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 | |
| Publication status | Published - 2016 Apr 5 |
| Externally published | Yes |
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Keywords
- Differential operators
- Numerical verification
- Solvability of linear problem
ASJC Scopus subject areas
- Analysis
Cite this
Norm bound computation for inverses of linear operators in Hilbert spaces. / Watanabe, Yoshitaka; Nagatou, Kaori; Plum, Michael; Nakao, Mitsuhiro T.
In: Journal of Differential Equations, Vol. 260, No. 7, 05.04.2016, p. 6363-6374.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Norm bound computation for inverses of linear operators in Hilbert spaces
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
UR - http://www.scopus.com/inward/record.url?scp=84958122143&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84958122143&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2015.12.041
DO - 10.1016/j.jde.2015.12.041
M3 - Article
AN - SCOPUS:84958122143
VL - 260
SP - 6363
EP - 6374
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 7
ER -