Seminars, 2019

Lectures:
  • February 5, 2019
    Globally Solving the Trust Region Subproblem Using Simple First-Order Methods
    Dr. Yakov Vaisbourd, Tel-Aviv University

    Abstract
    We consider the trust region subproblem which is given by a minimization of a quadratic, not necessarily convex, function over the Euclidean ball. Based on the well-known second-order necessary and sufficient optimality conditions for this problem, we present two sufficient optimality conditions defined solely in terms of the primal variables. Each of these conditions corresponds to one of two possible scenarios that occur in this problem, commonly referred to in the literature as the presence or absence of the ``hard case". We consider a family of first-order methods, which includes the projected and conditional gradient methods. We show that any method belonging to this family produces a sequence which is guaranteed to converge to a stationary point of the trust region subproblem. Based on this result and the established sufficient optimality conditions, we show that convergence to an optimal solution can be also guaranteed as long as the method is properly initialized. In particular, if the method is initialized with the zeros vector and reinitialized with a randomly generated feasible point, then the best of the two obtained vectors is an optimal solution of the problem in probability 1.
  • January 22, 2019
    Classification Problems of Infinite Type (in Differential Geometry)
    Dr. Ori Yudilevich, Catholic University of Leuven, Belgium

    Abstract
    The notions of a Lie groupoid and Lie algebroid (generalizations of the notions of a Lie group and Lie algebra) have proven to be powerful tools in differential geometry. One notable application, which takes its roots in the classical work of Élie Cartan, is their use in classifying geometric structures. For example, it was recently shown by Fernandes and Struchiner that the classification of so-called Kähler-Bochner manifolds, a result due to Robert Bryant (2001), can be understood as the integration of a certain Lie algebroid (that encodes the classification problem) to a Lie groupoid (that encodes the solutions). This method, however, is limited to a special class of classification problems known as problems of "finite type". Roughly speaking, this means that the space of solutions is finite dimensional. To address problems of "infinite type", one needs slightly more general tools.
    In this talk (based on joint work with Rui Loja Fernandes), I will present the notions of a Bryant groupoid and Bryant algebroid that can be used to tackle classification problems that are both "finite" and "infinite". These notions, which stem from the work of Robert Bryant, exhibit interesting behavior that brings together ideas from the theory of Lie groupoids and Lie algebroids, on the one hand, and from the theory of Partial Differential Equations, on the other. This talk is aimed at a broad mathematical audience and will not assume any previous knowledge of differential geometry.
  • January 15, 2019
    An embedded Cartesian scheme for the Navier-Stokes equations
    Prof. Dalia Fishelov, Afeka Tel Aviv Academic College of Engineering

    Abstract
    In this talk the two-dimensional Navier-Stokes system in stream function formulation is considered. 
    We describe a fourth-order compact scheme for regular domains in 2D. We then proceed to irregular domains.
    First, the irregular domain is embedded in a Cartesian grid. Then, an interpolating polynomial is built for regular elements inside the domain as well as for irregular elements near the boundary. 
    A compact high-order scheme is then constructed for the Navier-Stokes equations by applying the differential operators involved in the Navier-Stokes equations to the interpolating polynomial.
    Numerical results will be presented for various irregular domains. A particular attention is devoted to flows in elliptical domains. 
    In the case of the ellipse, we also demonstrate the ability of the scheme for computations of the eigenvalues and the eigenfunctions of the biharmonic problem on the ellipse.