Special Issue Devoted to Professor Ian Gladwell

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Pierluigi Amodio
Luigi Brugnano

Abstract

Dear SCPE readers,

This issue is devoted to Professor Ian Gladwell on the occasion of his retirement from the Mathematics Department of Southern Methodist University (SMU) in Dallas, Texas, USA. In his long career, Ian Gladwell was a member of the faculty of the University of Manchester (England) from 1967 to 1987 and a member of the faculty of SMU from 1987 to date. Whilst at SMU he was Chair of the Mathematics Department for two terms and served multiple terms as Director of Undergraduate Studies and as Director of Graduate Studies. Between 1975 and 2006 he supervised more than 20 Ph.D. students (see [1]) who are now working in the USA, Europe, and Asia.

Professor Gladwell has been very influential as an editor. This activity includes serving as Editor-in-Chief of the ACM Transactions on Mathematical Software from 2005 to date. He was also Associate Editor of the IMA Journal on Numerical Analysis; Scalable Computing: Practice and Experience; and the SIAM Journal on Numerical Analysis. He has served as editor for several books and special issues of journals. At present he serves as editor for the chapter in Scholarpedia devoted to Boundary Value Problems [2].

The many publications listed in [3] show that his research activity has focussed on the numerical integration of ordinary differential equations, quadrature, parallel computing, and mathematical software. His recent research has been concerned with the numerical solution of almost block diagonal (ABD) systems and applications. His most recent book is Solving ODEs with MATLAB, which he wrote with L.F. Shampine and S. Thompson. It was published by Cambridge University Press in 2003 [4].

Professor Gladwell was a pioneer in the development of mathematical software, especially software for the numerical solution of ordinary differential equations. Three software packages were published by the ACM and several of his programs were included in the NAG Fortran 77 library [5]. His association with NAG began in 1975 with his numerical ODE programs for the first NAG Library. He is a founder member of both the NAG Ltd. Technical Policy Committee and the NAG Inc. Advisory Panel. He has also been a long-term consultant for Texas Instruments.

The Special Issue contains five papers dealing with subjects related to the research activity of Ian Gladwell. Most of the authors had a fruitful collaboration with Ian in the past. We thank all of them to have agreed to our call.

The first paper Vectorized Solution of ODEs in Matlab is by L. F. Shampine, from the Southern Methodist University (USA). The author investigates a class of Runge-Kutta methods, able to efficiently exploit vectorization in the popular problem-solving environment Matlab. Local error estimates and continuous extensions that require no additional function evaluations are also derived. As a result, a (7,8) pair is derived and implemented in the program BV78 that well compares with the well-known Matlab ODE solver ode45, based on a (4,5) pair.

The second paper Conditioning and Hybrid Mesh Selection Algorithms for Two-Point Boundary Value Problems is by J. R. Cash, from the Imperial College, London (England), and F. Mazzia, from the University of Bari (Italy). The authors deal with the use of conditioning of the problem in the stepsize variation strategy. This allows to obtain very reliable algorithms, which are implemented in state of the art numerical codes for boundary value problems for ordinary differential equations. In particular, they speak about different choices of monitor functions that are used in the BVP codes and analyze the setting of the parameters in order to optimize the stepsize variation strategy.

The third paper Preconditioning of Implicit Runge-Kutta Methods is by L. O. Jay, from the University of Iowa, Iowa City (USA). A major problem in obtaining an efficient implementation of fully implicit Runge-Kutta (IRK) methods applied to systems of differential equations is to solve the underlying systems of nonlinear equations, usually obtained by application of modified Newton iterations with an approximate Jacobian matrix. In this article the author presents a cheap, and parallelizable, preconditioner for solving the linear systems with the approximate Jacobian matrix.

The fourth paper Parallel Numerical Solution of ABD and BABD Linear Systems Arising from BVPs is by P. Amodio, from the University of Bari (Italy), and G. Romanazzi, from the University of Coimbra (Portugal). The authors describe a parallel algorithm (based on the cyclic reduction) for the solution of linear systems with coefficient matrices having the ABD or the Bordered ABD (BABD) structures. They also report numerical tests involving parallel OpenMP versions of the Fortran 90 codes BABDCR and GBABDCR and compare them with COLROW. Finally, they discuss about the use of GBABDCR inside paralell version of BVP codes.

The fifth paper Parallel Factorizations in Numerical Analysis is by P. Amodio, from the University of Bari (Italy), and L. Brugnano, from the University of Florence (Italy). The authors review a number of parallel solvers for large, sparse, and structured linear systems through the use of the so called parallel factorizations, which provide parallel extensions of usual matrix factorizations. In particular, the paper is focused on the use of parallel factorizations for solving linear systems deriving from the numerical solution of ODEs. Moreover, the so called Parareal algorithm is derived within the framework of parallel factorizations.

Pierluigi Amodio,
Università di Bari,
Bari, Italy
Luigi Brugnano,
Università di Firenze,
Firenze, Italy

References

[1]. Ian Gladwell Graduate Student Supervision

[2]. Scholarpedia Boundary value problem

[3]. MathSciNet Publication results for Items authored by or related to Gladwell, Ian

[4]. L. F. Shampine, I. Gladwell, S. Thompson Solving ODEs with MATLAB Cambridge University Press

[5]. Nag Fortran library

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Section
Introduction to the Special Issue