Eigenvalue perturbation bounds are employed in a wide range
of theory and applications. I will first survey some of the
most important results in perturbation theory for symmetric
eigenvalue problems, including the max-min characterization
and Weyl's theorem. I will then discuss certain situations
in which known theory gives severely overestimated bounds
for the actual eigenvalue perturbation, and derive bounds
that are sharp in such cases.
It is a widespread but little-known phenomenon that the normwise
relative error of vectors x and y of floating point numbers can be
many orders of magnitude smaller than the unit roundoff. I will
discuss how this phenomenon arises and demonstrate the undesirable
effects it can have on a method for comparing the performance of
different algorithms. I will then present a transformation that
overcomes these problems whilst still preserving the desirable
properties of the comparison approach.
Computer simulation of the electromagnetic behaviour of a device is a common
practice in modern engineering. Maxwell's equations are solved on a
computer with the help of numerical methods. Contemporary devices constantly
grow in size and complexity. Therefore, new numerical methods should be
highly efficient. Many industrial and research applications of numerical
methods need to account for the frequency dependent materials.
The Finite-Difference Time-Domain (FDTD) method is one of the most
widely adopted algorithms for the numerical solution of Maxwell's
equations. A major drawback of the FDTD method is the interdependence of
the spatial and temporal discretisation steps, known as the
Courant-Friedrichs-Lewy (CFL) stability criterion. Due to the CFL
condition the simulation of a large object with delicate geometry will
require a high spatio-temporal resolution everywhere in the FDTD grid.
Application of subgridding increases the efficiency of the FDTD method.
Subgridding decomposes the simulation domain into a number of subdomains
with different spatio-temporal resolutions.
The research project described in this presentation uses the Huygens
Subgridding (HSG) method. The frequency dependence is included with the
Auxiliary Differential Equation (ADE) approach based on the one pole
Debye relaxation model. The main contributions of this work are (i)
extension of the one-dimensional (1D) frequency-dependent HSG method to
three dimensions (3D), (ii) implementation of the frequency-dependent
HSG method, termed the dispersive HSG, in Fortran 90, (iii)
implementation of the radio environment setting from the PGM-files, (iv)
simulation of the electromagnetic wave propagating from the
defibrillator through the human torso and (v) analysis of the
computational requirements of the dispersive HSG program.
Currently available Linear Algebra software packages rely on parallel implementations of the Basic Linear Algebra Subroutines (BLAS) to take advantage of multiple execution units. This solution is characterized by a fork-join model of parallel execution, which may result in suboptimal performance on current and future generations of multi-core processors since it introduces strict dependencies due to the presence of non parallelizable portions of code. The PLASMA project aims to overcome the shortcomings of this approach by introducing a pipelined model of parallel execution.