototyping of algorithms, and numerical methods for partial differential equations.

In this online presentation we will give a concise introduction into the basic usage of Mathematica, its data structures and syntax, and go through concrete examples of case studies. The Mathematica notebook used in the course will be distributed at the beginning so everybody can try things out as we go.

Students of EDDy are encouraged to ask for a free Mathematica license+download at office@acom.rwth-aachen.de to have their own installation ready. All others are invited to download an official trial version or simply watch.

Link to Zoom room:

https://rwth.zoom.us/j/99255902507?pwd=TEp3c2VEekd0YTZxaG5IZHpWMjRHQT09