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Monday, 28.05.2018, 14:00 (WIAS-ESH)
Joint Research Seminar on Nonsmooth Variational Problems and Operator Equations / Mathematical Optimization
Dr. F. J. Romero Hinrichsen, ETH Zurich, Schweiz:
Dynamical super-resolution with applications to ultrafast ultrasound
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal

Abstract
Recently there has been a successful development in ultrasound imaging, increasing significantly the sampling rate and therefore enhancing this imaging's capacities. In particular, for vessel imaging, the use of microbubble tracking allows us to super-resolve blood vessels, and by estimating the particles' speeds inside them, it is possible to calculate the vessels' diameters. In this context, we model the microbubble tracking problem, formulating it in terms of a sparse spike recovery problem in the phase space (the position and velocity space), that allows us to obtain simultaneously the speed of the microbubbles and their location. This leads to an L1 minimization algorithm for point source tracking, that promises to be faster than current alternatives.

Further Informations
Joint Research Seminar on Mathematical Optimization / Non-smooth Variational Problems and Operator Equations

Host
Humboldt-Universität zu Berlin
WIAS Berlin
Tuesday, 29.05.2018, 13:30 (WIAS-ESH)
Seminar Numerische Mathematik
Prof. G. Panasenko, University of Lyon, Frankreich & University of Chile:
Asymptotic analysis of wave propagation in a laminated beam with contrasting stiffness of the layers
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal

Abstract
The wave equation in a thin laminated beam is considered in the case when the ratio of the thickness and the length is a small parameter e, while the ratio of the stiffness coefficients of the layers is a great parameter w. The question of an approximation of such a beam by the one dimensional models is discussed. It is shown that the classical homogenized approximation is valid for the case of sufficiently small value of the product of e square by w. If this product is not small then the classical model should be replaced by a multicomponent homogenization model analogous to [1, 2]. The waves in such multicomponent beam model can have multiple wave velocities. This result generalizes [3] Section 2.6.

[1] Panasenko, G.P. “Averaging of processes in strongly in homogeneous media”, Doklady Akademii Nauk SSSR, 1988, 298, 1, 76--79 (in Russian). English transl. in Soviet Phys. Dokl., 1988, 33.
[2] Panasenko, G.P. “Multicomponent homogenization of processes in strongly non-homogeneous structures” Mathematics USSRSbornik, 1990, 181, 1, 134--142 (in Russian); English transl. in Math. USSR Sbornik, 1991, 69, 1, 143--153.
[3] Panasenko, G.P. “Multi-Scale Modelling for Structures and Composites”, Springer, Dordrecht, 2005, 398 pp.

Host
WIAS Berlin
Wednesday, 30.05.2018, 10:00 (WIAS-ESH)
Forschungsseminar Mathematische Statistik
F. Schäfer, California Institute of Technology, USA:
Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal

Abstract
Many popular methods in machine learning, statistics, and uncertainty quantification rely on priors given by smooth Gaussian processes, like those obtained from the Matern covariance functions. Furthermore, many physical systems are described in terms of elliptic partial differential equations. Therefore, implicitely or explicitely, numerical simulation of these systems requires an ecient numerical representation of the corresponding Green's operator. The resulting kernel matrices are typically dense, leading to (often prohibitive) O (N^2) or O(N^3) computational complexity. In this work, we prove rigorously that the dense N x N kernel matrices obtained from elliptic boundary value problems and measurement points distributed approximately uniformly in a d-dimensional domain can be Cholesky factorised to accuracy  in computational complexity O(N log^2(N) log^2d(N/e)) in time and O (N log(N) log d(N/e)) in space. For the closely related Matern covariances we observe very good results in practise, even for parameters corresponding to non-integer order equations. As a byproduct, we obtain a sparse PCA with near-optimal low-rank approximation property and a fast solver for elliptic PDE. We emphasise that our algorithm requires no analytic expression for the covariance function. Our work is inspired by the probabilistic interpretation of the Cholesky factorisation, the screening effect in spatial statistics, and recent results in numerical homogenisation.

Further Informations
Forschungsseminar “Mathematische Statistik”

Host
Humboldt-Universität zu Berlin
Universität Potsdam
WIAS Berlin
Wednesday, 30.05.2018, 15:15 (WIAS-ESH)
Berliner Oberseminar „Nichtlineare partielle Differentialgleichungen” (Langenbach-Seminar)
Prof. J. F. Rodrigues, University of Lisbon, Portugal:
Evolutionary quasi-variational and variational inequalities with constraints on the derivatives
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal

Abstract
In this talk I shall present a general framework for the study of the existence of quasi-variational and variational solutions to a class of nonlinear evolution systems in convex sets of Banach spaces describing constraints on a linear combination of partial derivatives of the solutions, which is a joint work with Lisa Santos and Fernando Miranda. The quasi-linear operators are of monotone type, but are not required to be coercive for the existence of weak solutions, which is obtained by a double penalisation/regularisation for the approximation of the solutions. In the case of time-dependent convex sets that are independent of the solution, we show also the uniqueness and the continuous dependence of the strong solutions of the variational inequalities, extending previous results to a more general framework and including several models that will be surveyed.

Further Informations
Oberseminar “Nichtlineare Partielle Differentialgleichungen” (Langenbach Seminar)

Host
WIAS Berlin
Humboldt-Universität zu Berlin
Tuesday, 12.06.2018, 15:00 (WIAS-406)
Seminar Modern Methods in Applied Stochastics and Nonparametric Statistics
Dr. P. Dvurechensky, WIAS Berlin:
Distributed calculation of Wasserstein barycenters
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
We study the semi-discrete optimal transport problem of decentralized distributed computation of a discrete approximation for regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. Particularly, we assume that there is a network of agents/machines/computers where each agent holds a private continuous probability measure, and seek to compute the barycenter of all the measures in the network by getting samples from its local measure and exchanging information with its neighbors. Motivated by this problem, we develop a novel accelerated primal-dual stochastic gradient descent method for general stochastic convex optimization problems with linear equality constraints, and modify it for decentralized distributed optimization setting to generate a new algorithm for the distributed semi-discrete regularized Wasserstein barycenter problem. The proposed algorithm can be executed over arbitrary networks that are undirected, connected and static, using the local information only. Moreover, we show explicit non-asymptotic convergence rates in terms of the problem parameters. Finally, we show the effectiveness of our method on the distributed computation of the regularized Wasserstein barycenter of univariate Gaussian and von Mises distributions; as well as some applications to image aggregation.

Host
WIAS Berlin
Wednesday, 13.06.2018, 13:00 (WIAS-ESH)
Joint Research Seminar on Nonsmooth Variational Problems and Operator Equations / Mathematical Optimization
Dr. K. Welker, Universität Trier:
Constrained shape optimization problems in shape spaces
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal

Abstract
Shape optimization problems arise frequently in technological processes which are modelled in the form of partial differential equations (PDEs) or variational inequalities (VIs). In many practical circumstances, the shape under investigation is parametrized by finitely many parameters, which on the one hand allows the application of standard optimization approaches, but on the other hand limits the space of reachable shapes unnecessarily. In this talk, the theory of shape optimization is connected to the differential-geometric structure of shape spaces. In particular, efficient algorithms in terms of shape spaces and the resulting framework from infinite dimensional Riemannian geometry are presented. In this context, the space of H^1/2-shapes is defined. The H^1/2-shapes are a generalization of smooth shapes and arise naturally in shape optimization problems. Moreover, VI constrained shape optimization problems are treated from an analytical and numerical point of view in order to formulate approaches aiming at semi-smooth Newton methods on shape vector bundles. Shape optimization problems constrained by VIs are very challenging because of the necessity to operate in inherently non-linear and non-convex shape spaces. In classical VIs, there is no explicit dependence on the domain, which adds an unavoidable source of non-linearity and non-convexity due to the non-linear and non-convex nature of shape spaces.

Further Informations
Joint Research Seminar on Mathematical Optimization / Non-smooth Variational Problems and Operator Equations

Host
Humboldt-Universität zu Berlin
WIAS Berlin