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Time consistency of the mean-risk problem B 349 (Theresienstr. 39, 80333 München)

Multivariate risk measures appear naturally in markets with transaction costs or when measuring the systemic risk of a network of banks. Recent research suggests that time consistency of these multivariate risk measures can be interpreted as a set-valued Bellman principle. And a more general structure emerges that might also be important for other applications and is interesting in itself. In this talk I will show that this set-valued Bellman principle holds also for the dynamic mean-risk portfolio optimization problem. In most of the literature, the Markowitz problem is scalarized and it is well known that this scalarized problem does not satisfy the (scalar) Bellman principle. However, when we do not scalarize the problem, but leave it in its original form as a vector optimization problem, the upper images, whose boundary is the efficient frontier, recurse backwards in time under very mild assumptions. I will present conditions under which this recursion can be exploited directly to compute a solution in the spirit of dynamic programming and will state some open problems and challenges for the general case. (Joint work with Gabriela Kováčová)

An Integrated Model of Fire Sales and Default Contagion in Financial SystemsB 349 (Theresienstr. 39, 80333 München)

In (Detering et. al, 2016) and (Detering et. al, 2018) we developed a random graph model for 'default contagion' in financial networks and using 'law of large numbers' effects we were able to compute the size of the final default cluster induced by an arbitrarily given initial shock in a large system. Further, we were able to derive sufficient and necessary criteria for resilience of a system to small shocks. In that sense our model provides better insights than the popular Eisenberg-Noe model which is concerned with existence and uniqueness of a clearing vector (and hence the final state of the system) but gives no indication of favorable network structures and sufficient capital requirements to ensure resilience. The Eisenberg-Noe model, however, has proven to be flexible enough to be extended by contagion channels other than default contagion, the most important being 'fire sales' which describes contagion effects due to falling asset prices as institutions sell off their assets.

In this article, we first propose a model for fire sales that uses an Eisenberg-Noe like description for finite networks but allows to describe the final state of the system (size of the default cluster and final price impact) asymptotically. In particular, we are able to provide sufficient capital requirements that ensure resilience of the system. Furthermore, we integrate the channel of default contagion into our model applying results from (Detering et. al, 2018) and extending them to the non-continuous case induced by the fire sales. Finally, for this integrated setting, we provide criteria that determine whether a certain financial system is resilient or prone to small initial shocks and furthermore give sufficient capital requirements for financial systems to be resilient.

This is joint work with Thilo Meyer-Brandis, Konstantinos Panagiotou and Daniel Ritter

tbaMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Thermally induced lesions and signal modulation in cardiomyocyte networksMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

In most biological systems, global average temperatures are a highly regulated and controlled quantity. As such, the majority of previous research regarding the influence of temperature on biological systems uses variations of ambient temperatures. Recent discussions, however, have highlighted the possibility of highly localized temperature gradients as an important factor in biological systems. In our research, we employ microstructured conductors to generate temperature gradients on the micrometer scale. As a biological model system, we culture cardiomyocyte-like cells on our chips. These cells show propagating waves in intracellular calcium concentration. In my presentation, I will illustrate how we use fluorescent methods to analyze the signal propagation in these networks. Further, I will introduce the application of highly localized temperature gradients to interrupt the propagation, influence the signal velocity, and relocate the signal origin in these networks.

TBAB 252 (Theresienstr. 39, 80333 München)

TBA

Sharpness of the Phase Transition in the Contact ProcessB 252 (Theresienstr. 39, 80333 München)

The contact process is an interacting particle system which can be interpreted as the spread of an infection. In this talk, we focus on the contact process on the two-dimensional integer lattice and consider the percolation transition. That is, we examine the size of the occupied cluster of the origin subject to the upper invariant measure. It is well known that there is a critical value \lambda_c such that for all infection rates bigger than \lambda_c, the upper invariant measure is non-trivial. Furthermore, there is another critical value \lambda_p such that the probability of the aforementioned cluster being infinite is bigger than zero for all infection rates bigger than \lambda_p. However, if the infection rate is smaller than \lambda_p, the distribution of the size of the cluster has an exponential tail. We sketch a proof of this result using techniques as in the proof of a related result for confetti percolation. A short introduction to the contact process will be given at the beginning of the talk.

Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applicationsMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

We propose a new variational model in weighted Sobolev spaces with non-standard weights and applications to image processing. We show that these weights are, in general, not of Muckenhoupt type and therefore the classical analysis tools may not apply. For special cases of the weights, the resulting variational problem is known to be equivalent to the fractional Poisson problem. The trace space for the weighted Sobolev space is identified to be embedded in a weighted $L^2$ space. We propose a finite element scheme to solve the Euler-Lagrange equations, and for the image denoising application we propose an algorithm to identify the unknown weights. The approach is illustrated on several test problems and it yields better results when compared to the existing total variation techniques.

Spectral localization vs. homogenization in the random conductance modelB 252 (Theresienstr. 39, 80333 München)

We study the asymptotic behavior of the top eigenvectors and eigenvalues of the random conductance Laplacian in a large domain of Zd (d≥2) with zero Dirichlet conditions. Let the conductances w be positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. Then we show that the spectrum of the Laplacian displays a sharp transition between a completely localized and a completely homogenized phase. A simple moment condition distinguishes between the two phases. In the homogenized phase we can even generalize our results to stationary and ergodic conductances with additional jumps of arbitrary length. Here, our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence. The investigation of the homogenized phase is joint work with M. Slowik and M. Heida.

TBAB 252 (Theresienstr. 39, 80333 München)

TBA

The Gradient Discretisation Method: framework and tools for the numerical analysis of elliptic and parabolic modelsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

The gradient discretisation method (GDM) is a generic framework for the design and convergence analysis of numerical schemes for elliptic and parabolic PDEs. The GDM is built on the choice of a few discrete elements, together called a "gradient discretisation":

- a finite dimensional space X,

- an operator that reconstructs, from a vector in X, a function over the physical domain

- an operator that reconstructs, from a vector in X, a vector-valued function (the "gradient") over the physical domain

Substituting, in lieu of the corresponding continuous space and operators, these discrete elements in the weak formulation of the PDE gives rise to a numerical scheme called the "gradient scheme" (GS).

Three to five properties only (depending on the model) on sequences of GDs ensure that the corresponding GSs converge for a wide range of models, from linear elliptic and parabolic PDEs to non-linear models including p-Laplace equations, the Stokes and Navier–Stokes equations, degenerate parabolic equations, and optimal control problems with elliptic state equations.

Specific choices of GDs correspond to specific schemes, and the GDM analysis therefore seamlessly applies to many numerical methods, from finite elements (conforming, non-conforming and mixed, including mass-lumped versions), finite volume methods, to virtual element methods, etc.

In this talk, I will give an overview of the GDM, the numerical methods it encompasses, and the results it yields. I will also cover some generic compactness results developed alongside the GDM to facilitate the convergence analysis for non-linear time-dependent problems. Some of the noticeable results stemming from the GDM include a response to a long-standing conjecture on the super-convergence of the Two-Point Flux Approximation finite volume method (popular in petroleum engineering), and a novel uniform-in-time convergence result for degenerate parabolic equations.

Reference: [1] Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, and Raphaèle Herbin. The gradient discretisation method. 511p, 2018. Mathematics & Applications, Springer, Heidelberg. To appear. URL: https://hal.archives-ouvertes.fr/hal-01382358.

Compact Course: Basic principles of variational analysis and generalized differentiation with applications to nonsmooth optimization and optimal controlMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Methods of variational analysis and generalized differentiation play an increasing role in modern optimization and optimal control. The main goal of this course is to overview basic variational principles and advanced constructions of generalized differentiation that flourish a rather comprehensive theory with numerous applications to optimization and control. Among applications, we plan to consider some classes of nondifferentiable and bilevel programming and optimal control of the sweeping process.

For further details: https://igdk1754.ma.tum.de/IGDK1754/CCMordukhovich