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Max-Plus Singular Values
No full textIn this paper we prove a new characterization of the max-plus singular values of a max- plus matrix, as the max-plus eigenvalues of an associated max-plus matrix pencil. This new characterization allows us to compute max-plus singular values quickly and accurately. As well as capturing the asymptotic behavior of the singular values of classical matrices whose entries are exponentially parameterized we show experimentally that max-plus singular values give order of magnitude approximations to the classical singular values of parameter independent classical matrices.
We also discuss Hungarian scaling, which is a diagonal scaling strategy for preprocessing classical linear systems. We show that Hungarian scaling can dramatically reduce the d-norm condition number and that this action can be explained using our new theory for max-plus singular values
Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry
No full textFor Hamiltonian systems with spherical symmetry there is a marked difference between zero and non-zero momentum values, and amongst all relative equilibria with zero momentum there is a marked difference between those of zero and those of non-zero angular velocity. We use techniques from singularity theory to study the family of relative equilibria that arise as a symmetric Hamiltonian which has a group orbit of equilibria with zero momentum is perturbed so that the zero-momentum relative equilibrium are no longer equilibria. We also analyze the stability of these perturbed relative equilibria, and consider an application to satellites controlled by means of rotors
Block preconditioners for linear systems arising from multilevel RBF collocation
No full textSymmetric multiscale collocation methods with radial basis functions allow approximation of the solution of a partial differential equation, even if the right-hand side is only known at scattered data points, without needing to generate a grid. However, the benefit of a guaranteed symmetric positive definite block system comes at a high computational cost. In particular, the condition number and sparsity deteriorate with the number of data points. Therefore, we study certain block diagonal and triangular preconditioners. We investigate ideal preconditioners and determine the spectra of the preconditioned matrices before proposing more practical preconditioners based on a restricted additive Schwarz method with coarse grid correction (ARASM). Numerical results verify the effectiveness of the preconditioners
Sylvester's Influence on Applied Mathematics
No full textJames Joseph Sylvester coined the term ``matrix'' and contributed
much to the early development of matrix theory.
To mark the 200th anniversary of his birth I show how Sylvester's work
on matrices continues to influence applied mathematics today
Gordon's inequality and condition numbers in conic optimization
No full textThe probabilistic analysis of condition numbers has traditionally been approached from different angles; one is based on Smale's program in complexity theory and features integral geometry, while the other is motivated by geometric functional analysis and makes use of the theory of
Gaussian processes. In this note we explore connections between the two approaches in the context of the biconic homogeneous feasiblity problem and the condition numbers motivated by conic optimization theory. Key tools in the analysis are Slepian's and Gordon's comparision inequalities for Gaussian processes, interpreted as monotonicity properties of moment functionals, and their interplay with ideas from conic integral geometry
An algorithm for quadratic eigenproblems with low rank damping
Get PDFWe consider quadratic eigenproblems
, where all coefficient
matrices are real and positive semidefinite, is regular and is
of low rank. Matrix polynomials of this form appear in the analysis of
vibrating structures with discrete dampers. We develop an algorithm for
such problems, which first solves the undamped problem
and then accommodates for the low
rank term . For the first part, we develop a new algorithm based on a method proposed by Wang and Zhao [SIAM J. Matrix Anal. Appl. 12-4 (1991), pp.
654--660], which can compute all eigenvalues of definite generalized eigenvalue problems with semidefinite coefficient matrices in a backward stable and symmetry preserving manner. We use this new algorithm to compute the solution to the undamped problem, and then use this solution in order to compute all eigenvalues of the original problem, and the associated eigenvectors if requested. To this end, we use an Ehrlich-Aberth iteration that works exclusively with vectors and tall skinny matrices and contributes only lower order terms to the overall flop count.
Numerical experiments show that the proposed algorithm is both fast and
accurate. Finally we discuss the application to large
scale quadratics and the possibility of generalizations to other problems
Regularised GMRES-type Methods for X-Ray Computed Tomography
Get PDFSlowly converging iterative methods such as Landweber or ART, have long been preferred for reconstructing a tomographic image from a set of CT data. In the recent years, a fast-converging method named CGLS has received attention for reconstructing tomographic data. However, there is a large class of methods that give more reliable solutions, when compared to CGLS. In this paper, we are going to consider the merits of the GMRES-type methods when applied to the CT problem, introduce various strategies, and compare the results with CGLS
Kinetic modelling of metabolic pathways: Application to serine biosynthesis
Get PDFIn this chapter, we describe the steps needed to create a kinetic model of a metabolic pathway using kinetic data from both experimental measurements and literature review. Our methodology is presented by using the example of serine biosynthesis in E. coli
Free centre-by-metabelian Lie rings
Get PDFWe study the free centre-by-metabelian Lie ring, that is, the free Lie ring with the property that the second derived ideal is contained in the centre. We exhibit explicit generating sets for the homogeneous and fine homogeneous components of the second derived ideal. Each of these components is a direct sum of a free abelian group and a (possibly trivial) elementary abelian -group. Our generating sets are such that some of their elements generate the torsion subgroup while the remaining ones freely generate a free abelian group. A key ingredient of our approach is the determination of the dimensions of the corresponding homogeneous and fine homogeneous components of the free centre-by-metabelian Lie algebra over fields of characteristic other than . For that we exploit a -term exact sequence of modules over a polynomial ring that is originally defined over the integers, but turns into a sequence whose terms are projective modules after tensoring with a suitable field.
Our results correct a partly erroneous theorem in the literature
Probabilistic Merging Operators
No full textThe present work presents a general theoretical framework for the study of operators which merge partial probabilistic evidence from different sources which are individually coherent, but may be collectively incoherent. We consider a number of principles for such an operator to satisfy including a set of principles derived from those of Konieczny and Pino Perez which were formulated for the different context of propositional merging. Finally we investigate two specific such merging operators derived from the Kullback-Leibler notion of informational distance: the social entropy operator, and its dual, the linear entropy operator. The first of these is strongly related to both the multi-agent normalised geometric mean pooling operator and the single agent maximum entropy inference process, ME. By contrast the linear entropy operator is similarly related to both the arithmetic mean pooling operator and the limit centre of mass inference process, CM^infinity