Ben Noble: A Grand Legacy in Applied Mathematics

March 13, 2007

James Case

Former students, collaborators, associates, and friends of the late Ben Noble celebrated his life and career in a two-part minisymposium at SIAM's 2006 Annual Meeting. The event, organized by Jagdish Chandra, furnished opportunities for assessing progress in the many applied areas in which Noble worked. Speakers highlighted his contributions (both direct and indirect) to those areas, which include linear algebra, generalized inverses, variational methods, dual extremum problems and nonlinear optimization, singular integral equations---with applications to electromagnetics and fracture/failure mechanics---numerical methods, symbolic algebra, and (after his retirement) computational techniques for the design of antenna arrays for medical imaging.

Chandra opened the symposium with a recitation of the bare facts: Benjamin Noble was born on May 1, 1922, in a fishing village near Aberdeen, Scotland, and died on January 5, 2006, near the home at Broughton Mills in the English Lake District to which he and his wife, Denise, had retired in 1985. After obtaining a bachelor's degree in radiophysics from the University of Aberdeen during the early years of World War II, Noble served King and Country by doing experimental work in underwater acoustics at the Admiralty Research Laboratory. After the war he obtained a master's degree from Cambridge University, and stayed on at the newly established Cambridge Mathematical Laboratory, where the early digital computer known as EDSAC was still under construction. He left Cambridge for the Anglo-Iranian Oil Company (now BP), where he stayed until 1952, when the University of Keele opened its doors.

After three years at Keele, he moved to the Royal College of Science and Technology (later the University of Strathclyde) in Glasgow, where he could continue to teach while pursuing a DSc at Aberdeen University. In 1962 he accepted a visiting position at the U.S. Army Mathematics Research Center (AMRC) at the University of Wisconsin–Madison; two years later, he became a permanent member and was also appointed professor of mathematics and computer science in the university. In 1975 he succeeded J. Barkley Rosser as director of the MRC (the inflammatory "Army" having been dropped from the name of the center during the student unrest of the Vietnam era), and served in that capacity until 1979. A self-described "Jack of all trades, master of none," Noble wrote four books during his career and directed 14 doctoral students. He retired from Wisconsin in 1985, whereupon he was named professor emeritus and returned to England to live.

Speaking in the minisymposium of Noble's contributions to linear algebra, Gil Strang of MIT pointed out that the teaching of linear algebra has changed more, and at more colleges and universities, than any other topic in the undergraduate curriculum. Noble was at the forefront of the change: His book on the subject was among the first to emphasize ideas and computations over formal proofs, and to include an early chapter on applications. The ones he chose---pin-jointed elastic frame networks, Markov transition matrices, and least-squares estimation---all combine practical importance with accessibility to students.

John R. Whiteman of Brunel University in the UK, who also owns property in the English Lake District, remained in close touch with Noble until the very last. Before turning to technical matters in his talk, he mentioned his delight on learning that Noble had read one of his early papers, liked it, and wondered whether the author would care to visit the MRC during the 1967–68 academic year. Whiteman did visit, joining a group of postdoctoral visitors that included Bob O'Malley, Joe Jerome, Larry Shoemaker, Dahlard Lukes, Anton Zetl, Krish Athreya, Horst Becker, Gerhard Wanner, Jim Daniel, and others who went on to become leaders in various applied fields. Whitehead never forgot Noble's insistence on (i) working on problems of genuine practical importance and (ii) doing "good mathematics" in the process. The latter, as applied to the solution of PDEs, seemed to mean that no answer was complete without careful error estimates.

The problem Whiteman described, which is similar to those he worked on with Noble at the MRC, involves thermoforming, the process whereby hot sheet metal is stamped into various shapes. The automobile industry uses thermoforming to produce hoods, doors, roofs, trunk lids, and fenders. The process is also used in the production of metal containers, and it is here that Whiteman's problem arises. The sheets are often painted and lettered before being stamped into their final shapes; any lettering applied before stamping will appear in distorted form in the final product. The letters in the (unstamped) pre-image must therefore be distorted in such a way that they will appear undistorted on the (stamped) final product. The problem, like other inverse mapping problems, can be numerically unstable.

Rounding out the minisymposium, Peter Linz of UC Davis and Ram Srivastava of SUNY Stony Brook discussed integral equations, one of Noble's earliest interests. Zuhair Nashed of the University of Central Florida described the application of generalized inverses to the solution of such equations, a subject that he and Noble discussed on many occasions.

Lucio Tavernini of the University of Texas, San Antonio, considered numerical methods for the solution of hybrid dynamical systems involving impulsive and hysteretic effects, and Alistair Spence of Bath University discussed the advantages---first described to him by Noble in 1972---of using known results concerning Newton's method to investigate the convergence of inverse iterative techniques. Since then, Spence said, similar ideas have been used to derive numerically stable methods for solving nonlinear eigenvalue and bifurcation problems. Most recently, the same idea has been brought to bear on the solution---within prescribed tolerances---of large sparse matrix eigenvalue problems.

Moayyed Hussain of General Electric R&D ended the sessions with a talk on the use of high-frequency (terahertz) electromagnetic radiation for target identification. The necessary components, such as sources, mixers, converters, and detectors, are all under active development. Because such components are and promise to remain large and unwieldy, however, sparse component arrays are needed. Hussain's work on sparse component arrays has built to some extent on the work undertaken by Noble in his later years on the scaling of antenna arrays for medical imaging.

All in all, it was an interesting and informative minisymposium, as well as a pleasant reunion of Noble's former colleagues and mentees, and a fitting tribute to a pioneer in the use of computers to solve engineering problems.

James Case writes from Baltimore, Maryland.

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