Development of a general analysis and unfolding scheme and its application to measure the energy spectrum of atmospheric neutrinos with IceCube
 Sarkar, S European Physical Journal C volume C75 issue 3 116-116 (11 Mar 2015)
Mon, 20 Oct 2014

14:15 - 16:30
L5

Mirror symmetry for varieties of general type

Mark Gross
(Cambridge)
Abstract
I will discuss joint work with Ludmil Katzarkov and Helge Ruddat. Given a hypersurface X in a toric variety of positive Kodaira dimension, (with a certain number of hypotheses) we construct an object which we believe can be viewed as the mirror of X. In particular, it exhibits the usual interchange of Hodge numbers expected in mirror symmetry. This may seem puzzling at first. For example, a curve of genus g would be expected to have a mirror such that h^{0,0}=g, which is not possible for a variety. However, our mirror is a singular scheme Y along with a perverse sheaf F, whose cohomology carries a mixed Hodge structure. It then makes sense to compute Hodge numbers for F, and we find the traditional exchange of Hodge numbers.
Thu, 04 Dec 2014

14:00 - 15:00
L5

Is the Helmholtz equation really sign-indefinite?

Dr Euan Spence
(University of Bath)
Abstract

The usual variational formulations of the Helmholtz equation are sign-indefinite (i.e. not coercive). In this talk, I will argue that this indefiniteness is not an inherent feature of the Helmholtz equation itself, only of its standard formulations. I will do this by presenting new sign-definite formulations of several Helmholtz boundary value problems.

This is joint work with Andrea Moiola (Reading).
 

Thu, 27 Nov 2014

14:00 - 15:00
Rutherford Appleton Laboratory, nr Didcot

Incomplete Cholesky preconditioners based on orthogonal dropping : theory and practice

Artem Napov
(Universite Libre de Bruxelles)
Abstract

Incomplete Cholesky factorizations are commonly used as black-box preconditioners for the iterative solution of large sparse symmetric positive definite linear systems. Traditionally, incomplete 
factorizations are obtained by dropping (i.e., replacing by zero) some entries of the factors during the factorization process. Here we consider a less common way to approximate the factors : through low-rank approximations of some off-diagonal blocks. We focus more specifically on approximation schemes that satisfy the orthogonality condition: the approximation should be orthogonal to the corresponding approximation error.

The resulting incomplete Cholesky factorizations have attractive theoretical properties. First, the underlying factorization process can be shown breakdown-free. Further, the condition number of the 
preconditioned system, that characterizes the convergence rate of standard iterative schemes, can be shown bounded as a function of the accuracy of individual approximations. Hence, such a bound can benefit from better approximations, but also from some algorithmic peculiarities. Eventually, the above results can be shown to hold for any symmetric positive definite system matrix.

On the practical side, we consider a particular variant of the preconditioner. It relies on a nested dissection ordering of unknowns to  insure an attractive memory usage and operations count. Further, it exploits in an algebraic way the low-rank structure present in system matrices that arise from PDE discretizations. A preliminary implementation of the method is compared with similar Cholesky and 
incomplete Cholesky factorizations based on dropping of individual entries.

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