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Reviving the Method of Particular Solutions
Get PDFFox, Henrici, and Moler made famous a "method of particular solutions" for computing eigenvalues and eigenmodes of the Laplacian in planar regions such as polygons. We explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties. The crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant. Similar methods may be used to improve other "mesh-free" algorithms for a variety of computational problems
An Interview with Peter Lancaster
No full textThis is the transcript of an interview with
Peter Lancaster recorded at the University of Manchester, March 15, 2005.
A shortened version of this interview appeared in SIAM News 38(6):5-6, Jul/Aug 2005.
See
http://www.siam.org/news/news.php?id=12
Universal connection and curvature for statistical manifold geometry
No full textStatistical manifolds are representations of smooth families of
probability density functions
that allow differential geometric methods to be applied to
problems in stochastic processes, mathematical statistics and
information theory. It is common to have to consider a number of
linear connections on a given statistical manifold and so it is
important to know the corresponding universal connection and
curvature; then all linear connections and their curvatures are
pullbacks. An important class of statistical manifolds is that
arising from the exponential families and one particular family is
that of gamma distributions, which we showed recently to have
important uniqueness properties in stochastic processes. Here we
provide formulae for universal connections and curvatures on
exponential families and give an explicit example for the manifold
of gamma distributions
Iterative Solution of a Nonsymmetric Algebraic Riccati Equation
No full textWe study the nonsymmetric algebraic Riccati equation whose four
coefficient matrices are the blocks of
a nonsingular -matrix or an irreducible singular
-matrix . The solution of practical interest is the minimal nonnegative
solution. We show that Newton's method with zero initial guess can be used to
find this solution without any further assumptions. We also present a
qualitative perturbation analysis for the minimal solution, which is
instructive in designing algorithms for finding more accurate approximations.
For the most practically important case, in
which is an irreducible singular -matrix with zero row sums,
the minimal solution is either stochastic or substochastic and
the Riccati equation can be tranformed into a unilateral matrix equation by
a procedure of Ramaswami. The minimal solution of the Riccati equation can
then be found by computing the minimal nonnegative solution of
the unilateral equation using
the Latouche--Ramaswami algorithm.
We show that the Latouche--Ramawami algorithm, combined with a shift
technique suggested by He, Mini, and Rhee,
is breakdown-free in all cases and is able to find the minimal solution
more efficiently and more accurately than the algorithm without a shift.
Our approach is to find a proper
stochastic solution using the
shift technique even if it is not the minimal solution. We show how we can
easily recover the minimal solution when it is not the computed stochastic
solution
Kinematic models for non-coaxial granular materials. Part I: theory
No full textThe purpose of this paper is to present a physically based plasticity model for non-coaxial granular materials. The model, which we shall call the double slip and rotation rate model (DSR2 model), is a pair of kinematic equations governing the velocity field. The model is based on a discrete micro-analysis of the kinematics of particles in contact, and is formulated by introducing a quantity called the averaged micro-pure rotation rate (APR) into the unified plasticity model which was proposed by one of the authors. Our macro-micro mechanical analysis shows that the APR is a non-linear function of, among other quantities, the macro-rotation rate of the major principal axis of stress taken in the opposite sense. The requirement of energy dissipation used in the double-sliding free-rotating model appears to be unduly restrictive as a constitutive assumption in continuum models. In the DSR2 model the APR tensor and the spin tensor are directly linked with non-coaxiality of the stress and deformation rate tensors. We also propose a simplified plasticity model based on the DSR2 model for a class of dilatant materials, and analyse its material stability. Copyright © 2005 John Wiley & Sons, Ltd
On Davis-Januszkiewicz homotopy types I; formality and rationalisation
Get PDFFor an arbitrary simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction (here called c(K)), which they showed to be homotopy equivalent to Davis and Januszkiewicz's examples. It is therefore natural to investigate the extent to which the homotopy type of a space is determined by having such a cohomology ring. We begin this study here, in the context of model category theory. In particular, we extend work of Franz by showing that the singular cochain algebra of c(K) is formal as a differential graded noncommutative algebra. We specialise to the rationals by proving the corresponding result for Sullivan's commutative cochain algebra, and deduce that the rationalisation of c(K) is unique for a special family of complexes K. In a sequel, we will consider the uniqueness of c(K) at each prime separately, and apply Sullivan's arithmetic square to produce global results for this family
The Polynomial Eigenvalue Problem
Get PDFIn this thesis, we consider polynomial eigenvalue problems. We extend results on
eigenvalue and eigenvector condition numbers of matrix polynomials to condition
numbers with perturbations measured with a weighted Frobenius norm. We derive
an explicit expression for the backward error of an approximate eigenpair of a
matrix polynomial written in homogeneous form. We consider structured eigenvalue
condition numbers for which perturbations have a certain structure such
as symmetry, Hermitian or sparsity. We also obtain explicit and/or computable
expressions for the structured backward error of an eigenpair.
We present a robust implementation of the HZ (or HR) algorithm for symmetric
generalized eigenvalue problems. This algorithm has the advantage of preserving
pseudosymmetric tridiagonal forms. It has been criticized for its numerical
instability. We propose an implementation of the HZ algorithm that allows stability
in most cases and comparable results with other classical algorithms for ill
conditioned problems. The HZ algorithm is based on the HR factorization, an
extension of the QR factorization in which the H factor is hyperbolic. This yields
us to the sensitivity analysis of hyperbolic factorizations