## Uppsala Hosts Workshop on Computation in Engineering

**July 23, 1999**

Guest of honor Bertil Gustafsson confers with Robert MacCormack, who gave the last of the invited talks on important computational problems in engineering.

**Per Lötstedt and Michael Thuné**

A one-day workshop, "Computational Methods in Engineering," was held on June 11 at Uppsala University in Uppsala, Sweden, in honor of Bertil Gustafsson, who celebrates his 60th birthday this summer. Gustafsson, a professor of numerical analysis at Uppsala University, is known to many readers of *SIAM News* for his work on boundary conditions and stability for difference methods.

About 150 participants listened to six invited speakers: Heinz-Otto Kreiss, UCLA; Margaret Wright, Bell Laboratories, Lucent Technologies; David Gottlieb, Brown University; Gene Golub, Stanford University; Björn Engquist, Royal Institute of Technology, Stockholm, and UCLA; and Robert MacCormack, Stanford. The speakers gave excellent review presentations on important computational problems in the engineering sciences. The organizers were Per Lötstedt, Lina Hemmingsson-Frändén, Anders Sjöberg, and Michael Thuné.

The starting point for Heinz Kreiss (who had been Bertil Gustafsson's graduate supervisor) was the observation that, in practice, we often solve partial differential equations numerically without knowing whether the underlying analytic problem has a solution. The question is: Can we use the numerical results to decide---or, at the least, make a plausible case---that the analytic problem has a solution that is close to the numerical solution? After treating some smaller examples, Kreiss turned to the Navier-Stokes equations for incompressible flows, where he discussed the dependence of the overall structure of the solution on phenomena occurring at small scales. His examples from recent turbulence calculations show that while the influence of these phenomena may be small in the short term, it grows in the long term.

In presenting examples of numerical optimization at Bell Labs, Margaret Wright described three important classes of algorithms: simplicial methods without derivatives, such as the Nelder-Mead method; sequential quadratic programming (SQP); and interior point methods. Each is a very active research area.

In general, Wright explained, the trend seems to be toward a more selective use of optimization methods---the quest to find *the* generally best method is being abandoned. Her talk, appropriately titled "A Smorgasbord of Computational Methods in Optimization," was interspersed with references to the many delicacies found in a traditional smorgasbord. ("Smorgasbord" is one of the few Swedish words that have been adopted in English.) The choice of an appropriate mix to make a perfect meal served as a metaphor for a selective approach in the use of optimization methods.

Electromagnetic calculations for solving Maxwell's equations have attracted much attention in recent years. David Gottlieb, an advocate of high-order methods of the spectral or finite difference type for these problems, explained that the numerical boundary conditions at the outer boundary must absorb outgoing waves so that they are not reflected back in an unphysical manner. He showed that a modification of the perfectly matched layer (PML) method of Berenger can be used to obtain a stable and efficient absorbing layer.

Gene Golub described various iterative methods for solving systems of linear equations. After a historic survey of the subject, beginning with early work of Young on the SOR method, and of Hestenes and Stiefel on the conjugate gradient method, he continued with a discussion of recent work on convergence acceleration. If the matrix has a special structure, he said, it should be exploited, e.g., in the construction of a preconditioner. One example is the symmetric but indefinite matrix that appears after discretization of Stokes's equation. Golub derived convergence criteria for methods separated into an inner and outer iteration and for iterations with random initial vectors.

Björn Engquist took the opportunity to remind the audience of Bertil Gustafsson's role in establishing contacts between Swedish industry and academic researchers in scientific computing. An early example of Gustafsson's efforts was a 1972 report with a survey of then current industrial problems involving the numerical solution of partial differential equations. Using that report as a point of departure, Engquist drew comparisons with industrial problems of today. His modern examples were from the joint industry/university projects of the Parallel and Scientific Computing Institute at the Royal Institute of Technology and Uppsala University, which was founded by Engquist and Gustafsson and is co-funded by Swedish industry. Some areas mentioned in the 1972 report, e.g., shock calculations and the ability to treat complicated boundaries, have matured, Engquist pointed out. Others---such as adaptivity---are active research areas today, as foreseen by Gustafsson. Giving his presentation a humorous touch, Engquist showed some of the crudely hand-drawn illustrations from the 1972 report---a demonstration, he said, of the vast improvement in computer graphics in 27 years!

The last speaker, Bob MacCormack, explained that integration of the compressible Navier-Stokes equations in time or to steady state for high Reynolds numbers is a very time-consuming computational task. MacCormack showed how implicit methods with clever factorizations of the Jacobian can improve the situation.

The subsequent banquet was a fine celebration of Bertil Gustafsson. There were several appreciative and warm-hearted speeches. The joyful atmosphere was promoted also by the Swedish tradition of community singing, and by scenes written and performed by some of Bertil's PhD students. Most of Bertil's 17 former students and all of his current students attended the banquet. In June, the nights in Sweden are light and warm, and the party continued with dancing until the early morning.

*Per Lötstedt and Michael Thuné are professors in numerical analysis at Uppsala University. *