# Mathematician connects functions, disciplines, people

August 13, 2000(Blacksburg, Va., Aug. 14, 2000) -- As a "jack of all trades," Layne T. Watson has spent 25 years building multidisciplinary teams to solve complex, real-life problems. As a master of two of those trades -- mathematics and computer science -- he converted the esoteric theory of homotopies, a way to mathematically represent families of functions, into a tool now widely used by industry to solve computationally intensive problems.

For his pioneering research leading to the wide-spread adoption of homotopy methods in the solution of industrial optimization problems, the Virginia Tech professor of mathematics and computer science has earned the university's Alumni Award for Research Excellence. The award will be presented Sept. 20 at the Fall Awards Convocation.

Watson's work is at the interface of computer science, mathematics, engineering, and science. "The overall goal is to provide sophisticated mathematical software, justified by rigorous mathematical analysis, to attack significant practical engineering and scientific problems," he says.

In the case of homotopies, Watson says, "It is an instance of something abstract having a practical application." He explains that he used a mathematical representation of a family of functions to obtain a mathematical representation of a family of problems, where the problems are defined by functions.

For example, a difficult problem that is described by a system of equations is approached by finding a similar problem that is simpler and for which there is a solution, then constructing a homotopy -- a (mathematical) description of a family of problems that includes the easy problem and the difficult problem and a description of the path between the two. "I start with the easy problem for which I have a solution, go to an intermediate problem and solve it, and work my way to the hard problem. I am computing solutions to the intermediate problems to reach the solution to the difficult problem," says Watson. "The practical side is devising computer programs to automate the process, and then applying the computer algorithm to engineering and scientific problems."

In 1987, he introduced HOMPACK, a suite of codes for globally convergent homotopy algorithms. The software was the culmination of a 10-year project to make mathematically obscure homotopy methods available to the typical engineer, and to provide production quality software to the user community. HOMPACK is now in use at over 200 sites from Australia to Yugoslavia, and it has been successfully applied to problems in fluid mechanics, solid mechanics, composite materials, chemical engineering, economics, biology, robotics, computer vision, optimal control, circuit design, and nonlinear optimization. HOMPACK90: a suite of FORTRAN 90 codes for globally convergent homotopy algorithms, was distributed in 1997.

Two companies using the technology are General Motors and Lucent Technologies. "General Motors has many different applications where they need an algorithm that will converge from an arbitrary initial guess, for example, the design of linkage mechanisms. A car has numerous mechanical linkages," Watson points out. Lucent Technologies, a major developer and consumer of analog circuits, has rewritten their in-house circuit simulation code to use globally convergent homotopy methods. Homotopy methods have also been used for optimal control problems and spacecraft rendezvous calculations.

Watson's research also provided the first rigorous proof of a multidisciplinary design optimization (MDO) strategy.

MDO attempts to integrate multiple disciplines to solve large scale engineering design problems. For example, aircraft design involves aerodynamics, structures, propulsion, control, materials, manufacturability, economics. Historically, an aircraft would be designed by iterating a design sequentially through teams in each of these specialties. "But, passing the designs back and forth takes a long time and results in a suboptimal design, because each group modifies the others' designs," says Watson. "The name of the game in industry is 'time to market'. MDO is a strategy to reduce time to market and produce a better product. But to do a single integrated optimization involving all the disciplines and constraints at once is a monstrous computational problem, even using parallel computing or supercomputers. So a direct implementation of the MDO strategy hasn't been possible."

He was a member of a team that received the best paper award in 1997 at the Sixth AIAA/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. "I was providing the mathematical and computer science part of the paper with a bunch of aircraft design engineers. It was an interesting strategy paper, but we hadn't proved mathematically that the approach would always work," he says.

So he and J.F. Rodriguez and J.E. Renaud, both of Notre Dame, provided the first rigorous proof of a mathematical strategy for coordinating the different components for MDO. "We included nonlinear constraints -- any meaningful design must satisfy numerous constraints -- and we made fruitful use of all the data of varying accuracies produced in the different disciplines. The tricky part is how to manage the optimizations within the individual disciplines. Each would like to do what is optimal for their area without worrying about the constraints imposed by others. If you do that, your aircraft will cost too much or be too heavy or use too much fuel... You have to create a scheme where each unit can work semi-independently while still integrating with the other units. We created an MDO algorithm, and proved that the algorithm converged." The researchers published their work, "Convergence of trust region augmented Lagrangian methods using variable fidelity approximation data," in Structural Optimization in 1998.

Richard Tapia, Noah Harding professor of computational and applied mathematics at Rice University, calls Watson "one of the top computational scientists in the world and a world expert in the theory and application of homotopy methods."

But the mathematician says his most important research contributions have been as a team builder. The nomination committee agrees: "Dr. Watson is distinguished by the range of disciplines in which his research has produced significant and notable influence. His research in linear complementarity theory, polynomial root location, and convergence behavior has provided solid results in fundamental mathematics, while his research in numerical analysis has led to the solution of problems in biology, computer vision, statistics, and several engineering disciplines. His contributions to all aspects of homotopy establishment -- theory, software, and applications, and his work on applications have earned him worldwide recognition."

The committee also noted that Watson's "strength as a team-builder is widely recognized, and his ability to translate and explain technical terminology crossing a number of scientific and engineering disciplines is acknowledged as remarkable."

An example is his present membership in a watershed management group that includes individuals from biology, economics, different areas of civil engineering, fisheries and wildlife, computer science, and math. "We are working on a system to help government planners assess land use. How will a building or a shopping center or a housing development affect land value, water runoff, streams, fish? What will be the requirements for roads and sewer lines and how will this infrastructure affect everything? It involves a whole lot of people, who generally don't talk to each other, working together to create a geographic information system and a computer program that government planners can use," Watson explains. "So someone in civil engineering talking about modeling bank erosion and shallow water equations will get blank stares from the people in computer science. And when the computer scientists talk about object oriented Web interfaces and Java beans, they get blank stares from the engineers. I translate one group's jargon into another group's jargon. After they've worked together for awhile, they understand each other and don't need me."

"Dr. Watson's ability to function as a catalyst has undoubtedly led to research results that otherwise would have remain undiscovered," the nominating committee noted.

He has collaborated with people in disciplines ranging from accounting, biology, and engineering to law, wood science, and zoology, and published almost 300 refereed publications with more than 100 different co-authors. His work, ranging from applied basic computer technology to abstract mathematics, has been continuously funded by both public and private sources since his days as a student. Recent sponsored research includes a $599,000 NASA grant to study variable-complexity multidisciplinary optimization on parallel computers; a $192,000 Department of Energy grant to develop parallel software for nonlinear systems of equations; a $75,000 grant from Michelin to study large sparse linear systems arising in tire design; and a Virginia Tech research division (ASPIRES) grant to develop leadership in problem solving environments for ecosystem assessment, management, and policy. Currently, his work is funded by a $102,000 NASA grant for MDO on a teraflop computer; a $161,000 USDA grant for modeling of transient effects in hot-pressing wood-based composites; a $80,000 NSF grant for homotopy theory; a $509,000 Air Force grant to develop innovative local-global methods for wing structural design; a million dollar NSF grant for a collaborative problem solving environment for modeling broadband wireless communication systems; and a $300,000 NSF grant to study modeling and simulation uncertainties in design optimization.

Watson, who grew up in Warrick County, Indiana, received undergraduate degrees in mathematics and psychology from the University of Evansville in 1969, and a Ph.D. in mathematics from the University of Michigan in 1974. He worked as a computer systems consultant and taught at the University of Michigan and Michigan State University until he joined the Virginia Tech computer science department in 1978. He has also been a visiting scholar at the University of Michigan and a guest professor at the University of Notre Dame in various engineering disciplines as well as mathematics and computer science.

Honors began early, with the freshman math award at the University of Evansville, recognition from Phi Kappa Phi and other honor societies in the 1980s, and election as a fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 1993.

He is active in numerous international professional organizations in several disciplines and has served as associate editor of the Society of Industrial and Applied Mathematics Journal on Optimization, Computational Optimization and Applications, The International Journal of Evolutionary Optimization, and the international journal, Engineering Computations.

Watson also teaches courses at all levels for computer science and math majors and, during his career, has introduced courses on nonlinear programming, advanced numerical analysis and nonlinear optimization, mathematical software, parallel computing, high performance computing, and discrete mathematics.

-end-

PR CONTACT: Susan Trulove 540-231-5646 STrulove@vt.edu

Virginia Tech

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