Das Mathematische Kolloquium ist eine gemeinsame wissenschaftliche Veranstaltung des gesamten Mathematischen Instituts. Es steht allen Interessierten offen und richtet sich neben den Mitgliedern und Mitarbeitern des Instituts auch an die Studierenden. Das Kolloquium findet dreimal im Semester am Donnerstag um 15:00 s.t. im Hörsaal II, Albertstr. 23b statt. Danach (gegen 16:15) gibt es Kaffee und Kekse, zu dem der vortragende Gast und alle Besucher eingeladen sind.
Numerical Analysis of Implicitly Constituted Incompressible Flow
Montag, 18.1.21, 08:30-09:30, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
In the classical theory of fluid mechanics a linear relationship between the shear stress\ntensor and the symmetric velocity gradient is often assumed. Even when a nonlinear\nrelationship is assumed between the shear stress and the symmetric velocity gradient,\nit is typically formulated in terms of an explicit relation. Implicit constitutive models\nprovide a theoretical framework that generalises this, allowing for an implicit\nconstitutive relation. In this talk, I will present some results dealing with the finite\nelement approximation of implicitly constituted incompressible flow, ranging from\nconvergence aspects to fast solvers. In particular, I will introduce a preconditioner\nbased on augmented Lagrangian stabilisation and a specialised multigrid algorithm\nthat exhibits robust behaviour even for models incorporating thermal effects.
The benefits of smoothness in Isogeometric Analysis
Montag, 18.1.21, 11:00-12:00, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
Splines are piecewise polynomial functions that are glued together with a given\nsmoothness. When using them in a numerical method, the availability of proper error\nestimates is of utmost importance. Classical error estimates for spline approximation\nare expressed in terms of\n\n(a) a certain power of the maximal grid spacing,\n\n(b) an appropriate derivative of the function to be approximated, and\n\n(c) a "constant" which is independent of the previous quantities but usually depends on the degree and smoothness of the spline.\n\nAn explicit expression of the constant in (c) is rarely available in the literature, because\nit is a minor issue in standard approximation analysis. There they are mainly\ninterested in the approximation power of spline spaces of a fixed degree. However, one\nof the most interesting features in the emerging field of Isogeometric Analysis is krefinement,\nwhich denotes degree elevation with increasing interelement smoothness.\nThe above mentioned error estimates are not sufficient to explain the benefits of\napproximation under k-refinement so long as it is not well understood how the\nconstant in (c) behaves.\nIn this talk we provide error estimates for k-refinement on arbitrary grids with an\nexplicit constant that is, in certain cases, sharp. These estimates are in fact good\nenough to cover convergence to eigenfunctions of classical differential operators. This\nforms a theoretical foundation for the outperformance of smooth spline discretizations\nof eigenvalue problems that has been numerically observed in the literature.\nSeite 2\nMoreover, we discuss how these error estimates can be used to mathematically justify the\nbenefits of spline approximation under k-refinement. Specifically, by comparing the constant\nfor spline approximation of maximal smoothness with a lower bound on the constant for\ncontinuous and discontinuous spline approximation, we show that k-refinement provides better\napproximation per degree of freedom in almost all cases of practical interest.\nThis talk is based on work performed in collaboration with Andrea Bressan and Carla Manni\nand Hendrik Speleers.
Multispecies kinetic modelling and mathematical theory for physical applications
Dienstag, 19.1.21, 08:30-09:30, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
My goal is to model certain physical problems that can be described by kinetic partial\ndifferential equations. Our models allow for efficient numerical simulations. Examples\nof the physics models are inertial confinement fusion and the re-entry problem of a\nspace vehicle. Using mathematical theory we verify well posedness and essential\nphysical properties of the model. From the modeling point of view this requires\nextending existing models in the literature by extending them to gas mixtures or by\nincluding degrees of freedom in internal energy, chemical reactions and quantum\neffects. This leads to new difficulties in their theoretical study.\nIn this talk, I will present recent results on existence and large time behaviour of\nsolutions to kinetic equations for gas mixtures; entropy inequality and large-time\nbehaviour of kinetic equations with degrees of freedom in internal energy and chemical\nreactions; and results on entropy minimization problems leading to equations with\nconservation of mass, momentum and energy when we deal with velocity dependent\ncollision frequencies.
Approximation of the Willmore energy by a discrete geometry model
Dienstag, 19.1.21, 11:00-12:00, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
In this joint work with Heiner Olbermann (UCLouvain), we study the discrete bending\nenergy of Grinspun et al defined for triangular complexes, and show that varying over\nall complexes with the Delaunay property, the minimal bending energy converges, as\nthe size of triangles tends to zero, to a version of the Willmore energy, in the sense of\nGamma-convergence. We show also that the Delaunay property is essential to\nguarantee the lower energy bound. Our article combines results from finite difference\nmethods, discrete geometry, and geometric measure theory.
Compensated Compactness and L1-estimates
Dienstag, 19.1.21, 14:00-15:00, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
In the first part of the talk we will review some recent results on the Murat--Tartar\nframework of Compensated Compactness Theory, by which we mean weak (lower\nsemi-)continuity of nonlinear functionals interacting with weakly convergent\nsequences of PDE constrained vector fields. We present improvements of the original\nwork of Murat and Tartar, as well as more recent work of Fonseca--Müller. We also\npresent answers to questions of Coifman--Lions--Meyer--Semmes and De Philippis.\nThe second part of the talk will concern properties of solutions of linear systems \( L u\n= \bmu \), where \( \bmu \) is a Radon measure, a borderline case not covered by\nCalderón--Zygmund Theory. We build on the fundamental work of Bourgain--Brezis--\nMironescu and Van Schaftingen towards surprising strong interior Sobolev estimates\nfor solutions. We also discuss the start of a theory towards estimates up to the\nboundary. The final part of the talk will cover fine properties of solutions and possible\nintersections with Geometric Measure Theory.
Numerical methods for conservation laws with nonlocal and discontinuous fluxes
Freitag, 22.1.21, 08:30-09:30, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
The field of hyperbolic conservation laws is a cornerstone in PDE theory. In my talk I\nwill present recent results pertaining nonlocal conservation laws as well as\nconservation laws with discontinuous flux.\nIn the first part of the talk I will present new regularity results for a class of nonlo-cal\nconservation laws. These results motivate the design of higher-order numerical\nschemes which are asymptotically compatible with the underlying local conservation\nlaw. I will detail the construction of a second-order numerical scheme that generalizes\nthe class of second-order reconstruction-based schemes for local conservation laws. It\ncan be shown that the second-order scheme converges towards a weak solution, and—\nunder certain assumptions on the nonlocal interaction kernel—even towards the\nunique entropy solution of the nonlocal conservation law. Such a result is currently\nout of reach for local conservation laws.\nIn the second part of the talk I will focus on conservation laws with discontinuous flux\nwhich has been an active research area during the last several decades. Many selection\ncriteria to single out a unique weak solution have been proposed in this context and\nseveral numerical schemes have been designed and analyzed in the literature.\nSurprisingly, the preexisting literature on convergence rates for such schemes is\npractically nonexistent. In this talk, focusing on so-called adapted entropy solutions,\nI will present the first-ever convergence rate results for finite volume and front tracking\nmethods as well as a flux-stability result. As an application, these results can be used\nfor uncertainty quantification in two-phase reservoir simulations for reservoirs with\nvarying geological properties.
Asymptotic Behavior of Gradient Flows and Nonlinear Spectral Theory
Freitag, 22.1.21, 11:00-12:00, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
In this talk I will discuss the gradient flow of absolutely p-homogeneous convex\nfunctionals on a Hilbert space and show that asymptotic profiles of the solution are\neigenfunctions of the subdifferential operator of the functional. This work applies, for\ninstance, to local and nonlocal versions of PDEs like p-Laplacian evolution equations,\nthe porous medium equation, and fast diffusion equations, herewith generalizing\nmany results from the literature to an abstract setting. Then I discuss the eigenvalue\nproblem associated to a infinity-Dirichlet energy in some more detail, show relations\nto distance functions, and speak about a discrete-to-continuum limit for this model\nusing Gamma-convergence. I conclude with perspectives and future work.
Space-time deep neural network approximations for high-dimensional PDEs
Freitag, 22.1.21, 14:00-15:00, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
It is one of the most challenging issues in applied mathematics to approximately solve\nhigh-dimensional partial differential equations (PDEs) and most of the numerical\napproximation methods for PDEs in the scientific literature suffer from the so-called\ncurse of dimensionality (CoD) in the sense that the number of computational\noperations employed in the corresponding approximation scheme to obtain\nan approximation precision \(\bvarepsilon >0\) grows exponentially in the PDE\ndimension and/or the reciprocal of \(\bvarepsilon\). Recently, certain deep learning\nbased approximation methods for PDEs have been proposed and various numerical\nsimulations for such methods suggest that deep neural network (DNN) approximations\nmight have the capacity to indeed overcome the CoD in the sense that the number of\nreal parameters used to describe the approximating DNNs grows at most polynomially\nin both the PDE dimension \(d \bin \bN\) and the reciprocal of the prescribed\napproximation accuracy \(\bvarepsilon >0\). There are now also a few rigorous\nmathematical results in the scientific literature which substantiate this conjecture by\nproving that DNNs overcome the CoD in approximating solutions of PDEs. Each of\nthese results establishes that DNNs overcome the CoD in approximating suitable PDE\nsolutions at a fixed time point \(T >0\) and on a compact cube \([a, b]^d\) but none of\nthese results provides an answer to the question whether the entire PDE solution on\n\([0, T] \btimes [a, b]^d\) can be approximated by DNNs without the CoD.\nIn this talk we show that for every \(a \bin \bR\), \( b \bin (a, \binfty)\) solutions\nof suitable Kolmogorov PDEs can be approximated by DNNs on the space-time region\n\([0, T] \btimes [a, b]^d\) without the CoD.
On the space of metrics with invertible Dirac operator
Donnerstag, 25.2.21, 15:00-16:00, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
Ammann, Dahl and Humbert showed that the property that a manifold admits a metric with invertible Dirac operator persists under the right surgeries. That is the Dirac-counterpart of the Gromov-Lawson construction on the question of existence of postive scalar curvature metrics and has also implications on this question. \nWe consider now the question whether we can also obtain a homotopy equivalence statement for spaces of metrics with invertible Dirac operator under surgery in the spirit of the positive scalar curvature result by Chernysh/Walsh. This is joint work with N. Pederzani.\n