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

5 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 - 2016 Apr 5
Externally published	Yes

Keywords

Differential operators
Numerical verification
Solvability of linear problem

ASJC Scopus subject areas

Analysis

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AU - Watanabe, Yoshitaka

AU - Nagatou, Kaori

AU - Plum, Michael

AU - Nakao, Mitsuhiro T.

PY - 2016/4/5

Y1 - 2016/4/5

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.

KW - Differential operators

KW - Numerical verification

KW - Solvability of linear problem

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JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 7

ER -

Norm bound computation for inverses of linear operators in Hilbert spaces

Abstract

Keywords

ASJC Scopus subject areas

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