Simulations Tackle Ever More Complex Problems
By Rolf Jeltsch - Swiss Federal Institute of Technology (ETH), Zürich, Switzerland
Professor & Head of the Seminar for Applied Mathematics, Swiss Federal Institue of Technology (ETH), Zurich; President of GAMM (the International Association of Applied Mathematics and Mechanics); and Congress Director of the 6th International Congress on Industrial an Applied Mathematics (ICIAM 07)
In 1949, Professor Stiefel - my "academic grandfather" and founder of the Seminar for Applied Mathematics - rented the Zuse Z4 for complex calculations. It used 2200 relays and a mechanical memory of 64 words. He first used this machine to solve a 4th-order partial differential equation related to the damming of a water reservoir and to solve a system of eight ordinary differential equations to calculate rocket flight. But it was also this machine for which Stiefel developed the conjugate gradient method, and Rutishauser was led through numerical experiments for stability analysis of linear multistep methods.
Computers developed quickly, and research has meanwhile concentrated on developing fast, reliable algorithms for basic numerical problems such as solving systems of ODEs along with quadrature, linear system, and eigenvalue problems. Those early days saw the creation of perfectly coded algorithms. They were originally written in Algol, but in the 70s they were converted into FORTRAN and extended to become EISPACK and LINPACK.
Enter: easy-to-use software
The 90s saw the emergence of easy-to-use software such as COMSOL Multiphysics, which could be used in standard situations by engineers at least for problems that could be mathematically well described.
In the meantime, computers have increased their speed dramatically. While the Z4 needed 0.5s for an addition operation and 6 s for a division operation, the fastest computer today produces 367 teraFLOPS, which is 2*1012 faster. As for memory, when I asked for 4 GB in 1990, the device that ETH installed was a 1.20 meter cube in an air-conditioned room�and today memory sticks have the same capacity.
These developments have brought the ability to solve engineering problems of high complexity. You are no longer limited to computing just a simple flow, but a simulation today can deal with, for instance, chemical reactions in combustion. With tools such as COMSOL Multiphysics, engineers can develop all sorts of better high-tech products, such as making cars more fuel efficient and safer.
All this would be impossible without the speedy implementation of novel algorithms. One good example is the PARDISO solver for shared-memory multiprocessors, developed in a PhD project at ETH Zurich just a few years ago. Today it is an integral part of COMSOL Multiphysics. In the near future even laptop PCs will have multiple processors, and so that engineers can take advantage of them, many new algorithms will appear or have to be rewritten.
Meanwhile, fundamental research is tackling even more complex problems. For instance, my colleagues are investigating simulations for high current arc plasmas. On the algorithm front we are studying better finite-element methods, meshing, linear algebra, and stochastic differential equations. In particular, with algorithms we are focusing on "breaking the curse of dimensionality" such as in sparse tensor product methods for radiative transfer. In order to solve ever more demanding problems, a key feature seems to be the development of algorithms focusing on the reduction of complexity. We have seen enormous progress here recently, but it will not be simple to put them into practice for 3D engineering problems.
Going after the big problems
With large memory, high computational speeds, and new algorithms we can approach problems much larger and more complex than ever before. It's even possible today to model the world's climate with a crude model. Yet there is a long way to bring such models to perfection. As for other fields, we have started modeling in nanotechnology, biology, and medicine. Biomolecular simulations still need an increase in computer power, memory, and algorithm speed. In engineering, new algorithms must be developed; for instance, for high-current switches where there is still no off-the-shelf software and in fact even the physical model is not quite clear.
All these application areas are so complex that mathematicians can no longer prove convergence and we need techniques to convince ourselves of the validity of the calculations. In our project on magnetohydrodynamic (MHD) solvers in real 3D geometries we are trying to test the quality of methods, such as Riemann solvers, and programs through the development of multidimensional test examples.
I look forward with fascination to the new problems that challenge us on the research front. To get a taste of where research in algorithms and numerics is heading, I invite you to attend ICIAM 07, being held from July 16-20 in Zurich, Switzerland. We meet only every four years, and I promise that this year´s event will have plenty of excitement and stimulating discussions.
References
- http://www.epemag.com/zuse/
- A.A. Grau, U. Hill, H. Langmaack, Handbook for Automatic Computation: Translation of Algol 60 (Grundlehren der mathematischen Wissenschaften, Vol 137), Springer, 1965.
- M. Gutknecht, "Numerical Analysis in Zurich - 50 Years Ago," Intelligencer, Springer 2007 (to appear).
- http://www.iciam07.ch