Seminars, 2015

  • December 22, 2015
    Division Algebras  

    Dr. Adam Chapman, Michigan State University, USA

    Division algebras (aka division rings or skew fields) are fascinating algebraic structures that proved to be useful in Physics and Electric Engineering. We shall discuss the construction and classification of the most popular division algebras, the quaternion algebras, and their tensor products. We shall present both classical and new results.
  • December 15, 2015
    A generalized risk measure for the problem of optimal Portfolio selection and its explicit solution  

    Prof. Zinoviy Landsman, Department of Statistics, University of Haifa
  • October 27, 2015
    About the General Method of Lyapunov Functionals Construction  

    Prof. Leonid Shaikhet

    The general method of Lyapunov functionals construction for stability investigation of stochastic differential equations with delays is considered. It is shown that the standard procedure of Lyapunov functionals construction allows to construct different regions of stability for considered equation in space of the equation parameters. The method can be applied for linear and nonlinear stochastic differential equations with discrete, variable and distributed delays, for constant and variable coefficients. The method can be applied also for difference equations with discrete and continuous time and for differential equations with partial derivatives.
    The method was used for investigation of different known applied mathematical models in mechanics, biology, ecology, sociology: model of inverted pendulum, SIR-epidemic model, predator-prey model, Nicholson's blowflies model, model of alcohol consumption, model of social obesity epidemic etc.
    Some unsolved problems of stability and optimal control for stochastic difference and differential equations are proposed.
  • September 16, 2015
    Solving Quasi-Variational Inequalities for Adaptive Total Variation Regularization 

    Dr. Frank Lenzen, University of Heidelberg

    Due to its property to encourage piecewise constant solutions, Total Variation (TV) plays an important role as regularizer for various image restoration tasks.
    To even further improve the restoration quality of TV regularization, various adaptive methods have been proposed in literature. Among these, we find the class of direction-dependent TV methods, which we refer to as anisotropic TV methods.
    The main focus of the adaptive methods are e.g. the preservation of contrast, edges, corners, slopes and fine image structures.
    To steer the adaptivity, additional information is required. We therefore can distinguish between data-driven approaches, where this information is taken from the input image or additional data, and solution-driven approaches, where the adaptivity is depending on the solution of the problem.
    We also remark that, while the classical TV regularization is convex, some of the proposed approaches in literature yield non-convex problems. While in view of the restoration quality these non-convex methods show advantages, their theoretical and numerical treatment comes with some difficulties.
    In our talk we discuss a method which implements solution driven adaptivity in terms of a fixed point problem, where the unknown on the one hand is the sought solution to the problem and on the other hand steers the adaptivity.
    Reformulating the fixed-point problem as a Quasi-Variational Inequality Problem (QVIP) enables us to provide theoretical results for existence and uniqueness of the fixed point. Deriving a solution of the QVIP amounts to solving a sequence of convex sub-problems.
    Therefore, our strategy combines advantages of convex and non-convex approaches, since on the one hand we can built on the convexity of the sub-problems, on the other hand achieve a performance similar to non-convex regularizers.
    As applications, we exemplarily consider image denoising, (non-blind) deblurring and inpainting.
  • June 16, 2015
    A Rigorous Proof of the Maxwell-Claussius-Mossotti Formula 

    Prof. Yaniv Almog, Louisiana State University, USA

    We consider a large number of identical inclusions (say spherical), in a bounded domain, with conductivity different than that of the matrix. In the dilute limit, with some mild assumption on the first few marginal probability distribution (no periodicity or stationarity are assumed), we prove convergence in H1 norm of the expectation of the solution of the steady state heat equation, to the solution of an effective medium problem, which for spherical inclusions is obtained through the Maxwell-Clausius-Mossotti formula. Error estimates are provided as well.
  • May 18, 2015
    Quadrature domains and fluid dynamics

    Prof. Erik Lundberg, Florida Atlantic University, USA

    We will discuss a nonlinear moving boundary problem where a domain grows by the normal derivative of its own harmonic Green's function.  A celebrated theorem of S. Richardson reduces this problem to a parallel investigation of quadrature domains, special domains that generate an exterior gravitational potential equivalent to that of finitely many point masses.  We review these connections and discuss exact solutions in the plane.  Then we consider the higher-dimensional case.
  • April 14, 2015
    On the Optimality of Averaging in Distributed Statistical  Learning

    Dr. Jonathan Rosenblatt, Weizmann Institute

    A common approach to statistical learning on big data is to randomly split it among m machines and calculate the parameter of interest by averaging their m individual estimates.
    Focusing on empirical risk minimization, or equivalently M-estimation, we study the statistical error incurred by this strategy.
    We consider two asymptotic settings: one where the number of samples per machine n->inf but the number of parameters p is fixed, and a second high-dimensional regime where both p,n-> inf with p/n-> kappa.
    Most previous works provided only moment bounds on the error incurred by splitting the data in the fixed p setting. In contrast, we present for both regimes asymptotically exact distributions for this estimation error. In the fixed-p setting, under suitable assumptions, we thus prove that to leading order, averaging is as accurate as the centralized solution.
    In the high-dimensional setting, we show a qualitatively different behavior:
    data splitting does incur a first order accuracy loss, which we quantify precisely.
    In addition, our asymptotic distributions allow the construction of confidence intervals and hypothesis testing on the estimated parameters.
    Our main conclusion is that in both regimes, averaging parallelized estimates is an attractive way to speedup computations and save on memory, while incurring a quantifiable and typically moderate excess error.
  • February 24, 2015
    On the number of eigenvalues of linear operators on Banach spaces
    Prof. Franz Hanauska, Technical University of Clausthal, Germany

    Let L0 be a bounded operator and K a compact operator, both defined on a Banach space. We will estimate the number of discrete eigenvalues of  L:= L0 + K in subsets of the unbounded component of the spectrum of L0, in terms of the approximation numbers of the perturbing operator K. Our method employs complex analysis and a finite-dimensional reduction, allowing us to avoid using the existing theory of determinants in Banach spaces,which would require strong restrictions on K.
    Joint work with M. Demuth, M. Hansmann and H. Katriel.