415 research outputs found
Non-recursive equivalent of the conjugate gradient method without the need to restart
A simple alternative to the conjugate gradient(CG) method is presented; this
method is developed as a special case of the more general iterated Ritz method
(IRM) for solving a system of linear equations. This novel algorithm is not
based on conjugacy, i.e. it is not necessary to maintain overall
orthogonalities between various vectors from distant steps. This method is more
stable than CG, and restarting techniques are not required. As in CG, only one
matrix-vector multiplication is required per step with appropriate
transformations. The algorithm is easily explained by energy considerations
without appealing to the A-orthogonality in n-dimensional space. Finally,
relaxation factor and preconditioning-like techniques can be adopted easily.Comment: 9 page
KRYLOV SUBSPACE METHODS FOR SOLVING LARGE LYAPUNOV EQUATIONS
Published versio
Model-order reduction techniques for the numerical solution of electromagnetic wave scattering problems
It is the aim of this work to contribute to the development of model-order reduction (MOR) techniques for the field of computational electromagnetics in relation to the electric field integral equation (EFIE) formulation. The ultimate
goal is to enable a fast-sweep analysis. In a fast-sweep problem, some parameter on which the original problem depends is varying and the problem must be solved as the parameter changes over a desired parameter range. The complexity of the original model prohibits its direct
use in simulation to compute the results at every required point. However, one can use MOR techniques to generate reduced-order models (ROMs), which can be rapidly solved to characterise the parameter-dependent behaviour of the system over the entire parameter range. This thesis focus is to implement robust, fast and accurate MOR techniques
with strict error controls, for application with varying parameters, using the EFIE formulations. While these formulations result in matrices that are significantly
smaller relative to differential equation-based formulations, the matrices resulting from discretising integral equations are very dense. Consequently,
EFIEs pose a difficult proposition in the generation of low-order accurate reduced order models.
The MOR techniques presented in this thesis are based on the theory of Krylov projections. They are widely accepted as being the most flexible and computationally efficient approaches in the generation of ROMs. There are three
main contributions attributed to this work.
² The formulation of an approximate extension of the Arnoldi algorithm to produce a ROM for an inhomogeneous contrast-sweep and source-sweep analysis.
² Investigation of the application of the Well-Conditioned Asymptotic Waveform Evaluation (WCAWE) technique to problems in which the system matrix has a nonlinear parameter dependence for EFIE formulations.
² The development of a fast full-wave frequency sweep analysis using the WCAWE technique for materials with frequency-dependent dielectric properties
O(N) methods in electronic structure calculations
Linear scaling methods, or O(N) methods, have computational and memory
requirements which scale linearly with the number of atoms in the system, N, in
contrast to standard approaches which scale with the cube of the number of
atoms. These methods, which rely on the short-ranged nature of electronic
structure, will allow accurate, ab initio simulations of systems of
unprecedented size. The theory behind the locality of electronic structure is
described and related to physical properties of systems to be modelled, along
with a survey of recent developments in real-space methods which are important
for efficient use of high performance computers. The linear scaling methods
proposed to date can be divided into seven different areas, and the
applicability, efficiency and advantages of the methods proposed in these areas
is then discussed. The applications of linear scaling methods, as well as the
implementations available as computer programs, are considered. Finally, the
prospects for and the challenges facing linear scaling methods are discussed.Comment: 85 pages, 15 figures, 488 references. Resubmitted to Rep. Prog. Phys
(small changes
A structure preserving shift-invert infinite Arnoldi algorithm for a class of delay eigenvalue problems with Hamiltonian symmetry
In this work we consider a class of delay eigenvalue problems that admit a
spectrum similar to that of a Hamiltonian matrix, in the sense that the
spectrum is symmetric with respect to both the real and imaginary axis. More
precisely, we present a method to iteratively approximate the eigenvalues of
such delay eigenvalue problems closest to a given purely real or imaginary
shift, while preserving the symmetries of the spectrum. To this end the
presented method exploits the equivalence between the considered delay
eigenvalue problem and the eigenvalue problem associated with a linear but
infinite-dimensional operator. To compute the eigenvalues closest to the given
shift, we apply a specifically chosen shift-invert transformation to this
linear operator and compute the eigenvalues with the largest modulus of the new
shifted and inverted operator using an (infinite) Arnoldi procedure. The
advantage of the chosen shift-invert transformation is that the spectrum of the
transformed operator has a "real skew-Hamiltonian"-like structure. Furthermore,
it is proven that the Krylov space constructed by applying this operator,
satisfies an orthogonality property in terms of a specifically chosen bilinear
form. By taking this property into account during the orthogonalization
process, it is ensured that even in the presence of rounding errors, the
obtained approximation for, e.g., a simple, purely imaginary eigenvalue is
simple and purely imaginary. The presented work can thus be seen as an
extension of [V. Mehrmann and D. Watkins, "Structure-Preserving Methods for
Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils",
SIAM J. Sci. Comput. (22.6), 2001], to the considered class of delay eigenvalue
problems. Although the presented method is initially defined on function
spaces, it can be implemented using finite dimensional linear algebra
operations
Computation methods for the eigenvalue analysis of large structures by component synthesis
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