OBSERVATIONS ON COMPUTATIONAL The International Symposium on Computational Mathematics, held at INTERNATIONAL SYMPOSIUM MATHEMATICS Introduction by David K. Kahaner and C.T. Kelley (Kahaner attended a small portion of ISCM was important because it The International Symposium on Computational Mathematics (ISCM) was held at the ANA Hotel in Matsuyama, Japan, from 30 August to 4 September 1990. Matsuyama is on the island of Shikoku, across the Inland Sea from Hiroshima on the main island of Honshu. The local organizing committee of T. Yamamoto (chair), T. Kitagawa, and M.-T. Noda from Ehime University in Matsuyama and X. Chen from Xi'an Jiaotong University in China worked for 2 years to plan and attract support for the meeting. A number of corporate and government sponsors from Matsuyama and Japan supported the meeting and most of the overseas visitors. Approximately 100 invited scientists attended, including about a dozen from the United States, Discussion seven from China, and others from Germany, Austria, The Netherlands, France, Switzerland, Finland, and Czechoslovakia. The meeting was timed to immediately follow the very large (4,000 attendees) International Congress of Mathematicians in Kyoto. The sessions began with an opening address by Prof. R. Glowinski of the University of Houston on domain decomposition methods. The formal sessions covered topics including eigenvalue problems in numerical linear algebra, preconditioning methods for solution of linear and nonlinear equations, acceleration methods for iterative schemes, continuation and bifurcation, inverse problems, monotonicity methods, boundary element methods, complex variable methods and numerical conformal mapping, quadrature rules, ordinary differential equations, finite element computations, computational fluid dynamics, interval methods, and many other topics in numerical analysis and scientific computations. A list of titles and authors is available by writing to D.K. Kahaner. Special informal sessions were held in the evenings on nonlinear equations, harmonic methods, and interval arithmetic. These informal sessions were well attended and lasted until quite late. Some of the most spirited discussions took place at this time. There were a number of talks and one informal session organized by Prof. T. Torii (Nagoya Univ.) on Durand-Kerner (DK) like methods for the computation of roots of polynomials. This is a fascinating topic that has engaged scientists for more than 100 years. The DK scheme essentially applies Newton's method to the symmetric functions of the roots as expressed by their coefficients. It was first pro posed by Weierstrass and has reappeared every few years since. Generally, DK-like methods were considered rather extensively during the whole meeting. From a point of view of pure numerical analysis, there were several interesting results, particularly on the behavior of the methods in the presence of multiple zeros and on interconnections with approximation theory. Other than for intellectual reasons, though, why should one study methods for polynomial zeros anymore? One reason for the Japanese interest is that M.-T. Noda (Ehime University) has been considering combined symbolic and numeric computation, a very good idea in our opinion. One of his prototype problems is evaluation of integrals and he demonstrated convincingly that for some problems numeric quadrature is inefficient or incapable, and pure symbolic computation is theoretically insufficient. At the moment these hybrid methods are still very primitive, but the concept of combining these techniques has been a goal for many years. Some symbolic systems, such as Mathematica, revert to numerical approximation whenever they cannot make progress, but Noda's idea is to develop new hybrid algorithms that will combine the best features of both approaches. With respect to DK methods, though, some of the Western scientists felt that more attention was paid to them than they deserved, and that numerical continuation methods, such as global Newton methods, are suited to handle more general problems more efficiently. On the other hand, one of us (Kahaner) has used DK methods over the years and they have performed very well in practice, whereas continuation methods are often troublesome and frequently require special tweaking. Another informal session was on "Methods for Enclosing Zeros of Systems of Nonlinear Equations," organized jointly by A. Frommer and G. Alefeld (Univ. of Karlsruhe). They were requested to organize this session by several scientists at the meeting, and, correspondingly, the audience was large, indicating that there was a good deal of interest. There were three lectures in this session: • "On the Convergence and Divergence of Krawczyk-Like Methods" (G. Alefeld) • "Efficient Modifications of the Interval Newton's Method" (A. Frommer) • "Centered Newton's Method" (K. Tanabe, Inst. of Stat. Math., Tokyo) The first two lectures tried to give an introduction to the basics of enclosure methods for systems of nonlinear equations as well as some advanced results on the convergence, speed of convergence, and computational complexity of state-of-the-art algorithms in this field. One of the things that appears to be fundamental for all enclosure schemes to work properly is that precise floating point arithmetic must be available. At the plenary lectures of H. Stetter (Tech. Univ. of Vienna) and Alefeld, it was pointed out that this has actually happened in several languages such as FORTRAN-SC and -XSC, PASCAL-SC and -XSC (developed in Karlsruhe by Kulisch), and ACRITHXSC (an IBM product). Tanabe's lecture discussed a modification of Newton's method. The principal idea is that in the case of an initial guess which is far from a zero, instead of reducing the step-length of the Newton-correction, it can be very advantageous to perform a step of full length, but in an altered direction which is determined in a well-defined manner through the solution of an easy mini mization problem. Several examples shown by Tanabe demonstrated that this modification of Newton's method is particularly useful when solving nonlinear programming problems. Many of the speakers from Japan and China spoke on topics related to the solution of nonlinear equations when the Jacobian is singular at the root. This was an active research area in the West in the late 1970s-early 1980s and some activity continues in East Germany. This topic was directly addressed in N. Osada's talk (Nagasaki Inst. of Applied Sci.) on extrapolation methods and indirectly discussed in the talks on bifurcation by X. Sun (Harbin Institute of Electrical Technology) and T. Tsuchiya (Kyushu Univ.). Discussions with X. Chen and Sun on this area made us aware that there is substantial Japanese and Chinese literature on this topic. The technical issues center on the fact that if the Jacobian is singular at the root, then (1) the convergence rates of Newton-like methods are degraded and (2) the set from which the initial iterate can be taken is no longer a ball about the solution but a special set whose form depends on the structure of the singularity. Research problems are (1) to describe the regions of attraction and convergence rates for Newtonlike methods and (2) to accelerate their convergence. In Japan, Osada, Tsuchiya, N. Yamamoto (Kyushu Inst. of Tech.), and T. Yamamoto (Ehime University) have worked on this. In China, Dr. X. Chen, Prof. X. Sun, and Prof. Z. Pan, all from the Harbin Institute of Electrical Technology, have published in Chinese. The work by X. Chen or T. Yamamoto was new to us. Tsuchiya's and N. Yamamoto's work is described below after a visit to Kyushu University. Osada's work uses general sequence acceleration methods and his results look promising, but he has only considered a limited class of singular prob lems. The Chinese work follows work of Decher and Kelley in the United States. These references, all in Chinese, are: 1. X. Sun and Z. Pan, "King-Wevener's method at singular points,” HIET Journal 9, 269-273 (1986). 2. Z. Pan and S. Zhou, "Chord method for solving non-linear equations at singular points," HIET Journal 12, 302-309 (1986). 3. X. Sun, Z. Pan, and S. Zhou, "Newton-Moser's method for a class of singular problems," HIET Journal 13, 208-212 (1990). HIET is an abbreviation for Harbin Institute of Electrical Technology. It is not clear if the HIET Journal is externally refereed. Space does not permit a summary of all the talks. Here we only mention a few that were particularly interesting to us. There were a number of papers related to boundary element methods. Two of the best were by K. Onishi (Fukuoka Univ.) and M. Sakakihara (Okayama Univ. of Sci.) according to E. Allgower. Onishi's, in particular, seemed fundamental. There were several excellent talks on computational fluid dynamics, by R. Krasny (Univ. of Michigan) describing computational experiments studying vortex sheets, by M. Tabata (Univ. of Electrocommunications) on a third-order upwind finite element scheme for incompressible flow at high Reynolds numbers, and by T. Ushijima (Univ. of Electrocommunications) on a very complete mathematical analysis of the linear water wave problem. M. Mori (Univ. of Tokyo) described an extension of the double integral transformation that is suitable for the evaluation of oscillatory and slowly damped integrals, and his results compared very well with Western computational methods, such as each time the system is to be solved the individual subsystems may be solved by LU decomposition of some iterative method. There are many generalizations of this idea and the range of applications are extensive. This approach ought to be better known in the West. Quadpack. Mori has been working on One simple but effective talk was by K. Murota (Univ. of Tokyo), who writes mostly in English, gave a summary of his work on combinatorial canonical form. This was new to us but seemed to be potentially very useful. The idea is that in solving certain systems of linear equations repeatedly, say, for example, the equations for Newton's method, the actual matrix elements vary but the zero/nonzero structure remains fixed. Using graph theoretic methods it is possible to rearrange the equations and the variables to obtain a block-triangular form of the matrix, or a hierarchical decomposition of the system of equations. Then Several points were clearly established to us by the Japanese (and Chinese) papers. First, these researchers had really done their homework. Although they occasionally were unaware of some work in the West, by and large they were very well versed in Western literature and had read the relevant papers with exceptional care. Second, they occasionally seemed to go off in directions that were questionable, but often this perseverance pays off. Third, the best work is on a par with first rate research elsewhere; excellence knows no national borders. Although adopting a low profile, the Japanese were nonetheless quite evident in revealing how well they have penetrated to the very core of the foundations of algorithmic, mathematical, and numerical analysis. Finally, another meeting perhaps emphasizing computation would be very worthwhile. Virtually every participant agreed that ISCM was one of the best (if not the best) organized meetings they had ever experienced, and that Prof. Yamamoto and his staff really did an outstanding job! SITE VISITS BY KELLEY Kyushu University, Fukuoka My host for this visit was Dr. Hidemi Kawasaki. I am familiar with some of his work on nonsmooth optimization that has appeared in the Western literature. While at Kyushu University I also met with Dr. Takuya Tsuchiya. Both Kawasaki and Tsuchiya are assistants in the Mathematics Department. Kawasaki received his Ph.D. in 1988 under the direction of Prof. Furukawa of Kyushu University. Tsuchiya received his Ph.D. under the direction of Prof. Babushka at the University of Maryland. The Mathematics Department at Kyushu University has about 30 faculty (professors and assistants) and 45 graduate students. There are roughly 10,000 students in the university. I was given tours of the computing facilities in the Mathematics Department and in the computing center during my visit. The Mathematics Department has several IBM-PC compatible computers, several Macintosh computers, and a SUN Sparcstation. The computing center has a Fujitsu VP-200 vector supercomputer. Dr. Kawasaki's work is in theoretical optimization. His goal is to formulate necessary conditions for optimality for general nonlinear constrained optimization problems with very weak differentiability assumptions. In addition to Dr. Kawasaki and Prof. Furukawa at Kyushu, others in Japan working in this area include Prof. Fukushima at Kyoto University, Prof. Kojima at Tokyo Technological University, Prof. Tanaka at Hirosaki University; and an assistant, Mr. Shiraishi, who is working on a doctorate under Prof. Furukawa and is now at Toyama University. Kawasaki's work is related to several efforts in the West in nonsmooth optimization and semi-infinite programming. Ioffe in the Soviet Union and Zowe and Hettich in Germany work on related problems. My discussions with Kawasaki focused on his work on directional derivatives for sup-type functions and the application of that work to problems in semi-infinite programming. This work is based on more fundamental ideas in Reference 1 and has appeared (or will appear) in References 2 and 3. To briefly discuss the technical issues, let $f(x,t)$ be a smooth function of the let $f(x,t)$ be a smooth function of the variables Sx\in R^n$ and $t\in TS, where $TS is a compact metric space. Define $$_S(x)=sup\{f(x,t),\t\in \T\}_$$ The problem of minimizing S, a non- In a similar way, semi-infinite pro- Kawasaki's work has appeared in the Western literature in such journals as Mathematical Programming and Mathematics of Operations Research. His work is of high quality and provides valuable references to Japanese literature. Dr. Tsuchiya's degree is in the area of finite element methods for partial differential equations. In particular, he is interested in finite element methods for path following. I am not an expert in finite elements, butI do have some knowledge of path following and bifurcation, and my comments on this work will therefore be brief and focus on the path following aspects. thereby obtaining an approximate solution arc, $x_h(t)$. He obtains SH^1$ and $W^{1,p}$ error estimates for this arc near points where there is no bifurcation. I had a more detailed discussion with Dr. Tsuchiya about some work he did at Kyushu University before he went to Maryland for his doctoral thesis. This work (Ref 4) represents an approach to bordering methods for singular systems of nonlinear equations. Tsuchiya's research on singular systems (Ref 4) is a reformulation and extension of a method called "bordering" in the Western literature. In this approach additional equations are added to the system to eliminate the singularity of the Jacobian at the solution. Prof. N. Yamamoto of Kyushu Institute of Technology in Fukuoka began the Japanese work in this area. The Japanese research on bordering extends work by Griewark and others in the West by admitting the possibility of high order singularities. Their algorithms add more equations if the first pass of bordering fails to resolve the singularity. Tsuchiya's work gives a mathematical characterization of the number of terms that must be added. Other work on singular systems in Japan is being done by Prof. T. Yamamoto and Dr. X. Chen at Ehime University. Dr. Chen is a visitor from Xi'an Jiaotong University in China. Prof. X. Sun of the Harbin Institute of Electrical Technology in China is also active in this area. In the West, Decker, Griewark, Keller, Kelley, and Osburne did a good deal of work in the early 1980s, and groups led by Werner and Schwetlik in Germany are active now. Consider a map $F(x,g)$ from My host for this visit was Prof. |