Make a bridge between
mathematics and engineering with
numerical computations with
Q. Please describe your specialty of verified numerical computations.
For example, anyone can imagine that the computation of pi, the ratio of a circle's circumference to its diameter, continues endlessly. When we are processing an indivisible number like pi using a computer, we cannot let the computer continue the computation forever just because the solution continues. What we do is that we stop the computation at some digit. I used pi as an example, but 1/10=0.1 is also an indivisible number. This phenomenon is caused because of the difference between decimal numbers and binary numbers. We used to find an answer to the 15th decimal place that was the limit that the binary number could express. Numbers smaller than that were considered insignificant errors. However, there was an incident in which the error resulted in a serious problem. It happened during the Gulf War in 1990. Iraq was attacking the U.S. military with scud missiles. The U.S. military tried to intercept the scud missiles using the Patriot missile system, the latest missile defense system at that time. This system, however, did not hit the scud missile and caused more than 100 deaths and injuries. Errors in numerical computation were the cause of this incident. People were aware that numerical computation with computers had errors, and the errors would eventually cause serious trouble. Yet, this incident drove people to really do something about the error.
Verified precision for numerical computation, like a warranty for home appliances
We are using a method called interval arithmetics to reduce the error as much as possible. This means that we set the upper and lower limits to a number to obtain and express the number between the limits. In other words, we set a range of error and try to keep the error within the range. We also use the fixed-point theorem to mathematically warrant the existence of a solution to an equation. I give you an example, there is an equation called the Lorenz equation that expresses the stagnation of air. The comparison between Matlab, the most commonly used software for numerical computations and the numerical computations with verified precision we developed revealed that there was little difference between the two shortly after the start of the computation. Yet, the error gradually started to become large in Matlab, and the number eventually became unreliable. Matlab is by no means low-precision software. Yet, it generates a large error when it processes a large number of computations because errors are cumulative. Mathematics is a science of being skeptical of everything. Therefore, “probably correct” is not acceptable in mathematics. The software we developed is the most precise and fastest in the world at this point. Our dream and vision are to add verified precision to all numerical computations like home appliances that come with warrantees.
Mathematics will die if it is left alone.
Q. Please describe the Department of Applied Mathematics
More faculties of the Department of Applied Mathematics are doctors of engineering than science. This means that there are many engineers with mathematical talent in this Department. Therefore, students can learn from both of the talents. I believe that mathematics and physics were inseparable in the time of Newton. These fields then gradually became abstract and separated with time. Mathematics will die if it is left alone. I believe that applied mathematics can be the bridge that connects mathematics to various fields such as engineering. Of course, we need to have deep insight and knowledge of mathematics to create the bridge. We proved a theorem called three-dimensional geometry and topology concerning hyperbolic geometry using verified numerical computations. We developed a program for high-speed numerical computations with verified precision using the supercomputer TSUBAME of the Tokyo Institute of Technology. With this program, we checked the existence of all solutions to 5,646,646 equations of multivariable complex simultaneous nonlinear equations. You can guess from the enormous number; it is nearly impossible to solve each of the equations. This is a good example where the technology of applied mathematics became effective for mathematical proofs. It is also a model case where we created a bridge between different fields.
Keep growing with the best and the brightest colleagues.
Q. What are good points of Waseda University?
Waseda University is a large school with many people, and we can be stimulated by all of them. This is the same for students and faculties alike. All students are bright and active. They perform wonderfully when I give them small hints. It is often said that people can only live among people. Therefore, I want students to make many friends and absorb various ideas while engaging in research and facing challenges with them. I believe my role is to be there for them as a supporter.