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. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the simultaneous tridiagonalization of two symmetric matrices

We discuss congruence transformations aimed at simultaneously reducing a pair of symmetric matrices to tridiagonal–tridiagonal form under the very mild assumption that the matrix pencil is regular. We outline the general principles and propose a unified framework for the problem. This allows us to gain new insights, leading to an economical approach that only uses Gauss transformations and orthogonal Householder transformations. Numerical experiments show that the approach is numerically robust and competitive.     

On the equivalence of the continuous Adams–Bashforth method and Nordsieck’s technique for changing the step size

Recent research has raised the question of whether Nordsieck’s technique for changing the step size in the Adams–Bashforth method is equivalent to the explicit continuous Adams–Bashforth method. This work provides a complete proof that the two approaches are indeed equivalent.     

On the stability of functionally fitted Runge–Kutta methods

Classical collocation RK 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 RK (FRK) 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 FRK method targeted at such a problem. However, FRK methods lead to variable coefficients that depend on the parameters of the problem, the time, the stepsize, 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 how to characterize the stability function of the methods in this particular class. We illustrate this explicitly with an example and we provide further insight for separable methods with symmetric collocation points. AMS subject classification (2000) 65L05, 65L06, 65L20, 65L60     

Fast generalized cross validation using Krylov subspace methods

The task of fitting smoothing spline surfaces to meteorological data such as temperature or rainfall observations is computationally intensive. The generalized cross validation (GCV) smoothing algorithm, if implemented using direct matrix techniques, is O(n ³) computationally, and memory requirements are O(n ²). Thus, for data sets larger than a few hundred observations, the algorithm is prohibitively slow. The core of the algorithm consists of solving series of shifted linear systems, and iterative techniques have been used to lower the computational complexity and facilitate implementation on a variety of supercomputer architectures. For large data sets though, the execution time is still quite high. In this paper we describe a Lanczos based approach that avoids explicitly solving the linear systems and dramatically reduces the amount of time required to fit surfaces to sets of data.      

Analysis of trigonometric implicit Runge–Kutta methods

Using generalized collocation techniques based on fitting functions that are trigonometric (rather than algebraic as in classical integrators), we develop a new class of multistage, one-step, variable stepsize, and variable coefficients implicit Runge–Kutta methods to solve oscillatory ODE problems. The coefficients of the methods are functions of the frequency and the stepsize. We refer to this class as trigonometric implicit Runge–Kutta (TIRK) methods. They integrate an equation exactly if its solution is a trigonometric polynomial with a known frequency. We characterize the order and A-stability of the methods and establish results similar to that of classical algebraic collocation RK methods.     

OpenMG: A new multigrid implementation in Python

In many large‐scale computations, systems of equations arise in the form Au = b, where A is a linear operation to be performed on the unknown data u, producing the known right‐hand side, b, which represents some constraint of known or assumed behavior of the system being modeled. Because such systems can be very large, solving them directly can be too slow. In contrast, a multigrid method removes different components of the error at different resolutions using smoothers that reduce high‐frequency components of the error more readily than low. Here, we present an open‐source multigrid solver written only in Python. OpenMG is a pure Python experimentation environment for testing multigrid concepts, not a production solver. The particular restriction method implemented is for ‘standard’ multigrid. By making the code simple and modular, we make the algorithmic details clear. The resulting solver is tested on an implicit pressure reservoir simulation problem with satisfactory results.Copyright © 2013 John Wiley & Sons, Ltd.     

On functionally-fitted Runge–Kutta methods

Functionally-fitted methods are generalizations of collocation techniques to integrate an equation exactly if its solution is a linear combination of a chosen set of basis functions. When these basis functions are chosen as the power functions, we recover classical algebraic collocation methods. This paper shows that functionally-fitted methods can be derived with less restrictive conditions than previously stated in the literature, and that other related results can be derived in a much more elegant way. The novelty in our approach is to fully retain the collocation framework without reverting back into derivations based on cumbersome Taylor series expansions. AMS subject classification (2000) 65L05, 65L06, 65L20, 65L60     

Numerical methods for the chemical master equation and applications to stochastic models or receptor oligomerisation

The chemical master equation is an important model for studying chemical kinetics, especially in the context of systems biology where noise is known to play an important role and so a discrete and stochastic framework is required. However due to the high dimensional nature of the equations it has often been felt to be too difficult to tackle numerically, although recently much progress has been made. Here some novel numerical methods are presented and applied to stochastic models of receptor oligomerisation, which have previously been studied only by simulation. This gives a novel perspective on these models and suggests some insights into the phenomena of the role they play in buffering cell signals. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)