## Research

### Knowledge Co-Creation - Profiles of researchers

#### In Pursuit of the Garden of Mathematical Study--

Mastering Precision-Guaranteed Numerical Calculation

Shinichi Ooishi

Professor at Waseda University, Faculty of Science and Engineering

##### Learning Thoroughly About Areas of Interest through Self-Study

Since high school, I was a precocious student who studied very keenly the subjects I was interested in, but felt that school lessons were too easy, and boring. I read piles of books, thinking to myself, "I want to be a writer," and studied German passionately. After a year, I reached the point where I could read the school's German textbook and supplementary reading materials in their entirety on the day they were handed out, without having to check any words in the dictionary, so I figured that I had achieved enough with German, and thereafter became engrossed in mathematics.

My perception of mathematics was that it consisted of creating "a box of magic for unraveling the mysteries of numbers." I pushed myself and read difficult books such as university textbooks and treatises aimed at researchers, but thinking about it now, I do not think they were of any use in creating magic boxes.

When the time came for me to enter university, I could not overcome the pressure from my parents, as they insisted that "Studying literature or mathematics won't set you up to make a living," so I entered the Department of Electronics and Communication Engineering, School of Science and Engineering. Because lessons were easy, I was able to understand the information covered in them immediately, and I studied the things that interested me outside of class. During my second year, I became absorbed in quantum mechanics, and studied everyday in the library.

During this time, I began to read the latest significant theses concerning the discoveries of quantum mechanics, in their original English and German versions. I studied the theses written by Shinichiro Tomonaga, Paul Dirac and Richard Feynman - all recipients of the Nobel Prize for Physics - and copied every single word of their theses into my notebook by hand.

In time, I became a third-year student, and had to decide on a theme for my bachelor's thesis. I thought to myself, "I want to do something relating to communications incorporating quantum mechanics!" but there was no laboratory that engaged in such work in the Department of Electronics and Communication Engineering at that time. At the very least, I wanted to solve a classical mechanics problem incorporating quantum mechanics. While thinking this, I came to learn about an extremely interesting nonlinear wave model known as 'soliton', and felt that I really wanted to try and take on this problem.

During a seminar for bachelor's thesis, I was under the guidance of Mr. Kazuo Horiuchi (a professor in the Department of Electronics and Communication Engineering, School of Science and Engineering of Waseda University at the time). Mr. Horiuchi was one of the first people to bring functional analysis into research within the realm of electronics, information and communication, and held a high level of interest in mathematical research. For his seminar, Mr. Horiuchi would invite Mr. Takeyasu Kotera from Tokyo University of Education (now Tsukuba University) every Saturday, and hold a seminar on nonlinear mathematics. Mr. Kotera was actually engaged in research on soliton, and was a pupil of Mr. Morikazu Toda, who had authored "Toda's Lattice Theory," one of the famous soliton theories. Through this connection, I was able to participate in Mr. Toda's seminars at Tokyo University of Education.

With the guidance of Mr. Horiuchi, and that of these teachers, I was able to write my bachelor's thesis with soliton as its theme. Given that I wrote my bachelor's thesis in an area of physical mathematics, while belonging to the Department of Electronics and Communication Engineering, you could say that I was quite a maverick.

##### The Garden of Soliton Turns into a Wilderness

I continued with my research on soliton after entering graduate school. During the first year of my master's program, I discovered a new theorem that certain class equations contained secondary soliton solutions, solutions which show that two solitons can safely interact, and this was highly praised by Mr. Toda, but in order to gain academic recognition, I needed to express it in an English thesis. However, I cannot write well in English. I suffered in agony for two years, and after advancing to my doctoral program, I was finally able to compile a more sophisticated English thesis.

During the second year of my doctoral program, my thesis was chosen to feature in the Physical Society of Japan's journal, the Journal of Physical Society of Japan. Once it was published, I received dozens of requests by postcard from researchers around the world, asking if I could "please send the offprint." Because this was a time when emails and copying machines still did not exist, all these requests came through mail. I have the postcards stored carefully in a scrapbook of mine.

I have also written many introductory books on mathematics and mathematical information science for science and engineering students. Left: "Fourier Analysis (Introductory Mathematics Course 6 for Science and Engineering)" (Iwanami Bookstore, 1989), Right: "Case-based Introduction to Information Theory" (Kodansha, 1993)

I went through growing pains, and wrote a series of eight English theses. I was ready and eager, thinking that "I have finally descended onto the garden of mystery that is soliton!" though unfortunately this feeling did not last long. This was because Mr. Mikio Sato, a talented mathematician, then classified solitons into several types through an algebraic approach. Because he gave an extremely smart explanation, saying that "Only this type of soliton appears," in an instant the mysterious garden turned into a wilderness where not even a single blade of grass grew. There were aspects of Mr. Sato's research that took my research as a starting point, but went much further forward, and soliton was thereafter no longer an appealing subject of research for me.

In 1981, after writing my doctoral dissertation, I had the idea to do a complete about-turn with the direction of my research, and delve into the world where the algebraic approach could not offer solutions. However, it was of absolute importance that I gain exact solutions. Wondering exactly what the world would be like where solutions could not be provided through mathematical formulae, I entered into the world of the computer's "numerical calculation." I achieved various results, but after ten years, in 1990 the mathematician Mr. Mitsuhiro Nakao read his survey thesis on "Precision-Guaranteed Numerical Calculations," which was contributed to the Journal of the Information Processing Society and Iwanami Bookstore's "Mathematics," and I knew intuitively that the direction in which my work would come to bear fruit was in this area.

##### The Precision of Abandoned Numerical Calculations

Computers are surprisingly lenient when it comes to accuracy in calculations. With calculations exceeding a certain number of digits, they compute while rounding or cutting off and, therefore, as the error builds up with each computation, depending on the scale it can increase to a size that cannot be disregarded.

For instance, the homotopy method - a constitutive solution method incorporating nonlinear equations - is used as a method to attain solutions for LSI circuits. It first solves simple problems, and then solves problems that have been modified so as to slightly approach the real problem... and in this way, it pursues solutions while gradually approaching the real problem, although it is necessary to give consideration to the ultimate impact of errors. If errors are disregarded, the accuracy of the final solution cannot be guaranteed, and as a result problems will arise with practical application, such as arriving at false solutions.

The remarkable results that exceeded my expectations have been gathered into one book. "Precision-Guaranteed Numerical Calculations" (Corona Publishing Co., Ltd., 1999)

In the world of numerical calculations, disregarding computational errors is referred to in technical language as "rounding," but at that time, disregarding rounding errors, and giving in and treating them as though they did not exist, was considered normal. Because I, and others like me, entered into this world, declaring "No, accuracy is something that ought to be guaranteed. There is no way I will give in," I think that I was seen as a maverick. However, as I wanted to attain strict solutions, giving in was not an option for me.

With help also from the unique ideas of students and foreign researchers in the laboratory, during the seven or eight years after that time, I made progress at a speed that greatly exceeded my expectations, and the precision-guaranteed numerical calculation became something that could be practically applied. At this time, during my deep discussions with foreign co-researchers, I came to decide that "Even if I can guarantee precision, if the length of time required for calculations is thousands or tens of thousands times more than usual, no one will use it. I absolutely must aim to guarantee precision while only roughly doubling the length of time required." This was difficult, but I suddenly thought of something from my nearly ten years of research. In fact, although it was thought that guaranteed-precision solution calculations would take 10,000 times the amount of time required for approximate solution calculations, through a simple method this could be reduced to merely twice as long.

Put simply, it broke free from the fixed idea that "computations are things that are done in order, one by one." According to conventional ideas, with a 100-part computation, this would be carried out with the error rounded off at each point. However, if one considers the idea of gathering all the 100 computations, and performing them in one go, this gives rise to the idea of also carrying out a single computation for all the rounding. Just like with Columbus' egg, I thought to myself, "Why did it take me ten years to think of this method?"

Using this concept, all that is necessary is to attach a program for guaranteed-precision to existing programs before they are used, with no manipulations of the existing program's contents required, and therefore in addition to speed, scalability will also be exceptionally high. Through the revolution of this method, research progressed instantly as practical application became entirely within range, and in 1999 I was able to put out a book entitled "Precision-Guaranteed Numerical Calculation." I also produced a proposal in a new area of research called "error-free transformation," and this became the sign for our laboratory.

##### Promoting Interaction in an Open Environment

Exclusive-use facilities with rows of computers carrying out high-speed parallel computation

Because precision-guaranteed calculations arrived in the world on a practical level, research activities underwent a great change. One after another, government grants were issued and facilities for parallel computation featuring lines of computers were prepared, vast numbers of young researchers and foreign researchers were hired, and dynamic places were developed.

For this project, we are renting a room outside of a university campus. We searched for a large room, which we are now using without partitions (photo). We were determined to create an open-space environment, where we can easily look over each other's work, and meetings can be watched by everyone. This is because we believe that by having foreign or external people interact and contribute ideas, we can find clues to problem solutions.

The laboratory of the project team headed by Professor Ooishi, with its un-partitioned, open-space environment

During these roughly twenty years, although the basic framework for precision-guaranteed numerical calculation has been achieved, many problems remain. While leaving the individual practical research to specialists, I hope to continue to carry out the theoretical research that will form the basis of mathematical analysis. In the next ten or twenty years I want to take on the problems that will be described as being "problems that are extremely important, with tremendous spillover effects, but extremely difficult."

The research initiative conceptual diagram, "Research on Precision Guarantees in Numeral Linear Simulations," adopted by JST Core Research for Evolutional Science and Technology (CREST)

#### Shinichi Ooishi

Professor at Faculty of Science and Engineering, Waseda University

He completed his doctoral program at Waseda University Graduate School in 1981(Doctor of Engineering). He was assistant lecturer at Waseda University's School of Science and Engineering from 1980, and full-time lecturer from 1982; assistant professor from 1984, and professor at said university from 1989. After working in the Department of Electronics and Communication Engineering, the Department of Information and Computer Science, and the Department of the Computer Science, in 2007 he was made Professor for the Department of Applied Mathematics in the Faculty of Science and Engineering. He has worked as research representative for JST Core Research for Evolutional Science and Technology (CREST)'s "Guaranteed Accuracy in Mathematical Linear Simulation" (2004-09), conducted specially-promoted research with the Ministry of Education, Culture, Sports, Science and Technology's scientific research fund for "Precision-Guaranteed Mathematical Calculation" (2005-09), and worked as research representative for JST Core Research for Evolutional Science and Technology (CREST)'s "Foundations of Nonlinear Precision-Guaranteed Mathematical Calculation Techniques and the Search for Error-Free Computational Engineering" (2009-15). He has received many academic awards, such as the Niwa Memorial Award, Okawa Publications Prize, Funai Prize for Science Promotion, and the Telecommunication Advancement Foundation's Telecom System Technology Award.

WASEDA UNVERSITY Research Promotion http://www.waseda.jp/rps/