Preface

Abstract

This special issue is dedicated to the memory of David M. Young, Jr. (October 20, 1923–December 21, 2008), who was an American mathematician and computer scientist affiliated with The University of Texas at Austin for most of his life. Dr. Young was one of the pioneers of modern numerical analysis. He was best known for establishing the mathematical foundation for the successive overrelaxation (SOR) method in his Harvard University doctoral research under the supervision of Professor Garret Birkhoff. In his 1950 dissertation, Iterative methods for solving partial differential equations of elliptic type, he showed that it was possible to automate the relaxation process for use on computing machines. Previously, it was believed that this could only be done using hand-computations on calculators by groups such as those under the direction of R. V. Southwell, for example. Furthermore, he derived an expression for the optimal relaxation factor under a condition, he named Property A. David Young held several jobs in industry, for instance, as a mathematician at the Computing Laboratory of the Aberdeen Proving Ground, Maryland, and as a manager of the Mathematical Analysis Department at the Ramo-Wooldridge Corporation, Los Angeles, California. He began his academic career at the University of Maryland, College Park. In 1958, he joined the faculty of the University of Texas in Austin as a Professor of Mathematics and as the first Director of the Computation Center. In 1970, he became the founding Director of the Center for Numerical Analysis, which is now a part of the University's Institute for Computational Engineering and Sciences. In 1982, David Young was awarded a distinguished professorship, an Ashbel Smith Professor of Mathematics and Computer Sciences, at The University of Texas at Austin. David Young is best known for his research on iterative solution methods for solving large sparse systems of linear equations. Such systems arise primarily in the numerical solution of partial differential equations, which had been discretized by either finite differences or finite elements. One of his early accomplishments was the analysis of the SOR method for consistently ordered matrices. Also, he was among the first to use the symmetric successive overrelaxation (SSOR) method, norms of matrices for iterative methods, Chebyshev polynomials for accelerating the iterative solution methods, adaptive procedures for determining iteration parameters, conjugate gradient methods with emphasis on generalizations for nonsymmetric systems, higher order discretization schemes for partial differential equations, and procedures for estimating the accuracy of the approximate solution of a linear system. Moreover, he studied vectorization and parallelization aspects of preconditioned iterative solution methods. David Young published numerous scientific papers and several books in numerical analysis. The two most relevant books for numerical linear algebra were Iterative Solution of Large Linear Systems, Academic Press, 1971, and Applied Iterative Methods, Academic Press, 1981, co-authored with Louis A. Hageman. Also, Professor Young wrote a two-volume book with the late Robert Todd Gregory, entitled A Survey of Numerical Mathematics, Addison-Wesley, 1972, 1973. (All of these books were republished by Dover.) David had not only a major impact on the development of scientific computing, but also on the lives of his students, colleagues, and friends. He was an international scientific leader during the first 50 years of the modern computer era. Professor Young was an inspiring teacher and afine gentleman. At conferences, David was friendly and generous with his time advising and counseling graduate students on both mathematical problems and personal issues. Dr. Young's spirit remains alive in the memories of all of those, like the undersigned, who had the privilege of knowing or working with him. The contents of this special issue reflect on David Young's life and research. All the authors of the papers have known and respected him for years. The initial contribution by David Kincaid, Richard Varga and Charles Warlick presents an outline of David Young's rich life, accomplishments, and work. It includes his family background in addition to his career and research. Next, Mary Wheeler and co-authors present a new method that deals with a topic that has been long been at the forefront of her research in Texas—the Darcy flow problems in porous media. A multi-scale preconditioning strategy is used that minimizes the computational costs associated with the construction of the multi-scale mortar basis functions. The contribution by Yuri Kuznetsov and coauthor deals with a new multi-level algebraic preconditioner for diffusion–reaction problems in highly heterogeneous media. The preconditioner is based on a special coarsening algorithm using an inner iterative procedure. This results utilizes the independence of the condition number of the preconditioned matrix on the diffusion coefficient. Owe Axelsson and Radim Blaheta present a general approach to construct preconditioners for finite element matrices applicable to symmetric and unsymmetric problems as well as for positive definite and indefinite problems. Special attention is devoted to problems of saddle point type. The paper by Yousef Saad and coauthors deals with an incremental LU factorization in applications with a slowly varying sequence of matrices that arise in the solution of time-dependent partial differential equations. To avoid wasteful recomputations of the entire LU factorization, various preconditioning techniques (such as based on approximate inverses) are used to enable cheaper updates of the incomplete factorizations. Tsun-Zee Mai and co-author present a treatment of interface lines in a strip-wise domain decomposition method leading to second-order accuracy. Also, the optimal overrelaxation parameter is estimated when this problem is solved using the SOR method. The contribution of Graham Carey and co-author discusses the influence of mesh quality on the condition numbers of the mass matrix, the stiffness matrix, and their computable bounds. In some cases, the mesh distortion causes the bounds on the condition number to be too big. The results are illustrated with several examples taken from practical applications. All contributions in this special issue deal with research topics of current interest that reflect, in various ways, the influence of David Young's work.