Talks

We show the existence of solutions of nonlinear non-autonomous Cauchy problems
\[
\partial_t u(t,x) - \nabla_x \cdot (a(t,x)\nabla_xu(t,x))= f(t,x,u(t,x),\nabla u(t,x)),
\qquad u(0,\cdot)=u_0
\]
for a bounded open set $\Omega\subseteq \mathbb R^n$.
The coefficient matrix $a$ is supposed to be symmetric, uniformly elliptic,
Lipschitz continuous w.r.t.\ $t\in(0,\tau)$ and measurable w.r.t.\ $x\in\Omega$;
the nonlinearity $f$ is required to satisfy a linear growth condition.
We show that, given $u_0\in H_0^1(\Omega)$, there exists $u\in L_2(0,\tau;H_0^1(\Omega))
\cap H^1(0,\tau;L_2(\Omega))$ solving the problem mentioned above.

The proof relies on Schaefer's fixed point theorem. In the
course of the proof one uses maximal regularity properties of solutions of
inhomogeneous linear problems and compact embeddings of vector-valued Sobolev spaces.

The result partly generalises [ArCh10].

The talk is based on joint work with Wolfgang Arendt and Jürgen Voigt.


[ArCh10] W. Arendt, R. Chill: Global existence for quasilinear
diffusion equations in isotropic nondivergence form. Ann. Sc. Norm. Super. Pisa
Cl. Sci. (5) IX, 523-539 (2010).

The question of possible generalisations of the operation of differentiation towards fractional powers can be traced back to a letter from L'Hospital to Leibniz in 1695 ([1]).
Since this time, mathematicians developed plenty of different approaches to fractional differentiation and integration generalising different aspects of the known theory.
The possibly most prominent examples are the fractional derivatives (and integrals) of Riemann-Liouville and Weyl ([4]).
Both can also be understood as instances of the sectorial functional calculus of sectorial operators as it was introduced in [2] and further promoted in [3].
Nonetheless, a direct use of the abstract techniques from operator theory seems to be rare in applications.
Therefore, the talk aims for introducing the audience in the basic principles of functional calculus and how to use it to recover the above mentioned instances of fractional derivatives.


$\mathbf{References}$

[1] B. Ross. The Development of Fractional Calculus 1695--1900. $\mathit{\text{Historia Math., 4(1):}}$ 75--89, 1977.

[2] A. McIntosh. Operators which have an $H_{\infty}$ functional calculus. $\mathit{\text{Miniconference on operator theory and partial differential equations:}}$ 210--231, 1986.

[3] M. Haase. $\mathit{\text{The Functional Calculus for Sectorial Operators,}}$ volume 169 of $\mathit{\text{Operator Theory: Advances and Applications.}}$ Birkhäuser Basel, 2006.

[4] K. S. Miller and B. Ross. $\mathit{\text{An Introduction to the Fractional Calculus and Fractional Differential Equations.}}$ John Wiley & Sons, 1993.

Im Maschinen- und Automobilbau werden für mechanisch extrem beanspruchte, temperatur- und chemikalienbeständige Bauteile Elastomerwerkstoffe benötigt. Typische Beispiele sind Reifen, Zahn- und Keilriemen, Motor- und Aggregatelager, Luftfedern und Fahrwerkbuchsen sowie statisch und dynamisch beanspruchte Dichtungen. Die Anforderungen bezüglich Zuverlässigkeit und Leistungsdichte sind speziell in Verbindung mit den hochsensiblen, sicherheitsrelevanten Funktionen der Bauteile besonders hoch.
Insbesondere im Rahmen der Digitalisierung in der Produktion gewinnen Simulationsmodelle verstärkt an Bedeutung. Viele Verarbeitungsschritte in der Herstellung von Elastomerbauteilen beginnend mit dem Mischen, dem Walzen und der Extrusion oder des Spritzgießens, über die Vulkanisation beeinflussen die endgültigen mechanischen Eigenschaften. Im Laufe ihres Einsatzlebens verändern sich diese Eigenschaften auf Grund von thermo-oxidativer Alterung, so dass auch Lebensdauervorhersagen zur einer Herausforderung werden. Die zuverlässige Erstellung von „Digitalen Zwillingen“ für Elastomerbauteile bedarf so einer Beschreibung vieler auch untereinander gekoppelter Effekte.
Dieser Vortrag bietet Einblicke in verschiedene Modellierungsansätze einzelner Abschnitte des Leben von Elastomeren. Hauptfokus ist hierbei die Beschreibung der mechanischen Eigenschaften unter Berücksichtigung der Vernetzung und Alterung.

Language of the talk is going to be either German or English depending on the audience preferences.

Positive Kernels provide methods to solve multivariate interpolation problems, but standard kernel methods usually lead to ill-conditioned linear systems. Therefore, a suitable choice of interpolation centers and basis functions reqiures particular care.

In this talk, i will give an introduction to kernel based approximation and discuss greedy point selection strategies, which will improve the stability of the interpolation method.

Compressed Sensing deals with reconstructing some unknown vector from few linear measurements in high dimension by additionally assuming sparsity, i.e. many entries are zero. Recent results guaranteed recovery even when just signs of the measurements are available (one-bit CS). A natural generalization of classical CS replaces sparse vectors by vectors lying on manifolds having low intrinsic dimension. In this talk I introduce the one-bit problem and proposes a tractable strategy to solve one-bit CS problems for data lying on manifolds. This is based on joint work with Johannes Maly and Mark Iwen.

Large-scale optimization problems governed by partial differential equations (PDEs) occur in a multitude of applications, for example in inverse problems for non-destructive testing of materials and structures, or in individualized medicine. Algorithms for the numerical solution of such PDE-constrained optimization problems are computationally extremely demanding, as they require multiple PDE solves during the iterative optimization process. This is especially challenging for transient problems, where methods working on the reduced objective functional are often employed to avoid a full spatio-temporal discretization of the associated optimality system. The evaluation of the reduced gradient then requires one solve of the state equation forward in time, and one backward-in-time solve of the adjoint equation. In order to tackle real-life applications, it is not only essential to devise efficient discretization schemes, but also to use advanced techniques to exploit computer architectures and decrease the time-to-solution, which otherwise is prohibitively long.

One approach is to utilize the increasing number of CPU cores available in current computers. In addition to more common spatial parallelization, time-parallel methods are receiving increasing interest in the last years. There, iterative multilevel schemes such as PFASST (Parallel Full Approximation Scheme in Space and Time) are currently state of the art and achieve significant parallel efficiency. In this talk, we investigate approaches to use PFASST for the solution of parabolic optimal control problems. Besides enabling time parallelism, the iterative nature of the temporal integrators within PFASST provides additional flexibility for reducing the cost of solving nonlinear equations, re-using previous solutions in the optimization loop, and adapting the accuracy of state and adjoint solves to the optimization progress. We discuss benefits and difficulties, and present numerical examples.

This is joint work with Michael Minion (Lawrence Berkeley National Lab).

Molecular-continuum methods, as referred to in my talk, employ a domain decomposition and compute fluid flow either by means of molecular dynamics (MD) or computational fluid dynamics (CFD) in the sub-domains. This enables multiscale investigations of nano- and microflows beyond the limits of validity of classical CFD.

In my talk, I will focus on latest developments in the macro-micro-coupling tool (MaMiCo). MaMiCo enables the coupling of arbitrary CFD and MD solvers, hiding the entire coupling algorithmics from the actual single-scale solvers. After a brief discussion of the limits of the MD method, I will focus on various aspects of the molecular-continuum coupling and its realization in MaMiCo, including parallelization, multi-instance sampling for MD (that is ensemble averaging) and filtering methods that extract smooth responses from the fluctuating MD description to enhance consistency on the side of the continuum solver. I will further present preliminary results from a study which aims to generate open boundary force models for MD using machine learning.