Norm bound computation for inverses of linear operators in Hilbert spaces

Yoshitaka Watanabe; Kaori Nagatou; Michael Plum; Mitsuhiro T. Nakao

doi:10.1016/j.jde.2015.12.041

Norm bound computation for inverses of linear operators in Hilbert spaces

Yoshitaka Watanabe, Kaori Nagatou, Michael Plum, Mitsuhiro T. Nakao

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

This paper presents a computer-assisted procedure to prove 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. By using some projection and constructive a priori error estimates, the invertibility condition together with the norm computation is formulated as an inequality based upon a method originally developed by the authors for obtaining existence and enclosure results for nonlinear partial differential equations. Several examples which confirm the actual effectiveness of the procedure are reported.

Original language	English
Pages (from-to)	6363-6374
Number of pages	12
Journal	Journal of Differential Equations
Volume	260
Issue number	7
DOIs	https://doi.org/10.1016/j.jde.2015.12.041
Publication status	Published - Apr 5 2016

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

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10.1016/j.jde.2015.12.041

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title = "Norm bound computation for inverses of linear operators in Hilbert spaces",

abstract = "This paper presents a computer-assisted procedure to prove 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. By using some projection and constructive a priori error estimates, the invertibility condition together with the norm computation is formulated as an inequality based upon a method originally developed by the authors for obtaining existence and enclosure results for nonlinear partial differential equations. Several examples which confirm the actual effectiveness of the procedure are reported.",

author = "Yoshitaka Watanabe and Kaori Nagatou and Michael Plum and Nakao, {Mitsuhiro T.}",

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AU - Watanabe, Yoshitaka

AU - Nagatou, Kaori

AU - Plum, Michael

AU - Nakao, Mitsuhiro T.

N1 - Funding Information: The authors heartily thank the anonymous referee for his/her thorough reading and valuable comments. This work was supported by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science, and Technology of Japan (No. 24340018 ). The computation was mainly carried out using the computer facilities at Research Institute for Information Technology, Kyushu University, Japan. Publisher Copyright: © 2015 Elsevier Inc.

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N2 - This paper presents a computer-assisted procedure to prove 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. By using some projection and constructive a priori error estimates, the invertibility condition together with the norm computation is formulated as an inequality based upon a method originally developed by the authors for obtaining existence and enclosure results for nonlinear partial differential equations. Several examples which confirm the actual effectiveness of the procedure are reported.

AB - This paper presents a computer-assisted procedure to prove 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. By using some projection and constructive a priori error estimates, the invertibility condition together with the norm computation is formulated as an inequality based upon a method originally developed by the authors for obtaining existence and enclosure results for nonlinear partial differential equations. Several examples which confirm the actual effectiveness of the procedure are reported.

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Norm bound computation for inverses of linear operators in Hilbert spaces

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