Evaluation of the performance of inexact GMRES

AbstractThe inexact GMRES algorithm is a variant of the GMRES algorithm where matrix–vector products are performed inexactly, either out of necessity or deliberately, as part of a trading of accuracy for speed. Recent studies have shown that relaxing matrix–vector products in this way can be justified theoretically and experimentally. Research, so far, has focused on decreasing the workload per iteration without significantly affecting the accuracy. But relaxing the accuracy per iteration is liable to increase the number of iterations, thereby increasing the overall runtime, which could potentially end up being greater than that of the exact GMRES if there were not enough savings in the matrix–vector products. In this paper, we assess the benefit of the inexact approach in terms of actual CPU time derived from realistic problems, and we provide cases that provide instructive insights into results affected by the build-up of the inexactness. Such information is of vital importance to practitioners who need to decide whether switching their workflow to the inexact approach is worth the effort and the risk that might come with it. Our assessment is drawn from extensive numerical experiments that gauge the effectiveness of the inexact scheme and its suitability for use in addressing certain problems, depending on how much inexactness is allowed in the matrix–vector products

On the stability function of functionally-fitted Runge--Kutta methods

Classical collocation Runge--Kutta methods are polynomially fitted in the sense that they integrate an ODE problem exactly if its solution is an algebraic polynomial up to some degree. Functionally fitted Runge--Kutta methods are collocation techniques that generalize this principle to solve an ODE problem exactly if its solution is a linear combination of a chosen set of arbitrary basis functions. Given for example a periodic or oscillatory ODE problem with a known frequency, it might be advantageous to tune a trigonometric functionally fitted Runge--Kutta method targeted at such a problem. However, functionally fitted Runge--Kutta methods lead to variable coefficients that depend on the parameters of the problem, the time, the step size, and the basis functions in a non-trivial manner that inhibits any in-depth analysis of the behavior of the methods in general. We present the class of so-called separable basis functions and show that it is possible to characterize the stability region of some special methods in this particular class. Explicit stability functions are given for some representative examples

An Implementation of The Method of Moments on Chemical Systems with Constant and Time-dependent Rates

Among numerical techniques used to facilitate the analysis of biochemical reactions, we can use the method of moments to directly approximate statistics such as the mean numbers of molecules. The method is computationally viable in time and memory, compared to solving the chemical master equation (CME) which is notoriously expensive. In this study, we apply the method of moments to a chemical system with a constant rate representing a vascular endothelial growth factor (VEGF) model, as well as another system with time-dependent propensities representing the susceptible, infected, and recovered (SIR) model with periodic contact rate. We assess the accuracy of the method using comparisons with approximations obtained by the stochastic simulation algorithm (SSA) and the chemical Langevin equation (CLE). The VEGF model is of interest because of the role of VEGF in the growth of cancer and other inflammatory diseases and the potential use of anti-VEGF therapies in the treatment of cancer. The SIR model is a popular epidemiological model used in studying the spread of various infectious diseases in a population

Lumpability Abstractions of Rule-based Systems

The induction of a signaling pathway is characterized by transient complex formation and mutual posttranslational modification of proteins. To faithfully capture this combinatorial process in a mathematical model is an important challenge in systems biology. Exploiting the limited context on which most binding and modification events are conditioned, attempts have been made to reduce the combinatorial complexity by quotienting the reachable set of molecular species, into species aggregates while preserving the deterministic semantics of the thermodynamic limit. Recently we proposed a quotienting that also preserves the stochastic semantics and that is complete in the sense that the semantics of individual species can be recovered from the aggregate semantics. In this paper we prove that this quotienting yields a sufficient condition for weak lumpability and that it gives rise to a backward Markov bisimulation between the original and aggregated transition system. We illustrate the framework on a case study of the EGF/insulin receptor crosstalk.Comment: In Proceedings MeCBIC 2010, arXiv:1011.005

Implementation of a variable block Davidson method with deflation for solving large sparse eigenproblems

The Davidson method is a preconditioned eigenvalue technique aimed at computing a few of the extreme (i.e., leftmost or rightmost) eigenpairs of large sparse symmetric matrices. This paper describes a software package which implements a deflated and variable-block version of the Davidson method. Information on how to use the software is provided. Guidelines for its upgrading or for its incorporation into existing packages are also included. Various experiments are performed on an SGI Power Challenge and comparisons with ARPACK are reported

Preface to the proceedings

This Special Part of the ANZIAM Journal (supplement) contains the refereed papers from the tenth Biennial Computation Techniques and Applications Conference (CTAC2001) held at the University of Queensland, Brisbane, during 16-18 July 2001. The CTAC series of conferences is held under the auspices of the Computational Mathematics Group of the Australian and New Zealand Industrial and Applied Mathematics (ANZIAM) division of the Australian Mathematics Society. They provide a forum for scientists, engineers, and mathematicians interested in the development of computational techniques and their application to problems of practical importance. Previous conferences in the CTAC series have been held at University of Sydney (1983), University of Melbourne (1985), University of Sydney (1987), Griffith University, Brisbane (1989), University of Adelaide (1991), Australian National University, Canberra (1993), Swinburne University of Technology, Melbourne (1995), University of Adelaide (1997), Australian National University, Canberra (1999). There were about 100 participants at the conference, five of whom gave keynote addresses as part of the general program for CTAC2001-these speakers and the titles of their presentations were: Real Time Optimisation and Nonlinear Model Predictive Control for Large DAE Models Prof Hans-Georg Bock, Interdisciplinary Center for Scientific Computing (IWR), Heidelberg, Germany; Some Computational Problems Arising in Continuous Time Finance Prof Carl Chiarella, School of Finance and Economics, University of Technology, Sydney, Australia; Matrix Decomposition Algorithms in Finite Element Methods for Poisson's Equation Prof Graeme Fairweather, Mathematical and Computer Sciences, Colorado School of Mines, Golden, USA. Computational Problems in Landscape Design for Conservation Prof Hugh Possingham, Departments of Zoology and Mathematics, University of Queensland, Brisbane, Australia; Iterative Methods for Nonsymmetric Ill-posed Problems Prof Lothar Reichel, Kent State University, Ohio, USA; In addition to these invited presentations, 74 contributed papers were presented at the conference. Of these, 48 were submitted for consideration for the CTAC2001 Proceedings, and 43 were accepted for publication after each was peer-reviewed by at least two referees and the papers revised to the satisfaction of the referees. Following the convention established in previous CTAC proceedings, the papers appearing here have not been grouped by topic; the abstracts of the five invited speakers are included, followed by the contributed papers in alphabetical order of the first author. As with previous CTAC Proceedings, the papers cover a wide variety of topics with the usual CTAC emphasis on both industrial applications and numerical techniques. A new feature is the marked number of presentations in the areas of financial mathematics and computational biology. The editors thank the referees for their time and assistance with the refereeing process, and the ANZIAM Journal of the (supplement) for agreeing to publish the proceedings as a journal special issue. Also, the editors would like to especially acknowledge the hard work of the Organising Committee and in addition, Jerard Barry, Markus Hegland and Yvette Zuidema. CTAC2001 Organising Committee A/Prof John A. Belward (Dept of Mathematics, University of Queensland), Prof K. Burrage (Chair, Dept of Mathematics, University of Queensland), Dr G. Chandler (Dept of Mathematics, University of Queensland), Dr K. Gates (Dept of Mathematics, University of Queensland), Dr Graeme J. Pettet (School of Mathematical Sciences, Queensland University of Technology), Dr Wayne Read (James Cook University), Prof Tony Roberts (Dept of Mathematics and Computing, University of Southern Queensland), Dr Roger B. Sidje (Dept of Mathematics, University of Queensland), Dr Elliot Tonkes (Department of Mathematics, University of Queensland). Acknowledgements Financial sponsorship from the following organisations is gratefully acknowledged: The University of Queensland Australian Partnership for Advanced Computing (APAC) Silicon Graphics, Inc. (SGI

Rational approximation to the Fermi-Dirac function with applications in Density Functional Theory

We are interested in computing the Fermi-Dirac matrix function in which the matrix argument is the Hamiltonian matrix arising from Density Function Theory (DFT) applications. More precisely, we are really interested in the diagonal of this matrix function. We discuss rational approximation methods to the problem, specifically the rational Chebyshev approximation and the continued fraction representation. These schemes are further decomposed into their partial fraction expansions, leading ultimately to computing the diagonal of the inverse of a shifted matrix over a series of shifts. We descibe Lanczos methods and sparse direct method to address these systems. Each approach has advanatges and disadvatanges that are illustrated with experiments

Functionally fitted explicit pseudo two-step Runge-Kutta methods

Explicit pseudo two-step Runge–Kutta (EPTRK) methods belong to the wider class of general linear multistep methods. The particularity of EPTRK methods is that they do not use the last two iterates as conventional two-step methods do. Rather, they predict the intermediate stage values and combine them with the last iterate to obtain the next iterate. EPTRK methods were initially designed to suit parallel computers, but they have been shown to achieve arbitrary high-order and thus can be useful as conventional explicit RK methods on sequential computers as well. Our contribution in this paper is to present a new family of functionally fitted EPTRK methods aimed at integrating an equation exactly if its solution is a linear combination of a chosen set of basis functions. We use a variation of collocation techniques to show that this new family, which we call FEPTRK, shares the same accuracy properties as EPTRK. The added advantage is that FEPTRK can use specific fitting functions to capitalize on the special properties of the problem that may be known in advance