Applications of reproducing kernels to the Tikhonov regularization

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540113
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Gunma University
• Principal Investigator: SAITOH Saburou 群馬大学, その他部局等, 名誉教授 (10110397)
• Co-Investigator(Kenkyū-buntansha): MATSUURA Tutomu 群馬大学, 大学院理工学府, 准教授 (80181692) ; WATANABE Syuuji 群馬大学, 大学院理工学府, 教授 (90222405)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 再生核 / 積分変換 / 数値解法 / 初値問題 / チコノフ正則化法 / 偏微分方程式 / 積分方程式 / ゼロ除算

Outline of Final Research Achievements

We were able to establish the Aveiro discretization method that gives many numerical solutions of partial differential and integral equations by applying the theories of reproducing kernels and of Tikhonov regularization, and we published papers as applications. In particular, we got a general concept of representation of fractional functions and we found the division by zero as the typical case. We also succeeded in the numerical solutions for some convolution integral equations which involve non-linear equations.
Incidentally, we found a general theory among initial value problems in partial differential equations, reproducing kernels, eigenfunctions, integral transforms, and special function theory.
For the publication plan of a general theory of reproducing kernels, we complete a book manuscript with about 420 pages and we are discussing its publication with some publishers over about 2 years, however, at this moment we were not able to determine its publication, unfortunately.

Free Research Field

応用数学