Optimal estimation for inverse of infinite-dimensional operator by self-validating numerical computations and its applications

2015 Fiscal Year Final Research Report

• Project/Area Number: 24340018
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kyushu University
• Principal Investigator: Yoshitaka Watanabe 九州大学, 情報基盤研究開発センター, 准教授 (90243972)
• Co-Investigator(Kenkyū-buntansha): Nagatou Kaori 九州大学, マス・フォア・インダストリ研究所, 准教授 (40326426)
• Co-Investigator(Renkei-kenkyūsha): Nakao Mitsuhiro T. 佐世保工業高等専門学校, 校長 (10136418)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 精度保証付き数値計算 / 偏微分方程式 / 計算機援用証明 / 関数解析 / 無限次元固有値問題 / 不動点定理

Outline of Final Research Achievements

"Self-validating numerical computations" stands for a computer-assisted approach to prove the existence of the solutions and its guaranteed error bounds for a given problem. By using self-validating numerical computations, this reserch proposed a computer-assisted procedure to assure the invertibility of a linear operator which is the sum of an unbounded bijective and a bounded operator in a Hilbert space, and to compute a bound for the norm of its inverse. We also showed that our bounds are expected to converge to the exact operator norm and to provide accurate and efficient enclosure results for the solution of nonlinear problem by infinite-dimensional Newton-type methods.

Free Research Field

計算数学