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Kaori Nagatou

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Numerical Verification Method and its Applications

Kaori Nagatou
Degree: Doctor of Mathematical Science (Kyushu University)
Research Interests: Numerical Verification Method
Unit:Number & Equation

Report

A self-validating numerical method has recently received much attention as a computation method in science and technology from the viewpoint of reliability of computation results. This method guarantees the error between the exact solution to a problem and the approximate solution obtained by a computer. In addition, this method is gaining importance as a way to prove the existence of a solution using a computer (numerical verification method, computer assisted proof) even for problems in which the theoretical proof for the existence of solutions is difficult.

My group has been conducting research to enclose the exact eigenvalues (and eigenfunctions) of infinite dimensional operators. This can be considered as a theoretical research by a computer for nonlinear elliptic equations.

Our achievements in this area include the proposal of a numerical verification method to a coupling type eigenvalue problem of a non-commutative harmonic oscillator, proof of stability of a bifurcating solution in Kolmogorov problem, proposition of a verification method for a stationary solution for the two-dimensional Navier-Stokes equation, and proof of the existence of a stationary solution for the Driven Cavity problem for a large Reynolds number.

The eigenvalue enclosing method, which is the basis of our research, guarantees with mathematical rigor the existence or non-existence of a certain eigenvalue of infinite dimensional operators by a computer. Although this method provides reliability in the obtained eigenvalue, the great advantage is that the evaluation of the eigenvalue can be used to prove related problems, such as the proof of stability of bifurcating solutions. Hence, this is very significant from the standpoint of mathematical rigorousness, and is the biggest feature of our research. The several eigenvalue enclosing methods for self-adjoint operators have been proposed for many years. However, very little research has been conducted on the non-self-adjoint operator. It is expected that our method will be applicable to specific issues, such as a fluid problem so that new knowledge may be obtained in mathematical models.

Furthermore, our proposal is the only method in Japan and elsewhere that guarantees a range of non-existence of eigenvalues, particularly in the case of partial differential operators, except for cases where it is obvious from an analytical discussion. Mathematically rigorous proof of existence or non-existence of eigenvalues within a certain range is an important problem in solid-state physics, and for example, it is an issue in designing semiconductor devices. For this reason, we strive to apply our work to real world problems by developing an efficient implementation method in collaboration with corporations participating in the MI program.

Photonic Crystal Model

A German research group led by Prof. Dr. Plum has recently initiated a research project on the existence of a band gap structure in a photonic crystal. The possibility of a joint research with this group is likely using our eigenvalue excluding method. A photonic crystal is fabricated using nanotechnology, and it is known that light is absorbed at all frequencies, which do not exist in the band gap. Its mathematical model can be described by Maxwell’s equations, and the band gap in a photonic crystal corresponds to the spectral gap. Hence, rigorous analysis of the existence and non-existence of eigenvalues of the corresponding operators has significance from a practical engineering point of view.

My group aims to discover a breakthrough in this field by producing scientists who are well-versed with such rigorous mathematical analysis through our education and training at our graduate school, and by strengthening the collaborative relationship with participating corporations in the MI program. This breakthrough encompasses two areas: One is on the corporation side and uses our mathematical theories, while the other is on the mathematical side by constructing mathematical theories capable of satisfying practical requirements.

A further significance of this research project is that it may become a launching pad for a new direction called computer assisted proof, in which a computer is used to logically and rigorously prove theorems rather than only using a pen and paper. Another significance is that this research project could become the first step in pioneering a new scientific field by providing a constitutive proof to a non-constitutive existence theorem in mathematics as well as packaging such a proof so that many people can use it.

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