D. M. Young, Alter- Pos

comm-young.qxp 4/17/98 8:39 AM Page 1446 Garrett Birkhoff and Applied Mathematics David M. Young Introduction The Automation of Relaxation Methods Garrett Birkhoff contributed to many areas of With the advent of high-speed computers in the mathematics during his long and distinguished 1940s Birkhoff became very interested in their career. He is, of course, very well known for his possible use for obtaining numerical solutions to work in algebra and in lattice theory. However, in problems involving elliptic PDE. Many such problems could be “reduced”, by the use of finite dif- this article we will focus on his work in applied ference methods, to the solution of a (usually) mathematics, including the numerical solution of large system of linear algebraic equations where elliptic partial differential equations, reactor cal- the matrix was very sparse. However, because of culations and nuclear power, and spline approxi- the relatively low speeds and the very limited mem- mations. We will also give a very brief discussion ory sizes of computers which were then available, of his work on fluid dynamics. Additional infor- the direct solution of such systems was usually out mation on Birkhoff’s work in applied mathemat- of the question. ics can be found in many of the publications listed On the other hand, many such large linear sys- below; see especially [11]. tems were actually being solved by R. V. Southwell The author gratefully acknowledges the contri- and his associates in England using relaxation butions of Richard Varga and Carl de Boor. Varga methods and without using computers; see [39]. contributed the section entitled “Reactor Calcula- Relaxation methods involve first choosing an ini- tions and Nuclear Power”, and de Boor contributed tial guess for the unknown solution, u, at each grid the section entitled “Spline Approximations”. point and then computing at each point the “residual”, i.e., a number which measures the amount by The Numerical Solution of Elliptic Partial which the linear equation for that point fails to be Differential Equations satisfied. One can eliminate, or “relax”, the resid- In this section we describe two aspects of Birkhoff’s ual at a given grid point by suitably modifying the work on the numerical solution of elliptic partial value of u at that point. (If one “overcorrects” or differential equations (PDE), his role in the au- “overrelaxes”, then the sign of the residual is tomation of “relaxation methods”, and his work on changed.) Of course the residuals at nearby grid the dissemination of information on the numeri- points are also changed when the value of u at a cal selection of elliptic PDE. Additional work of Birk- particular grid point is changed. By repeated use hoff in this area is described in the section enti- of relaxation a skilled person could soon achieve tled “Reactor Calculations and Nuclear Power”. a situation where all of the residuals were very small and where the values of u at the grid points David M. Young is the Ashbel Smith Professor of Math- provided a satisfactory solution to the problem. ematics and Computer Sciences at The University of Texas In the late 1940s when I asked Birkhoff for a the- at Austin. His e-mail address is [email protected]. sis topic, he suggested that I work on the 1446 NOTICES OF THE AMS VOLUME 44, NUMBER 11 comm-young.qxp 4/17/98 8:39 AM Page 1447 “automation” of relaxation methods. Actually there tains a wealth of information and is recommended was already a systematic iteration procedure avail- reading for anyone interested in working in this able, namely, the “Liebmann method” [34] (which area. is a special case of the Gauss-Seidel method). How- The two conferences provided, among other ever, the Liebmann method is often exceedingly things, forums for discussions about the ELLPACK slow. Another method that was available at the time software package that was being developed at Pur- was Richardson’s method [36]. This method in- due University by John Rice and his associates. Con- volves the use of a number of parameters. How- tributions to ELLPACK were made by a number of ever, at the time it was not obvious how the para- other institutions. For example, several iterative meters should be chosen. (It was discovered later programs were contributed by The University of that by a suitable choice of the parameters, which Texas. could be found using Chebyshev polynomials, one David Kincaid and David Young, who directed could obtain very rapid convergence; see, e.g., [38] the development at The University of Texas of the and [43].) ITPACK software package for solving large sparse Largely as a result of the stimulus, encourage- linear systems by iterative methods, regard Birk- ment, and many useful suggestions provided by hoff as the “godfather” of the project. For several Birkhoff, I was able to develop a method which is years he had been patiently but seriously sug- now called the “successive overrelaxation” (SOR) gesting that such a package be developed. The im- method and which is described in [41, 42]. (The SOR plementation of his idea was delayed in part by un- method was developed independently by Frankel certainty as to how to choose the iteration [31], who called it the “extrapolated Liebmann” parameters, such as omega for the SOR method, method.) The SOR method provides an order-of- and how to decide when to terminate the iteration magnitude improvement in convergence as com- process. Eventually, as described in the book by pared to the Gauss-Seidel method for many linear Hageman and Young [32] and in the paper by Kin- systems corresponding to the numerical solution caid et al. [33], these and other obstacles were of elliptic PDE. Thus, for a class of problems cor- largely overcome and the ITPACK software pack- responding to the Dirichlet problem the number age was completed. of iterations required for convergence with the SOR method is proportional to h-1, where h is the Reactor Calculations and Nuclear Power grid size, as compared with h-2 as required with Garrett Birkhoff was intimately associated with the Gauss-Seidel method. reactor computations which played an essential The SOR method, with generalizations, modifi- role in the design of nuclear power reactors. This cations, and extensions (see, e.g., Varga [40]), was arose primarily from his role as a consultant to the used extensively for engineering and scientific Bettis Atomic Power Laboratory from 1955 through computations for many years. Eventually it was su- the early 1960s. perseded by other methods, such as precondi- As a brief background, analytical models of nu- tioned conjugate gradient methods and methods clear reactors were brand new in the early 1950s, based on the use of Chebyshev polynomials. unlike the case of analytical fluid dynamics, which Further discussion of Birkhoff’s role in the au- had enjoyed two hundred years of development. tomation of relaxation methods can be found in Fortuitously, high-powered digital computers were [11]. also making their appearance in the early 1950s. Dissemination of Information on the Numerical Because building full-scale nuclear reactors was Solution of Elliptic PDE both expensive and very time consuming, it was Birkhoff was very active in the dissemination of in- prudent and farsighted then to look to digital com- formation on the numerical solution of elliptic puters to numerically solve the associated nuclear PDE. This activity included the preparation of a reactor models. Even more fortuitous was the si- book with Robert Lynch (see [16]) and playing a multaneous emergence in 1950 of David M. Young’s leading role in the arranging of two conferences thesis [41], which contained an analytic treatment on “Elliptic Problem Solvers”. The first of these con- of the SOR iterative method for numerically solv- ferences was held in Santa Fe in 1980 and led to a ing second-order elliptic boundary problems. publication; see [37]. The second conference was In that exciting period when nuclear reactors held in Monterey in 1982 and also led to a publi- were first being considered for naval ships, Bettis cation; see [20]. hired in 1954 five new Ph.D.s—Harvey Amster, The book with Lynch provides an excellent sur- Elis Gelbard, and Stanley Stein in physics, and vey of many topics, including formulations of typ- Jerome Spanier and Richard Varga in mathematical elliptic problems and classical analysis, dif- ics—all of whom made contributions to various as- ference approximations, direct and iterative pects of nuclear reactor theory. There is no doubt methods, variational methods, finite element meth- that detailed discussions with the energetic con- ods, integral equation methods, and a description sultant, Garrett Birkhoff, helped solidify many of of the ELLPACK software package. The book con- their emerging ideas. Garrett loved the challenge DECEMBER 1997 NOTICES OF THE AMS 1447 comm-young.qxp 4/17/98 8:39 AM Page 1448 of working in new research areas, and his enthu- els. The idea was to determine the free parameters siasm was infectious! in a suitably flexible mathematical model so as to But Garrett’s contributions to reactor theory fit closely to measurements taken from the finished and reactor computations were much more than physical model of the car. There was also the hope just the random discussions of a consultant with that eventually the design process itself could be Bettis people. Three solid contributions of his carried out entirely on computers. stand out. Early on he saw the relevance of non- Birkhoff was quick to recommend the use of negative matrices (or, more generally, operators cubic splines (i.e., piecewise cubic polynomial func- which leave a cone invariant) to nuclear reactor the- tions with two continuous derivatives) for the rep- ory, and this can be seen in his publications [3] and resentation of smooth curves.

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