Kategorien-Filter ist aus: keine Filterung nach Kategorien.

The heat and the landscape(using zoom) (Parkring 11, 85748 Garching-Hochbrück)

If lengths 1 and 2 are assigned randomly to each edge in the planar grid, what are the fluctuations of distances between far away points? This problem is open, yet we know, in great detail, what to expect. The directed landscape, a universal random plane geometry, provides the answer to such questions. In some models, such as directed polymers, the stochastic heat equation, or the KPZ equation, random plane geometry hides in the background. Principal component analysis, a fundamental statistical method, comes to the rescue: BBP statistics can be used to show that these models converge to the directed landscape.

Monge-Ampère equations and inverse problems in opticsonline (Zugangsdaten auf Anfrage , 0 (using zoom))

Non-imaging optics is a field of optics which is interested in designing optical components, such as mirrors or lenses, that transfer a given source light to a prescribed target. The goal is not to simulate the trajectory of the light through an optical component, which would be the direct problem, but instead to build the shape of a mirror or a lens that transfers a source light to a given target light. This inverse problem amounts in different settings to solving Monge-Ampère type equations. In this talk, I will show how these equations are connected to optimal transport and can be solved using a geometric discretization called semi-discrete. I will also present the design of different kinds of mirrors or lenses that allow to transfer any point or parallel source light to any target. This work is in collaboration with Quentin Mérigot and involves Jocelyn Meyron.

To join the Zoom Meeting: https://tu-berlin.zoom.us/j/61131079394?pwd=MWlKaGp6emg1ZEptM0FwaUxQcVpsQT09 Meeting ID: 611 3107 9394 Passcode: 023904

Oberseminar : Mathematical modeling for the prevention of neonatal cerebral hemorrhagePasscode 101816 (https://tum-conf.zoom.us/j/96536097137 , 0 (using zoom))

Intracerebral hemorrhage is the major complication in the development of premature infants. The early childhood cerebral hemorrhage can lead to the clinical picture of cerebral palsy characterized by disorders of motor development and posture as well as other partial performance disorders such as deficits in speech, perception, learning disabilities and epilepsy. Cerebral hemorrhage is often triggered by fluctuations of cerebral blood flow (CBF). Therefore, regular monitoring of CBF is an important task in medical care of preterm infants. Although several measuring techniques have been developed during last years, they are still not the part of the clinical routine. We present a mathematical model for the calculation of CBF. The model takes into account peculiarities of the immature cerebral circulation, such as presence of a specific area in the developing brain with a highly fragile blood vessel network, called germinal matrix, and an impaired ability of the cerebral blood vessels to vary their diameter in response to fluctuations of main medical parameters. Furthermore, finite element analysis of critical stresses, exerted on vessels’ walls, will be presented based on calculated CBF. Additionally, machine learning models for identifying of preterm infants at risk of cerebral hemorrhage and a viability theory based approach to control impaired cerebral autoregulation will be outlined.

Regularity for non-uniformly elliptic equations and applications in stochastic homogenizationonline (Zugangsdaten auf Anfrage , 0 (using zoom))

I will discuss regularity properties for solutions of linear second order non-uniformly elliptic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results due to Trudinger from the 1970s. As an application of the local boundedness result, we deduce a quenched invariance principle for random walks among random degenerate conductances. If time permits I will discuss further regularity results for nonlinear non-uniformly elliptic variational problems.

Minimizers in the liquid drop modelonline (Zugangsdaten auf Anfrage , 0 (using zoom))

The liquid drop model goes back to Gamow’s theory for atomic nuclei in the 1930s. Recently, it has gained renewed interest in the mathematics literature and several interesting questions remain open. I will discuss some rigorous results on the existence and nonexistence of minimizers with a general Riesz potential. The talk is based on recent joint work with Rupert L. Frank.

An Approximation Algorithm for Network Flow Interdiction with Unit CostsVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

In the network flow interdiction problem (NFI), an interdictor aims to remove arcs of total cost at most a given budget B from a graph with given arc costs and capacities such that the value of a maximum flow from a source s to a sink t is minimized. This problem is known to be strongly NP-hard, but, despite its broad applicability, only few polynomial-time approximation algorithms have been found so far. In this talk, I present an approximation algorithm for a special case of NFI where arcs have unit costs and capacities are restricted to two possible values 1 and u > 1. Afterwards, I will sketch how the algorithm can be extended to the more general version of NFI where arcs have arbitrary capacities. To the best of my knowledge, this is the first algorithm for a version of NFI whose approximation ratio only depends on the interdiction budget. On simple graphs, its approximation ratio dominates the previously best known approximation ratio.

TBA(using zoom) (Parkring 11, 85748 Garching-Hochbrück)

TBA

Oberseminar : Physics-aware, data-driven discovery of slow and stable coarse-grained dynamics in the Small Data regimePasscode 101816 (https://tum-conf.zoom.us/j/96536097137 , 0 (using zoom))

Despite recent successes from applications of machine learning tools in the field of computational physics, critical challenges persist in problems involving Small Data and multiscale systems. Given high-dimensional time-series’ data from a multiscale dynamical system, we present a probabilistic framework that learns an interpretable, lower-dimensional, coarse-grained model whose long-term stability is guaranteed. We include a layer of physically motivated latent variables and enable the incorporation of known physical constraints. Such domain knowledge can be extremely useful when training in the Small Data regime and for out-of-sample predictions. In contrast to existing schemes, the proposed model does not require the a priori definition of projection operators and, being fully probabilistic, is capable of capturing the predictive uncertainty due to the information loss resulting from model compression. We illustrate its performance in high-dimensional particle systems and demonstrate its efficacy and accuracy by generating extrapolative, long-term predictions with quantified uncertainty.

Nonparametric C- and D-vine based quantile regression(using Zoom, see http://go.tum.de/410163 for more details) (Parkring 11, 85748 Garching)

Quantile regression is a field with steadily growing importance in statistical modeling. It is a complementary method to linear regression, since computing a range of conditional quantile functions provides a more accurate modelling of the stochastic relationship among variables, especially in the tails. We introduce a novel nonrestrictive and highly flexible nonparametric quantile regression approach based on C- and D-vine copulas. Vine copulas allow for separate modeling of marginal distributions and the dependence structure in the data, and can be expressed through a graph theoretical model given by a sequence of trees. This way we obtain a quantile regression model, that overcomes typical issues of quantile regression such as quantile crossings or collinearity, the need for transformations and interactions of variables. Our approach incorporates a two-step ahead ordering of variables, by maximizing the conditional log-likelihood of the tree sequence, while taking into account the next two tree levels. We show that the nonparametric conditional quantile estimator is consistent. The performance of the proposed methods is evaluated in both low- and high-dimensional settings using simulated and real world data. The results support the superior prediction ability of the proposed models.

The Boundary of Bounded Motions in the Planar Circular Restricted Three-Body ProblemVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

The planar circular restricted three-body problem is a perturbation of the Kepler problem that describes the motion of a satellite orbiting two massive bodies. Recently, it has been shown that for any value of the mass ratio of the two bodies and values of the Jacobi constant large enough, the "manifolds of infinity" of the Kepler Problem intersect transversally, leading to a stochastic layer surrounding the split separatrices and oscillatory motions, in which the motion of the satellite is unbounded, but the satellite always returns to some bounded region. In a joint work with Amadeu Delshams, we prove the existence of a "furthest" invariant KAM torus which bounds the stochastic layer and so the region of bounded motions.

Non-existence of bi-infinite polymer Gibbs measures on Z^2(using zoom) (Parkring 11, 85748 Garching-Hochbrück)

To each vertex x\in Z^2 assign a positive weight \omega_x. A geodesic between two ordered points on the lattice is an up-right path maximizing the cumulative weight along itself. A bi-infinite geodesic is an infinite path taking up-right steps on the lattice and such that for every two points on the path, its restriction to between the points is a geodesic. Assume the weights across the lattice are i.i.d., does there exist a bi-infinite geodesic with some positive probability? In the case the weights are Exponentially distributed, we answer this question in the negative. We show an analogous result for the positive-temperature variant of this model. Joint work with Marton Balazs and Timo Seppalainen.