Adaptively estimating nonlinear single index models with monotonic link functionsMI 02.10.011 (Boltzmannstr. 3, 85748 Garching)

The single index model (SIM) is a popular tool for modeling data where the output depends on the features through a linear 1D projection. If data follows this model, efficient estimation at 1D minimax rates is possible. However, the model space is restricted by the assumed linearity. In this talk we introduce a generalized SIM that allows for projections onto 1D manifolds. We propose an estimator based on local linear regression, where the localization happens in the output domain instead of in the feature domain. Finally, we present theoretical guarantees and numerical studies on real data sets to support the usefulness of a nonlinear SIM.

Nonstationary Joint ExtremesBC1 2.02.01 (Parkring 11, 85748 Garching)

In this talk, I will discuss key ideas on time-changing extremal dependence structures. Extremal dependence between international stock markets is of particular interest in today’s global financial landscape. However, previous studies have shown this dependence is not necessarily stationary over time. We concern ourselves with modeling extreme value dependence when that dependence is changing over time, or other suitable covariate. Working within a framework of asymptotic dependence, we introduce a regression model for the angular density of a bivariate extreme value distribution that allows us to assess how extremal dependence evolves over a covariate. We apply the proposed model to assess the dynamics governing extremal dependence of some leading European stock markets over the last three decades, and find evidence of an increase in extremal dependence over recent years.

Pathwise Pricing-Hedging DualityBC1 2.02.01 (Parkring 11, 85748 Garching)

We discuss weak topologies on the Skorokhod space of cadlag functions. In particular, we study the weak* topology they induce on the family of probability measures on the canonical space and give applications to the pathwise pricing-hedging duality. We also discuss related open problems.

Financial risk measures for a network of individual agents holding portfolios of light-tailed objectsBC1 2.02.01 (Parkring 11, 85748 Garching)

In this talk we consider a financial network of agents holding portfolios of independent light-tailed risky objects with losses assumed to be asymptotically exponentially distributed with distinct tail parameters. The derived asymptotic distributions of portfolio losses refer to the class of functional exponential mixtures. We also provide statements for Value-at-Risk and Expected Shortfall measures as well as for their conditional counterparts. We establish important qualitative differences in the asymptotic behavior of portfolio risks under light tail assumption compared to heavy tail settings which should be accounted for in practical risk management. (joint work with Claudia Klüppelberg)

Yet another „proof“ that something is wrong with Bohmian mechanicsB 133 (Theresienstr. 39, 80333 München)

I discuss the recent no-go theorem of Frauchiger and Renner based on an ``extended Wigner's friend'' thought experiment which is supposed to show that any single-world interpretation of quantum mechanics leads to inconsistent predictions if it is applicable on all scales. I show that no such inconsistency occurs if one considers a complete description of the physical situation. I then discuss implications of the thought experiment that have not been clearly addressed in the original paper, including a tension between relativity and nonlocal effects predicted by quantum mechanics. My analysis applies in particular to Bohmian mechanics, showing that it provides a perfectly consistent description of the thought experiment.

Long time behaviour of a reduced model for two-phase flows in porous mediaMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

We consider a thin film approximation for two-phase porous media that is derived thanks to Dupuit’s approximation. The model then boils down to a degenerate cross diffusion system that can be seen as a generalisation of the porous medium equation. This system can be interpreted as the Wasserstein gradient flow of some energy. The solutions converge as time goes to infinity towards some self similar solutions, that can correspond to minimizers of some modified energy after proper rescaling. The shape of these minimizers strongly depends on the parameter range. Finally, we study numerically the convergence rate as time goes to infinity thanks to an energy diminishing finite volume scheme. (joint work with A. Ait Hammou Oulhaj, C. Chainais-Hillairet and P. Laurençot)

On a New Mixed Formulation of Kirchhoff-Love and Reissner-Mindlin Plates and ShellsGebäude 33, Raum 1431 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

In 2D linear elasticity it is well-known that a stress tensor field satisfying the homogeneous equilibrium equation can be expressed in terms of the Airy stress function provided the domain is topologically simple. There is a similar result for the bending moment tensor field in plate models, if the mid-surface of the plate is simply connected. While the stress tensor field in 2D inear elasticity can be written as a second-order differential operator applied to the Airy stress function, the bending moment tensor field in plate models is only a first-order differential operator applied to some 2D vector field. We will show how this result can be used to reformulate the Kirchhoff-Love and the Reissner-Mindlin plate and shell models as well-posed second-order systems. The reformulation of the plate and shell models as second-order systems allows for discretization methods in approximation spaces with continuous functions. This includes standard continuous Lagrangian finite element methods and spline spaces from isogeometric analysis on multi-patch domains with continuous patching only.

Critical points of quadratic low-rank optimization problemsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

The absence of spurious local minima in many non-convex minimization problems, e.g. in the context of compressed sensing, has recently triggered a lot of interest due to its important implications on the global convergence of optimization algorithms. One example is low-rank matrix sensing under rank restricted isometry properties. It can be formulated as a minimization problem for a quadratic cost function constrained to a low-rank matrix manifold, with a positive semidefinite Hessian acting like a perturbation of identity on cones of low-rank matrices. We present an approach to show strict saddle point properties and absence of spurious local minima for such problems under improved conditions on the restricted isometry constants. This is joint work with Bart Vandereycken.

Denoising and Completion of Structured Low-Rank Matrices via Iteratively Reweighted Least SquaresMI 02.10.011 (Boltzmannstr. 3, 85748 Garching)

In this talk, we propose and analyze a new Iteratively Reweighted Least Squares (IRLS) algorithm for the problems of completion and approximation of a matrix by low-rank matrices that are low-rank and also linearly structured. For the particular case of Hankel or Toeplitz matrix completion, which has applications for the super resolution or harmonic retrieval problem, we establish local convergence of the algorithm with a quadratic rate of convergence, under appropriate assumptions on the sample complexity. These assumptions on the sample complexity match the weakest ones available in the literature. At the same time, we provide numerical experiments demonstrating that our approach beats the state-of-the-art in terms of data efficiency. This is based on joint work with Claudio Mayrink Verdun.

Variational models for microstructures in shape-memory alloysMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Shape-memory alloys are special materials that undergo a martensitic phase transition, that is, a diffusionless first order solid-solid phase transformation. Pattern formation in shape-memory alloys is often studied in the framework of the calculus of variations. The formation of microstructures is typically explained as result of a competition between a bulk elastic energy and a higher order surface energy. I shall discuss some recent analytical results on such variational problems, in particular regarding needle-like microstructures and stress-free inclusions.

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Direct interaction along light cones at the quantum levelB 133 (Theresienstr. 39, 80333 München)

Here I explain the idea that direct interactions along light cones, similar to the Wheeler-Feynman formulation of electrodynamics, can be implemented on the quantum level using integral equations for multi-time wave functions. Multi-time wave functions are wave functions psi(x_1,...,x_N) with N spacetime arguments x_i for N particles. The crucial point is that the N time variables of the x_i make it possible to express interactions with time delay, as relativity requires. Starting from the integral formulation of the non-relativistic Schrödinger equation, I derive a covariant integral equation as an evolution equation for psi, and discuss its mathematical structure. It is shown that the equation correctly reduces to the Schrödinger equation with a Coulomb potential when time delay effects are neglected. The main mathematical results are existence and uniqueness theorems for a simplified version of the equation. This talk is partly about joint work with Roderich Tumulka.

Approximate curvatures of a varifoldMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Varifolds, i.e. Radon measures on the Grassmannian bundle of unoriented tangent d-planes of a Riemannian n-manifold M, represent a variational generalization of unoriented, d-dimensional submanifolds of M. By a suitable extension of classical variation operators, we introduce a notion of approximate second fundamental form that is well-defined for a generic varifold. Rectifiability, compactness, and convergence results are proved, showing in particular the consistency and stability of approximate curvatures with respect to varifold convergence. If restricted to the case of "discrete varifolds", this theory provides a general framework for extracting key features from discrete geometric data. Some numerical tests on point clouds (evaluation of curvatures and geometric flows, also in presence of noise and singularities) will be shown. We shall finally discuss some future perspectives and open problems. This is a joint research with Blanche Buet (Univ. Paris XI - Orsay) and Simon Masnou (Univ. Lyon 1).

A review on some mathematical results about crystallizationMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

In this talk, I will review some results on the crystallization conjecture, that is, the mathematical proof of the fact that, under appropriate conditions, interacting particles place themselves into periodic configurations. I will first review classical models at zero temperature, in which few results have been proved. Apart from the question of the emergence of periodicity, determination of the optimal lattice, in link with special functions, will be addressed. Then, some problems regarding the quantum case, in which the notion of crystalline order needs to be defined in a different way. Similarities with positive temperature classical models will be outlined. This is a joint work with M. Lewin (Univ. Paris Dauphine).

Numerical methods in stochastic homogenizationRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to numerical solutions, since the latter are nonlocal functions of the former. In this talk, I will also discuss about a new method for computing solutions of elliptic equations with random rapidly oscillating coefficients. This is a joint work with S. Armstrong, A. Hannukainen and J.-C. Mourrat.

Finite crystallization for ionic dimersRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

In this talk we will present some crystallization results of ionic dimers. In particular we consider finite discrete systems consisting of two different atomic types and investigate ground-state congurations for congurational energies featuring two- body short-ranged particle interactions. The atomic potentials favor some reference distance between different atomic types and include repulsive terms for atoms of the same type, which are typical assumptions in models for ionic dimers. Our goal is to show a two-dimensional crystallization result. More precisely, we give conditions in order to prove that energy minimizers are connected subsets of the hexagonal lattice where the two atomic types are alternately arranged in the crystal lattice. We also provide explicit formulas for the ground-state energy. Finally, we characterize the net charge, that means, the difference of the number of the two atomic types. Analyzing the deviation of congurations from the hexagonal Wulff shape, we prove that for ground states consisting of n particles the net charge is at most of order O(n^1/4) where the scaling is sharp.

Nonlocal PDE System arising in Vegetation Pattern Formation - How fairy circles appearMI 01.13.010 (Boltzmannstr. 3, 85748 Garching)

For a better understanding of vegetation patterns the idea of a description via systems of partial differential equations (PDE) - also with non-local terms - came up. This type of PDE had become more popular over the last decade, however, analysis results are still rare. In my Master Thesis I studied a special system and have shown existence and uniqueness results for a larger class of problems. I also established an approximation of their non-local term, which is specially geared to the transition from non-local to local PDE. In the end a numerical simulation was indeed capable of showing that so called “fairy circles” (a unexplainable vegetation pattern) do appear within the given vegetation-model.

TBA2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

TBA

Hard-core model on 2D latticesMI 03.10.011 (Boltzmannstr. 3, 85748 Garching)

It is well-known that in R2 the maximum-density configuration of hard-core (non-overlapping) disks of diameter D is given by a triangular/hexagonal arrangement (Fejes Tóth, Hsiang). If the disk centers are placed at sites of a lattice, say, a unit triangular lattice L2 or a unit square lattice Z2, then we get a discrete analog of this problem, with the Euclidean exclusion distance. I will discuss high-density Gibbs/DLR measures for the hard-core model on L2 and Z2 for a large value of fugacity z. According to the Pirogov-Sinai theory, the extreme Gibbs measures are obtained via a polymer expansion from dominating ground states. For the hard-core model the ground states are associated with maximally dense sublattices, and dominance is determined by counting defects in local excitations. On L2 we have a complete description of the extreme Gibbs measures for a large z and any D; a convenient tool here is the Eisenstein integer ring. For Z2, the situation is made more complicated by various (related) phenomena: sliding, non-tessellation etc. Here, some results are available; conjectures of various generality can also be proposed. A number of our results are computerassisted. This is a joint work with A. Mazel and Y. Suhov.