Abstract

Two new ways to couple scalar conservation laws on networks are presented. We first discuss a numerically motivated approach, in which a new scheme is derived using a relaxation system and taking the relaxation limit also at the nodes of the network. The scheme is mass conservative, allows for a second order extension and yields well defined and easy-to-compute coupling conditions for general networks. The second approach focuses on the traffic flow model by Lighthill, Witham and Richards in a network setting. The choice of coupling condition at the junction has been a major modeling challenge. Using car trajectory data obtained from drone photography over a German motorway we develop new data-driven coupling conditions, which we compare to classical approaches from literature.

Abstract

In standard methods for time-dependent partial differential equations, the time-dependence is resolved by individual steps that advance the time variable incrementally. In space-time methods, in contrast, an approximation of the solution on the entire time interval is computed simultanteously. In this talk, adaptive solvers based on different types of space-time variational formulations are discussed with focus on parabolic problems.

Abstract

We are interested in the feedback stabilization of a quantity which evolution can be described using a Hamilton-Jacobi equation in $R^n$. The mathematical treatment then leads to a system of $n$ hyperbolic transport PDEs for the perturbation of a desired state which should be stabilized. There exists a rich literature for the one dimensional case including different Lyapunov functions leading to exponential decay of the $L^2$ norm when suitable feedback controls are applied. Here we want to extend these results to the multi-dimensional case leading to a novel Lyapunov function with space dependent weight functions taking accounting for the multidimensional geometry. We show the exponential decay of the Lyapunov function provided that the weights and controls are chosen appropriately. The design feedback control is closely related to the precise choice of the Lyapunov function and its weights. We further present numerical experiments.

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

Diffusion-based generative models build a representation of the information contained in data by successively adding noise, but keeping track of the intermediate states. For the generative phase, this process can then be run backwards to create new samples. We will discuss an interpretation of this method in terms of gradient flows on the Wasserstein space of probability measures.

Abstract

We model heterogeneous elastic wires by closed planar curves with density. The associated elastic energy depends on a density-modulated stiffness. Working with the inclination angle function, the $L^2$-gradient flow corresponding to this energy is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness and global existence of solutions and study properties of the flow.

This talk is based on a joint work with Anna Dall’Acqua, Gaspard Jankowiak and Fabian Rupp.

Abstract

One of the main questions in the theory of the linear transport equation is whether uniqueness of weak solutions to the Cauchy problem holds in the case the given vector field is not smooth. In the talk I will provide an overview on some results obtained in the last few years, showing that even for incompressible, Sobolev (thus quite “well-behaved”) vector fields, uniqueness of solutions can drastically fail. This result can be seen as a counterpart to DiPerna and Lions’ well-posedness theorem.

Abstract

In this talk I am going to introduce the Dirichlet boundary value problem with $L^p$ boundary data for elliptic linear second order PDEs and discuss the Carleson condition as sufficient condition for solvability of this problem. In the last few years, the theory involving Carleson type conditions is continuously making progress, for example in related different problems like the real or complex valued Neumann or Regularity problem.

To start with we will motivate the Carleson condition and explain why it arises “naturally” as condition for solvability of the Dirichlet boundary value problem. Motivated by the Carleson condition we will discuss perturbation theory of operators without drift term. This perturbation theory allows us to extent the class of operators which a Carleson type condition can be applied to, and hence we get a wider class of operators for whom the $L^p$ Dirichlet boundary value problem is solvable. The perturbation theory and its application is joint work Martin Dindoš and Erik Sätterqvist.

Abstract

We derive a macroscopic limit for a sharp interface version of a model proposed by Komura, Shimokawa and Andelman to investigate pattern formation in biomembranes due to competition of chemical and mechanical forces. We identify sub- and supercrital parameter regimes and show with the introduction of the autocorrelation function that the ground state energy leads to the isoperimetric problem in the subcritical regime, which is interpreted to not form fine scale patterns.

This is joint work with Hans Knüpfer and Anna Marciniak–Czochra.

Abstract

There exist two-phase flow models such as Baer-Nunziato type models that exhibit non-conservative products. Dal Maso, Murat and LeFloch introduced a concept of weak stability for non-conservative products. In the presentation the basics of their framework will be summarized and applied to hyperbolic equations with non-conservative-products.

Abstract

I will introduce a new approach for constructing robust well-balanced (WB) finite-volume methods for nonconservative onedimensional hyperbolic systems of nonlinear PDEs. The WB property, namely, the ability of the scheme to exactly preserve physically relevant steady-state solutions is enforced using a flux globalization approach according to which a studied system is rewritten in an equivalent quasi-conservative form with global fluxes. To this end, one needs to incorporate nonconservative product terms into the global fluxes. The resulting system can then be solved using a Riemann-problem-solver-free central-upwind (CU) scheme. However, a straightforward integration of the nonconservative terms would result in a scheme capable of exactly preserving very simple smooth steady states only and failing to preserve discontinuous steady states naturally arising in the nonconservative models.

In order to ameliorate the flux globalization based CU scheme, we evaluate the integrals of the nonconservative product terms using a path-conservative technique. This results in a new WB flux globalization based path-conservative central-upwind scheme (PCCU) scheme, which is much more accurate and robust than its predecessors. This is illustrated on the nonconservative system describing fluid flows in nozzles with variable cross-sections and a variety of shallow water models including the two-layer thermal rotating shallow water equations.

Abstract

In models of porous media flows, for instance in hydrogeology, the porosity of the solid matrix is typically treated as a static quantity. However, under certain circumstances, such as in soft sedimentary rocks or in magma flows, the porosity of the solid material can evolve under the influence of fluid pressure. In particular, this can lead to the formation of solitary porosity waves and of higher-porosity channels. We consider a system of nonlinear PDEs for porosity and effective pressure, based on a poroviscoelastic model, which describes such phenomena. The focus of this talk is on the well-posedness of this PDE problem, which has been established in the literature only for initial porosities of high Sobolev smoothness. We discuss several preliminary results for porosities of low regularity, including cases with jump discontinuities that are of particular interest in geological applications. We then turn to some first results on a space-time adaptive numerical method based on a discontinuous Petrov-Galerkin approach for efficiently approximating the evolution of porosity channels. Finally, we discuss available measurement data and associated inverse problems.

This is joint work in progress with Simon Boisserée (Mainz), Lisa Maria Kreusser (Bath), and Evangelos Moulas (Mainz).

Abstract

We propose AEGD, a new algorithm for gradient-based optimization of stochastic objective functions, based on adaptive updates of quadratic energy. The method is shown to be unconditionally energy stable, irrespective of the step size. In addition, AEGD enjoys tight convergence rates, yet allows a large step size. The method is straightforward to implement and requires little tuning of hyper-parameters. Experimental results demonstrate that AEGD works well for various optimization problems: it is robust with respect to initial data, capable of making rapid initial progress, shows comparable and most times better generalization performance than SGD with momentum for deep neural networks.

Abstract

We consider a model for cell polarization given by a nonlocal reaction-diffusion system on a two-dimensional surface. We prove the convergence to a free boundary problem in a fast reaction limit and derive conditions for the onset of polarization. Furthermore we study regularity properties of the nonlocal parabolic obstacle problem that represents the asymptotic reduction.

(This is joint work with Anna Logioti, Barbara Niethammer and Juan Velazquez.)

Abstract

Ginzburg-Landau type functionals provide a relaxation scheme to construct harmonic maps in the presence of topological obstructions. They arise in superconductivity models, in liquid crystal models (Landau-de Gennes functional) and in the generation of cross-fields in meshing. For a general compact manifold target space we describe the asymptotic number, type and location of singularities that arise in minimizers. We cover in particular the case where the fundamental group of the vacuum manifold is nonabelian and hence the singularities cannot be characterized univocally as elements of the fundamental group. We obtain similar results for $p$-harmonic maps with $p$ going to $2$. The results unify the existing theory and cover new situations and problems.

This is a joint work with Antonin Monteil (Paris-Est Créteil, France), Rémy Rodiac (Paris–Saclay, France) and Benoît Van Vaerenbergh (UCLouvain).

Abstract

A variational model for the interaction between homogenization and phase separation is considered. The focus is on the regime where the latter happens at a smaller scale than the former, and when the wells of the double well potential are allowed to move and to have discontinuities. The zeroth and first order $\Gamma$-limits are identified. The topology considered for the latter is that of two-scale, since it encodes more information on the asymptotic local microstructure. As a corollary, the minimum of the mass constrained minimization problem is characterized in terms of discontinuities of the wells.

Abstract

Kinetic equations play a leading role in the modelling of large systems of interacting particles/agents with a recognized effectiveness in describing real world phenomena ranging from plasma physics to multi-agent dynamics. The derivation of these models has often to deal with physical, or even social, forces that are deduced empirically and of which we have limited information. Hence, to produce realistic descriptions of the underlying systems it is of paramount importance to consider the effects of uncertain quantities as a structural feature in the modelling process.

In this talk, we focus on a class of numerical methods that guarantee the preservation of main physical properties of kinetic models with uncertainties. In contrast to a direct application of classical uncertainty quantification methods, typically leading to the loss of positivity of the numerical solution of the problem, we discuss the construction of novel schemes that are capable of achieving high accuracy in the random space without losing nonnegativity of the solution [1,4]. Applications of the developed methods are presented in the classical RGD framework and in related models in life sciences. In particular, we concentrate on the interplay of this class of models with mathematical epidemiology where the assessment of uncertainties in data assimilation is crucial to design efficient interventions, see [2,3].

Bibliography:

[1] J. A. Carrillo, L. Pareschi, M. Zanella. Particle based gPC methods for mean-field models of swarming with uncertainty. Commun. Comput. Phys., 25(2): 508-531, 2019.

[2] G. Dimarco, B. Perthame, G. Toscani, M. Zanella. Kinetic models for epidemic dynamics with social heterogeneity. J. Math. Biol., 83, 4, 2021.

[3] A. Medaglia, G. Colelli, L. Farina, A. Bacila, P. Bini, E. Marchioni, S. Figini, A. Pichiecchio, M. Zanella. Uncertainty quantification and control of kinetic models of tumour growth under clinical uncertainties. Int. J. Non Linear Mech., 141: 103933, 2022.

[4] L. Pareschi, M. Zanella. Monte Carlo stochastic Galerkin methods for the Boltzmann equation with uncertainties: space-homogeneous case. J. Comput. Phys. 423:109822, 2020.

Abstract

In this talk, I introduce a data-driven approach to viscous fluid mechanics, in particular to the stationary Navier-Stokes equation. The essential idea is to replace the constitutive law by experimental data. More precisely, usually one takes experimental data and then extrapolates a relation (the viscosity) between the deviatoric stress $\sigma$ and the strain $\epsilon$, for example $\sigma(\epsilon) = \mu_0 \epsilon$ (Newtonian fluid) or $\sigma(\epsilon)= \mu_0 |\epsilon|^{\alpha-1} \epsilon$ (power-law fluid). This relation is then used to obtain the Navier-Stokes equation.

Instead of using a constitutive relation, we introduce a data-driven formulation that has previously been examined in the context of solid mechanics. The idea is to find a solution that satisfies the differential constraints, derived from first principles, and is as close as possible to the experimental data. We obtain a variational formulation which we analyse under the aspects of weak lower-semicontinuity, coercivity and relaxation/$\Gamma$-convergence.

This talk is based on joint work with Christina Lienstromberg (Stuttgart) and Richard Schubert (Bonn).

Abstract

We consider the area preserving Willmore evolution of surfaces $\phi$, that are close to a half sphere with small radius, sliding on the boundary $S$ of a domain $\Omega$ while meeting it orthogonally. We prove that the flow exists for all times and keeps a `half spherish' shape. Additionally we investigate the asymptotic behaviour of the flow and prove that for large times the barycenter of the surfaces approximately follows an explicit ordinary differential equation. If time allows we conclude by investigating the convergence of the flow.

Abstract

The primary visual cortex (V1) of mammals is an area of the brain that receives and interprets visual input. Neurons in V1 have the characteristic that they respond preferentially to a particular orientation angle of the edge of a visual stimulus. In mammals such as cats, V1 contains an ordered map of the orientation preference of each neuron where cells preferring similar angles reside close to one another. In mice, however, the map of orientation preference appears random and disordered, with little correlation between preferred orientation and location in cortical space. Though much is known about orientation-preference maps in adult mammals, the mechanism underlying the formation of these maps during development is still unknown. In particular, I am interested in understanding under which conditions does the map that forms appear ordered (like in cats) or disordered (like in mice). In this talk, I will discuss a mathematical model used to describe a neuronal network of V1 during development and suggest a testable hypothesis for the mechanism underlying the formation of either an ordered or disordered orientation-preference map.

Abstract

In this talk, I will describe a refined local singularity analysis for the Ricci flow developed jointly with R. Buzano. The key idea is to investigate blow-up rates of the curvature tensor locally, near a singular point. Then I will show applications of this theory to Ricci flows with scalar curvature bounded up to the singular time.

Abstract

Grad’s 13-moment equations describe transport in mildly rarefied gases well, but are not properly embedded into nonequilibrium thermodynamics since they are not accompanied by a formulation of the second law. In this work, the Grad-13 equations are embedded into the framework of GENERIC (general equation for the nonequilibrium reversible–irreversible coupling), which demands additional contributions in the equations to guarantee thermodynamic structure. As GENERIC building blocks, we use a Poisson matrix for the basic convection behavior and antisymmetric friction matrices to correct for additional convective transport terms. The ensuing GENERIC-13 equations completely match the Grad-13 equations up to second-order terms in the Knudsen number and fulfill all thermodynamic requirements.

Joint work with H.C. Öttinger, Zürich. Appeared in Phys. Fluids 34, 017105 (2022)

Abstract

In this course we consider projection methods for the numerical approximation of time-dependent incompressible fluid equations. Such schemes are based on the projection structure of the equations due to the incompressibility constraint. An operator splitting leads to a prediction and a correction step in each time step both of which are simpler problems. For this reason (high order) projection methods are of particular interest for large scale simulations. But on the other hand the splitting introduces additional challenges e.g. regarding the boundary conditions.

We start by reviewing projection methods for viscous flows (Navier-Stokes and Stokes equations) dating back to Chorin and Temam and compare them to mixed methods. Then we take a look at what has been done for inviscous equations (Euler). Finally we present the Green-Naghdi model that is derived from the incompressible free surface Euler equations. We discuss how its projection structure helps to pose meaningful boundary conditions.

Abstract

In this course we consider projection methods for the numerical approximation of time-dependent incompressible fluid equations. Such schemes are based on the projection structure of the equations due to the incompressibility constraint. An operator splitting leads to a prediction and a correction step in each time step both of which are simpler problems. For this reason (high order) projection methods are of particular interest for large scale simulations. But on the other hand the splitting introduces additional challenges e.g. regarding the boundary conditions.

We start by reviewing projection methods for viscous flows (Navier-Stokes and Stokes equations) dating back to Chorin and Temam and compare them to mixed methods. Then we take a look at what has been done for inviscous equations (Euler). Finally we present the Green-Naghdi model that is derived from the incompressible free surface Euler equations. We discuss how its projection structure helps to pose meaningful boundary conditions.

Abstract

In this course we consider projection methods for the numerical approximation of time-dependent incompressible fluid equations. Such schemes are based on the projection structure of the equations due to the incompressibility constraint. An operator splitting leads to a prediction and a correction step in each time step both of which are simpler problems. For this reason (high order) projection methods are of particular interest for large scale simulations. But on the other hand the splitting introduces additional challenges e.g. regarding the boundary conditions.

We start by reviewing projection methods for viscous flows (Navier-Stokes and Stokes equations) dating back to Chorin and Temam and compare them to mixed methods. Then we take a look at what has been done for inviscous equations (Euler). Finally we present the Green-Naghdi model that is derived from the incompressible free surface Euler equations. We discuss how its projection structure helps to pose meaningful boundary conditions.

Abstract

TBA

Abstract

Statistical inverse problems lead to complex optimisation and/or Monte Carlo sampling problems. Gradient descent and Langevin samplers provide examples of widely used algorithms. In my talk, I will discuss recent results on sampling algorithms, which can be viewed as interacting particle systems, and their mean-field limits. I will highlight the geometric structure of these mean-field equations within the, so called, Otto calculus, that is, a gradient flow structure in the space of probability measures. Affine invariance is an important outcome of recent work on the subject, a property shared by Newton’s method but not by gradient descent or ordinary Langevin samplers. The emerging affine invariant gradient flow structures allow us to discuss coupling-based Bayesian inference methods, such as the ensemble Kalman filter, as well as invariance-of-measure-based inference methods, such as preconditioned Langevin dynamics, within a common mathematical framework. Applications include nonlinear and logistic regression.