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.