After his visit to a Shell Research Laboratory, my high school teacher in math told us in class (now more than 40 years ago) that he was so happy with his education, because mathematics had helped him to understand the explanations and demonstrations that had been given by the Shell researchers. He said, "If you master mathematics then you potentially understand everything." That was certainly a slight exaggeration, but it nevertheless sounded like a golden message. Since I definitely wanted to have a better understanding of what was going on around me, mathematics seemed the obvious way to go. Also, if it was not much beyond high school math, then it was pretty easy in addition. What could one wish more? So I enrolled in Utrecht University in 1961. Pretty soon I discovered that mathematics was much more than a set of principles that helped one to solve intellectual riddles. It was not a finished system that one could aim to master after some limited time, but it was really a way of thinking, a means of expressing creativity: endless, an old established science, but still fresh and with undiscovered green meadows, nearby and far away.

I also learned that mathematics was more than merely an intellectual activity: it was a necessary tool for getting a grip on all sorts of problems in science and engineering. Without mathematics there is no progress. However, mathematics could also show its nasty face during periods in which problems that seemed so simple at first sight refused to be solved for a long time. Every researcher will recognize these periods of frustration and helplessness. My first position outside the academic world was with the Dutch Nuclear Research Center and it was an eye-opener for me in that mathematical techniques, in combination with computers, could be used for solving very complicated real-life problems, such as predicting and controlling the behavior of a nuclear reactor. I was deeply impressed by the numerical masterpieces of Jim Wilkinson and Dick Varga. They led the way in showing how one could overcome some serious limitations of computers for solving linear systems of equations. Such systems are immensely important, because most scientific computations lead in one way or another to the necessity to solve linear systems. Although the real world seems to be highly nonlinear, we have to linearize first in order to get insight and to produce meaningful solutions.

Many problems in physics, chemistry, engineering, earth sciences, etc., lead to very large systems, and it has always been a challenge for me to help shift our limits in tackling the computational complexity. It is very much the same as the drive of an athlete to try to break barriers. Research by many of us has now led to the ability to solve systems of, say, billions of unknowns. The increase in speed of computers and the human intelligence in discovering faster methods have contributed almost in equal part to the progress made since the early days of Gauss and Jacobi (who solved systems of order 7).

By doing this kind of research, one is highly rewarded for useful ideas. Other scientists use them to their advantage and report on their progress. This is what is hidden beneath the cool citation scores. It is a great feeling to realize that my work is used by so many other people.

My first success in research was in the mid-1970s, when I proposed, together with Koos Meijerink, the so-called incomplete LU decompositions of matrices, as a way to accelerate the convergence of the Conjugate Gradient method. Our ICCG method became a widely used tool. Then, around 1980, I became heavily inspired by the work of Lanczos, Paige and Saunders, Manteuffel, and others. This led to the completion of my Ph.D. thesis in 1982, at the age of 38. In 1984, I resumed a position in the academic word again: first as a professor at Delft University of Technology, and since 1990 as a professor at the University of Utrecht.

Being a professor I felt certainly obliged to something in return for the honor and I devoted much of my (spare) time to research. Together with my advisor Bram van der Sluis, I published a paper that helped to further the understanding of the Conjugate Gradients method ("The rate of convergence of conjugate gradients," Numer. Math., 48[5]: 543-60, 1986). Early ideas by Sonneveld (1984) for improvements in the bi-Conjugate Gradient (Bi-CG) method, for the solution of unsymmetric linear systems, intrigued me for a long time. Sonneveld had a brilliant idea for doubling the speed of convergence of Bi-CG for virtually the same computational costs: CGS. He also published a rather obscure method under the name of IDR. I doubt whether that paper got more than two or three citations altogether. The eventual understanding of that method and the reformulation of it, so that rounding errors had much less bad influence on its speed of convergence, led to the so frequently cited Bi-CGSTAB paper (1992). Since some of the more successful methods had been published in SIAM J. Sci. (for instance, GMRES, 1986), that journal was the obvious choice for me. Also the presentation of Bi-CGSTAB at one of the famous IBM workshops in Oberlech (what a pity that IBM stopped that activity!) was extremely helpful in making other scientists acquainted with the new technique.

Bi-CGSTAB is a surprisingly simple algorithm for the combination of two successful techniques: the fast but irregularly converging Bi-CG and the stabilizing effect of GMRES: some 15 lines of computer code. This has helped many people in research and industry solve their complicated computational problems. It has also stimulated further research in my own area. For instance, many new preconditioning techniques are used in combination with Bi-CGSTAB, which explains some of the many citations. Bi-CGSTAB has also been included in the popular mathematical research platform MATLAB.

Over the past few years, I have shifted my focus of attention to eigenvalue problems. This is also an exciting area, with many unsolved problems and with many applications in other sciences (plasma physics, astronomy, climate modeling, acoustics, mechanical engineering, etc.). My first success in this area has been the development of the Jacobi-Davidson method (with Gerard Sleijpen, 1996), which was awarded the SIAG-LA prize in 1998 (Sleijpen, G.L.G., and van der Vorst, H.A., "A Jacobi-Davidson iteration method for linear eigenvalue problems," SIAM J Matrix Anal. A., 17[2]: 401-25, April 1996). I trust that there are numerous nuggets to be uncovered in this research area and I hope to be able to find more of these. Meanwhile, I keep an eye on acceleration techniques. One never knows…

Henk A. van der Vorst
University of Utrecht
Mathematical Institute
Utrecht, The Netherlands