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Structural reconstruction using quadratic sparsity penalisationMI 02.08.020 (Boltzmannstr. 3, 85748 Garching)

Exact periodic stripes for minimizers of a local/non-local interaction functional in general dimension, MI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

In this talk we will consider a functional consisting of a perimeter term and a non-local term which are in competition. In the discrete setting such functional was introduced by Giuliani, Lebowitz, Lieb and Seiringer. We show that the minimizers of such functional are optimal periodic stripes for both the discrete and continuous setting. In the discrete setting, such behaviour has been shown by Giuliani and Seiringer using different techniques for a smaller range of exponents. One striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one dimensional. This model has many similarities with the celebrated Ohta-Kawasaki functional. In particular for Ohta-Kawasaki functional, the minimality of periodic stripes is conjectured. This work is in collaboration with Sara Daneri.

Local phenomena in random dynamical systems: bifurcations, synchronisation, and quasi-stationary dynamicsMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

We consider several related topics in the bifurcation theory of random dynamical systems: synchronisation by noise, noise-induced chaos and qualitative changes of fi nite-time behaviour. We study these phenomena with reference to a Hopf bifurcation subject to white noise and the model of a stochastically driven limit cycle on the cylinder. Furthermore, we investigate the bifurcation behaviour of unbounded noise systems in bounded domains, exhibiting the local character of random bifurcations which are usually hidden in the global analysis.

Stability of solutions and ergodicity for stochastic local and nonlocal $p$-Laplace equationsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift given as the subdifferential of a convex function and prove continuous dependence of the solutions with regard to random Mosco convergence of the convex potentials. To this aim, we identify the concept of stochastic variational inequalities (SVI) as a well-suited framework to study such stability properties. In particular, we provide an SVI treatment for stochastic nonlocal $p$-Laplace equations and prove their convergence to the respective local models. Furthermore, ergodicity for local and nonlocal stochastic singular $p$-Laplace equations is proven, without restriction on the spatial dimension and for all $p\in [1, 2)$. This generalizes previous results from [Gess, Tölle; J. Math. Pures Appl. (2014)], [Liu, Tölle; Electr. Comm. Probab. (2011)], [Liu; J. Evol. Equations (2009)]. In particular, the results include the multivalued case of the stochastic (nonlocal) total variation flow. Under appropriate rescaling, the convergence of the unique invariant measure for the nonlocal stochastic $p$-Laplace equation to the unique invariant measure of the local stochastic $p$-Laplace equation is proven.

The talk is based on joint work with Benjamin Gess (MPI Leipzig / Bielefeld University): [Gess, Tölle; J. Differential Equations (2016)], [Gess, Tölle; SIAM J. Math. Anal. (2016)].

Invariant manifolds in the space of tempered distributionsRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

In this presentation we deal with the existence of solutions to stochastic partial differential equations in scales of Hilbert spaces, and show how this is related to the existence of invariant manifolds. As a particular example, we will treat an equation in the space of tempered distributions; here the Hilbert scales are given by Hermite-Sobolev spaces.

Lagrangian Coherent Structures IMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Lagrangian coherent structures (LCSs) are often viewed as „sets that defy strong dispersion“ or „sets that don’t mix with their environment“ under the action of a dynamical system. As such, they can be intuitively understood as regions of regular motion, embedded in an otherwise chaotic/turbulent sea. Over the last two decades, many, mostly phenomenological, theoretical and computational approaches have been developed.

In this two-part lecture, I discuss a recently developed mathematical framework in which LCSs are recognized as almost-invariant sets (or, even stronger, as optimal barriers to diffusive transport) under advection-diffusion as observed from a rather non-standard Lagrangian/material viewpoint. The framework covers deterministic advection-diffusion in and stochastic perturbations of incompressible fluid flows.

Numerical homogenization beyond periodicity and scale separationMI HS 3 (Boltzmannstr. 3, 85748 Garching)

Physical processes in micro-heterogeneous media can often be modelled by partial differential equations (PDEs) with oscillatory coefficients that represent complex material microstructures. Given the multiscale nature of these processes, the construction of computable macroscopic (homogenized or effective) models is crucial for the efficient and reliable numerical simulation in this context. Numerical homogenization is a multiscale method for the derivation and simulation of such macroscopic models. At the example of a prototypical linear elliptic model diffusion problem, this lecture illustrates such an approach. It is based on the computation of operator-dependent subspaces with a quasi-local basis and approximation properties independent of oscillations and roughness of the diffusion coefficient. While constructive approaches in the mathematical theory of homogenization require strong structural assumptions such as (local) periodicity and scale separation of the coefficients, the new subspace decomposition approach to homogenization is applicable, reliable and accurate beyond these restrictions. This added value of the approach is demonstrated by theoretical and experimental results.

Asymptotic completeness in dissipative scattering theoryB 004 (Theresienstr. 39, 80333 München)

Signal Processing with 1-bit Sensor Array MeasurementsMI 02.08.020 (Boltzmannstr. 3, 85748 Garching)

The talk focuses on signal processing with measurements obtained by an array of sensors which feature a low-resolution digitization process. Employing a large number of antennas in conjunction with low-complexity analog-to-digital conversion is motivated via the technical requirements of exemplary future wireless systems. We outline the challenges associated with statistical processing and analysis under such a system architecture. Reducing the intractable probabilistic models arising under hard-limiting within the exponential family, we then conservatively approximate established information measures connected to signal processing performance. Finally, the presented methods are exploited to study the favorable design of high-performance wireless sensor systems with low hardware complexity.

Lagrangian Coherent Structures IIMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Lagrangian coherent structures (LCSs) are often viewed as „sets that defy strong dispersion“ or „sets that don’t mix with their environment“ under the action of a dynamical system. As such, they can be intuitively understood as regions of regular motion, embedded in an otherwise chaotic/turbulent sea. Over the last two decades, many, mostly phenomenological, theoretical and computational approaches have been developed.

In this two-part lecture, I discuss a recently developed mathematical framework in which LCSs are recognized as almost-invariant sets (or, even stronger, as optimal barriers to diffusive transport) under advection-diffusion as observed from a rather non-standard Lagrangian/material viewpoint. The framework covers deterministic advection-diffusion in and stochastic perturbations of incompressible fluid flows.

Invariant densities for randomly switched ODEsB 252 (Theresienstr. 39, 80333 München)

We consider random dynamical systems characterized by Poissonian switching between finitely many deterministic vector fields on a smooth manifold. Under a hypoellipticity condition at an accessible point on the manifold, the invariant measure of the associated Markov semigroup, if it exists, is unique and absolutely continuous. Unlike in the case of hypoelliptic diffusions, the density of the invariant measure need not be globally smooth due to possible contraction at critical points. Understanding the behavior of the density is then a challenging problem. The talk aims to give an introduction to the ergodic theory of systems with random switching. It is based on work with Yuri Bakhtin, Sean Lawley and Jonathan Mattingly.

Optimization in spaces of vector measuresGebäude 33, Raum 3401 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

In this talk we focus on optimization problems in which we discuss the existence of optimizers, first order optimality conditions as well as sufficient conditions for the existence of a sparse optimizer, i.e. an optimal solution which is only supported on finitely many points in the spatial domain. For the numerical solution we propose and analyze a conditional gradient algorithm for which a sublinear rate of convergence is proven. We introduce additional acceleration steps that lead to a geometric rate of convergence under suitable conditions. The talk is completed by several concrete applications, including the identification of acoustic point sources and the optimal placement of measurement sensors for the identification of an unknown parameter entering a partial differential equation.

A density result in GSBDp with applications to the approximation of brittle fracture energies MI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

The seminar concerns the proof, obtained in collaboration with Antonin Chambolle, that any function in GSBDp is approximated by functions in SBV, bounded, whose jump is a finite union of smooth hypersurfaces. The approximation takes place in the sense of Griffith-type energies, the study of which motivates the definition of the space GSBDp, and it is also in Lp outside a sequence of sets whose measure tends to 0.

I will describe also how to apply this density result to deduce Γ-convergence approximation à la Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are a priori not even integrable.

On entropic regularization of optimal transport: theory, numerics and applicationsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

In this talk, I will describe some aspects of entropy minimization related to optimal transport. Even if the problem can be traced back to the seminal work of Schrödinger in the 1930’s, it has been greatly revitalized in the recent period (in particular thanks to the work of Christian Léonard who connected both theories through large deviations principles) and a striking aspect of Schrödinger’s problem is that it is amenable to very efficient numerical schemes. I will also describe applications of these ideas to incompressible fluid dynamics and the theory of mean-field games.

Dynamical Sampling on Finite Index SetsMI 02.08.020 (Boltzmannstr. 3, 85748 Garching)

Dynamical Sampling aims to subsample solutions of linear dynamical systems at various times. In a real world scenario one might think of sensors on the ground measuring tem\-perature or other quantities. Instead of installing a dense network of sensors, the idea is to use less sensors and exploit the dynamical process which the measured quantity is subject to. A natural mathematical model of this framework consists of considering inner products (the samples) of the form \( <h,A^n f_i> \), where \(h\) is the signal (temperature distribution etc.), \((f_i)_{i\in I}\) a system of fixed vectors (corresponding to the locations of the sensors), and \(A\) a linear evolution operator which is connected with the dynamical system. The objective of the general Dynamical Sampling problem is to figure out for which operators \(A\) and which families \( (f_i)_{i\in I} \) the (arbitrary) signal \(h\) can be stably recovered from the sampling data \( \{<h,A^n f_i>\}_{n,i} \).

Here, we only consider finite index sets \(I\). We start off with the finite-dimensional situation for which we prove a characterization theorem. In the infinite-dimensional case we begin by only allowing normal operators \(A\). Here, we prove that the operator \(A\) is necessarily a diagonal operator with eigenvalues \( \lambda_j \) of multiplicity at most \(|I|\) in the open unit disk. Moreover, the eigenvalue sequence \( ( \lambda_j)_{j\in \mathbb{N}}\) must be a finite union of so-called uniformly separated sequences. We will complete this list of necessary conditions to a characterization of the problem.

In the case where the operator \(A\) is not assumed to be normal, we also provide a (less explicit) characterization which allows for deducing necessary conditions on the spectrum of the operator \(A\). In the case of one iterated vector, we find a more explicit characterization in terms of inner functions in the unit disk.

The talk is based on joint works with C. Cabrelli, O. Christensen, M. Hasannasab, U. Molter, and V. Paternostro.

Dynamic Networks and Applications in Social Media AnalysisMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Social media services have introduced new opportunities in different areas of marketing, politics, economics and public communication. Simultaneously, these services have empowered different interest groups in tampering and manipulating the public opinion. Interesting examples are the spread of fake news during Trump's campaign and its consequences and the effect of influential users on public opinion. In our research, we apply concepts of dynamic temporal networks on Facebook Data in order to investigate the impact of influential users. More specifically, we use DTN in order to detect the users that significantly alter Political Networks in Facebook.

Exact $L_2$-small ball probabilities for finite-dimensional perturbations of Gaussian processes: spectral method.2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

I consider the problem of small ball probabilities for Gaussian processes in $L_2$-norm. I focus on the processes which are important in statistics (e.g. Kac-Kiefer-Wolfowitz processes), which are finite dimentional perturbations of Gaussian processes. Depending on the properties of the kernel and perturbation matrix I consider two cases: non-critical and critical.

For non-critical case I prove the general theorem for precise asymptotics of small deviations. For a huge class of critical processes I prove a general theorem in the same spirit as for non-critical processes, but technically much more difficult. At the same time a lot of processes naturally appearing in statistics (e.g. Durbin, detrended processes) are not covered by those general theorems, so I treat them separately using methods of spectral theory and complex analysis.

First I will give an introduction to the topic and a historical review, and then I will tell about my results.

Random matrices with slow correlation decayMI 03.10.011 (Boltzmannstr. 3, 85748 Garching)

We first present a short introduction to random matrix theory and its motivations from quantum physics. In the main part of the talk we review some recent results on the local eigenvalue statistics of various random matrix models generalising the classical Wigner random matrices with independent identically distributed zero mean entries. We demonstrate that the celebrated Wigner-Dyson-Mehta universality conjecture also extends to correlated random matrices with a finite polynomial decay of correlations and arbitrary expectation. Our proof relies on a quantitative stability analysis of the matrix Dyson equation (MDE) as well as on a systematic diagrammatic control of a multivariate cumulant expansion.

POD-based model order reduction in multiobjective optimal control techniquesMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

In most optimal control problems, the cost function contains two components: The cost of the energy used to control the system and a defect of the state variable against a desired target. While these terms are often summed up using weighting factors, the talk will introduce concepts from multiobjective optimization to compute the Pareto front - a set of optimal compromises between the two. The hereby increased computational effort is countered using Proper Orthogonal Decomposition (POD) to reduce the dimension of state and adjoint equations. Two multiobjective strategies will be introduced: The reference point method to deal with high-dimensional control spaces and a set-oriented method for a high number of objective functions. Rigorous and heuristic error estimators are used to track the quality of the current reduced-order model.