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FUNM_QUAD: An implementation of a stable quadrature-based restarted Arnoldi method for matrix functions
Get PDFThis short note gives an overview of the FUNM_QUAD Matlab code which implements the restarted Arnoldi algorithm described in [A. Frommer, S. Güttel, M. Schweitzer: Efficient and stable Arnoldi restarts for matrix functions based on quadrature, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 1602--1624]
Detecting and Reducing Redundancy in Alarm Networks
Get PDFAlarm systems are vital for the safe operation of almost all large-scale industrial and technical installations, such as chemical plants or power stations. The optimization of alarm systems has great potential to improve the safety of these installations, and also to increase their profitability through the reduction of automated shut-downs and suboptimal operation modes.
In this work we present a new approach to alarm system optimization through the identification of redundant alarms. Our approach is based on a ranking of alarms by their connectivity in the alarm network. We also propose an overall redundancy measure for the alarm system which can be used to monitor performance improvements after redundant alarms have been removed. We present an example demonstrating that our ranking technique provides operational staff with useful information, allowing them to enhance the effectiveness of their existing alarm systems
Matching Exponential-Based and Resolvent-Based Centrality Measures
Get PDFThe relative importance of nodes in a network can be quantified via
functions of the adjacency matrix. Two popular choices of function are the
exponential, which is parameter-free,
and the resolvent function, which yields the Katz centrality measure.
Katz centrality can be the more computationally efficient,
especially for large directed networks,
and has the benefit of generalizing naturally to
time-dependent network sequences,
but it depends on a parameter.
We give a prescription for selecting the Katz parameter
based on the objective of matching the centralities of the
exponential counterpart.
For our new choice of parameter the resolvent can be very ill conditioned,
but we argue that the centralities computed in floating point arithmetic can
nevertheless reliably be used for ranking.
Experiments on \revised{six} real networks show that the new choice of Katz parameter
leads to rankings of nodes that \revised{generally} match
those from the exponential centralities well in practice
Matching Exponential-Based and Resolvent-Based Centrality Measures
Get PDFThe relative importance of nodes in a network can be quantified via
functions of the adjacency matrix. Two popular choices of function are the
exponential, which is parameter-free,
and the resolvent function, which yields the Katz centrality measure.
Katz centrality can be the more computationally efficient,
especially for large directed networks,
and has the benefit of generalizing naturally to
time-dependent network sequences,
but it depends on a parameter.
We give a prescription for selecting the Katz parameter
based on the objective of matching the centralities of the
exponential counterpart.
For our new choice of parameter the resolvent can be very ill conditioned,
but we argue that the centralities computed in floating point arithmetic can
nevertheless reliably be used for ranking.
Experiments on \revised{six} real networks show that the new choice of Katz parameter
leads to rankings of nodes that \revised{generally} match
those from the exponential centralities well in practice
Scaled and squared subdiagonal Padé approximation for the matrix exponential
Get PDFThe scaling and squaring method is the most widely used algorithm for computing the exponential of a square matrix A. We introduce an efficient variant that uses a much smaller squaring factor when ||A||>>1 and a subdiagonal Padé approximant of low degree, thereby significantly reducing the overall cost and avoiding the potential instability caused by overscaling, while giving forward error of the same magnitude as the standard algorithm. The new algorithm performs well if a rough estimate of the rightmost eigenvalue of A is available and the rightmost eigenvalues do not have widely varying imaginary parts, and it achieves significant speedup over the conventional algorithm especially when A is of large norm. Our algorithm uses the partial fraction form to evaluate the Padé approximant, which makes it suitable for parallelization and directly applicable to computing the action of the matrix exponential exp(A)b, where b is a vector or a tall skinny matrix. For this problem the significantly smaller squaring factor has an even pronounced benefit for efficiency when evaluating the action of the Padé approximant
Dynamic Network Analysis in Julia
Get PDFThis paper introduces EvolvingGraphs, a Julia software package for the creation, manipulation, and study of dynamic networks. We describe the underlying model of EvolvingGraphs and discuss the implementations
of components and centrality in the case of dynamic
networks. We make particular use of the parameterizable
type system and multiple dispatch in Julia. Users can work on a variety of graph types with nodes and timestamps of any Julia type
A model for Gaussian perturbations of graphene
Get PDFGraphene consists nominally of a regular planar hexagonal carbon lattice monolayer. However,
its structure experiences perturbations in the presence of external influences,
whether from substrate properties, thermal or electromagnetic fields, or ambient fluid movement.
Here we give an information geometric
model to represent the state space of perturbations as a Riemannian pseudosphere with
scalar curvature close to -1/2. This would allow the
representation of a trajectory of states under a given ambient or process change,
so opening the possibility for geometrically formulated
dynamical models to link structural perturbations to
the physics
Octad Orbits for certain Subgroups of M_{24}.
Get PDFUsing Curtis's MOG we display the orbits and orbit
representatives for various subgroups of the Mathieu group acting on the octads of the Steiner system . This information is deployed to study a graph associated with the largest simple Fischer group
Efficient block preconditioning for a C1 finite element discretisation of the Dirichlet biharmonic problem
Get PDFWe present an efficient block preconditioner for the two-dimensional biharmonic Dirichlet problem discretised by C1 bicubic Hermite finite elements. In this formulation each node in the mesh has four different degrees of freedom (DOFs). Grouping DOFs of the same type together leads to a natural blocking of the Galerkin coefficient matrix. Based on this block structure, we develop two preconditioners: a 2x2 block diagonal preconditioner (BD) and a block bordered diagonal (BBD) preconditioner. We prove mesh independent bounds for the spectra of the BD-preconditioned Galerkin matrix under certain conditions. The eigenvalue analysis is based on the fact that the proposed preconditioner, like the coefficient matrix itself, is symmetric positive definite and is assembled from element matrices. We demonstrate the effectiveness of an inexact version of the BBD preconditioner, which exhibits near optimal scaling in terms of computational cost with respect to the discrete problem size. Finally, we study robustness of this preconditioner with respect to element stretching, domain distortion and non-convex domains
Geometry in a Fréchet Context A Projective Limit Approach
No full textMany geometrical features of manifolds and fibre bundles modelled on Fréchet spaces either cannot be defined or are difficult to handle directly. This is due to the inherent deficiencies of Fréchet spaces; for example, the lack of a general solvability theory for differential equations, the non-existence of a reasonable Lie group structure on the general linear group of a Fréchet space, and the non-existence of an exponential map in a Fréchet-Lie group. In this book, the authors describe in detail a new approach that overcomes many of these limitations by using projective limits of geometrical objects modelled on Banach spaces. It will appeal to researchers and graduate students from a variety of backgrounds with an interest in infinite-dimensional geometry. The book concludes with an appendix outlining potential applications and motivating future research.
Features:
Proposes a new approach that overcomes many complications of the geometric theory.
Self-contained chapters and detailed proofs help the reader progress systematically through the book.
Includes an extensive introduction to the geometry of Banach manifolds and bundles.
Provides a number of suggestions for further research in the geometry and for applications, notably in physical field theory