{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:40:42Z","timestamp":1740123642766,"version":"3.37.3"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,1,11]],"date-time":"2023-01-11T00:00:00Z","timestamp":1673395200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,1,11]],"date-time":"2023-01-11T00:00:00Z","timestamp":1673395200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000646","name":"Japan Society for the Promotion of Science London","doi-asserted-by":"publisher","award":["18K03434","15H03637"],"award-info":[{"award-number":["18K03434","15H03637"]}],"id":[{"id":"10.13039\/501100000646","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003382","name":"Core Research for Evolutional Science and Technology","doi-asserted-by":"publisher","award":["JPMJCR14D4"],"award-info":[{"award-number":["JPMJCR14D4"]}],"id":[{"id":"10.13039\/501100003382","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2023,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper describes some numerical verification procedures to prove the invertibility of a linear operator in Hilbert spaces and to compute a bound on the norm of its inverse. These approaches improve on previous procedures that use an orthogonal projection of the Hilbert space and its a priori error estimations. Several verified examples which confirm the effectiveness of the new procedures are presented.<\/jats:p>","DOI":"10.1007\/s10915-023-02097-6","type":"journal-article","created":{"date-parts":[[2023,1,11]],"date-time":"2023-01-11T05:02:48Z","timestamp":1673413368000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Efficient Approaches for Verifying the Existence and Bound of Inverse of Linear Operators in Hilbert Spaces"],"prefix":"10.1007","volume":"94","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6520-3552","authenticated-orcid":false,"given":"Yoshitaka","family":"Watanabe","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Takehiko","family":"Kinoshita","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mitsuhiro T.","family":"Nakao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,1,11]]},"reference":[{"issue":"3","key":"2097_CR1","doi-asserted-by":"publisher","first-page":"1033","DOI":"10.1007\/s00205-010-0309-7","volume":"197","author":"G Arioli","year":"2010","unstructured":"Arioli, G., Koch, H.: Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto\u2013Sivashinski equation. Arch. Ration. Mech. Anal. 197(3), 1033\u20131051 (2010). https:\/\/doi.org\/10.1007\/s00205-010-0309-7","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"1","key":"2097_CR2","doi-asserted-by":"publisher","first-page":"448","DOI":"10.1016\/j.jde.2011.08.014","volume":"252","author":"G Arioli","year":"2012","unstructured":"Arioli, G., Koch, H.: Non-symmetric low-index solutions for a symmetric boundary value problem. J. Differ. Equ. 252(1), 448\u2013458 (2012). https:\/\/doi.org\/10.1016\/j.jde.2011.08.014","journal-title":"J. Differ. Equ."},{"issue":"1","key":"2097_CR3","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1137\/17M1155624","volume":"51","author":"JB van den Berg","year":"2019","unstructured":"van den Berg, J.B., Williams, J.: Rigorously computing symmetric stationary states of the Ohta\u2013Kawasaki problem in three dimensions. SIAM J. Math. Anal. 51(1), 131\u2013158 (2019)","journal-title":"SIAM J. Math. Anal."},{"issue":"4","key":"2097_CR4","doi-asserted-by":"publisher","first-page":"1398","DOI":"10.1137\/050645968","volume":"45","author":"S Day","year":"2007","unstructured":"Day, S., Lessard, J.P., Mischaikow, K.: Validated continuation for equilibria of PDEs. SIAM J. Numer. Anal. 45(4), 1398\u20131424 (2007). https:\/\/doi.org\/10.1137\/050645968","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"2097_CR5","doi-asserted-by":"publisher","first-page":"1070","DOI":"10.1137\/16M1073777","volume":"16","author":"JL Figueras","year":"2017","unstructured":"Figueras, J.L., Gameiro, M., Lessard, J.P., de la Llave, R.: A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations. SIAM J. Appl. Dyn. Syst. 16(2), 1070\u20131088 (2017). https:\/\/doi.org\/10.1137\/16M1073777","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"2097_CR6","unstructured":"Grisvard, P.: Elliptic problems in nonsmooth domains. In: Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston, MA (1985)"},{"key":"2097_CR7","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/s000210050002","volume":"1","author":"JG Heywood","year":"1999","unstructured":"Heywood, J.G., Nagata, W., Xie, W.: A numerically based existence theorem for the Navier\u2013Stokes equations. J. Math. Fluid Mech. 1, 5\u201323 (1999)","journal-title":"J. Math. Fluid Mech."},{"issue":"299","key":"2097_CR8","doi-asserted-by":"publisher","first-page":"1427","DOI":"10.1090\/mcom\/3046","volume":"85","author":"A Hungria","year":"2016","unstructured":"Hungria, A., Lessard, J.P., Mireles James, J.D.: Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach. Math. Comp. 85(299), 1427\u20131459 (2016). https:\/\/doi.org\/10.1090\/mcom\/3046","journal-title":"Math. Comp."},{"issue":"2","key":"2097_CR9","doi-asserted-by":"publisher","first-page":"526","DOI":"10.1016\/j.cam.2009.12.044","volume":"234","author":"T Kinoshita","year":"2010","unstructured":"Kinoshita, T., Nakao, M.T.: On very accurate enclosure of the optimal constant in the a priori error estimates for $$H^2_0$$-projection. J. Comput. Appl. Math. 234(2), 526\u2013537 (2010). https:\/\/doi.org\/10.1016\/j.cam.2009.12.044","journal-title":"J. Comput. Appl. Math."},{"issue":"9","key":"2097_CR10","doi-asserted-by":"publisher","first-page":"5431","DOI":"10.1016\/j.jde.2018.10.027","volume":"266","author":"T Kinoshita","year":"2019","unstructured":"Kinoshita, T., Watanabe, Y., Nakao, M.T.: An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces. J. Differ. Equ. 266(9), 5431\u20135447 (2019)","journal-title":"J. Differ. Equ."},{"issue":"2\u20133","key":"2097_CR11","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1007\/BF03186545","volume":"26","author":"K Nagatou","year":"2009","unstructured":"Nagatou, K.: Numerical verification method for infinite dimensional eigenvalue problems. Jpn. J. Ind. Appl. Math. 26(2\u20133), 477\u2013491 (2009)","journal-title":"Jpn. J. Ind. Appl. Math."},{"issue":"1","key":"2097_CR12","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1016\/j.jde.2019.08.008","volume":"268","author":"K Nagatou","year":"2019","unstructured":"Nagatou, K., Plum, M., McKenna, P.: Orbital stability investigations for travelling waves in a nonlinearly supported beam. J. Differ. Equ. 268(1), 80\u2013114 (2019)","journal-title":"J. Differ. Equ."},{"issue":"1","key":"2097_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00607-004-0111-1","volume":"75","author":"MT Nakao","year":"2005","unstructured":"Nakao, M.T., Hashimoto, K., Watanabe, Y.: A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems. Computing 75(1), 1\u201314 (2005). https:\/\/doi.org\/10.1007\/s00607-004-0111-1","journal-title":"Computing"},{"key":"2097_CR14","doi-asserted-by":"crossref","unstructured":"Nakao, M.T., Plum, M., Watanabe, Y.: Numerical verification methods and computer-assisted proofs for partial differential equations. In: Springer Series in Computational Mathematics, vol.\u00a053. Springer Singapore (2019)","DOI":"10.1007\/978-981-13-7669-6"},{"key":"2097_CR15","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/0377-0427(94)00090-N","volume":"60","author":"S Oishi","year":"1995","unstructured":"Oishi, S.: Numerical verification of existence and inclusion of solutions for nonlinear operator equations. J. Comput. Appl. Math. 60, 171\u2013185 (1995)","journal-title":"J. Comput. Appl. Math."},{"key":"2097_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2019.112620","volume":"372","author":"S Oishi","year":"2020","unstructured":"Oishi, S.: Numerical inclusion of exact periodic solutions for time delay Duffing equation. J. Comput. Appl. Math. 372, 112620 (2020)","journal-title":"J. Comput. Appl. Math."},{"key":"2097_CR17","first-page":"19","volume":"110","author":"M Plum","year":"2008","unstructured":"Plum, M.: Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance. Jahresber. Dtsch. Math. Ver. 110, 19\u201354 (2008)","journal-title":"Jahresber. Dtsch. Math. Ver."},{"key":"2097_CR18","doi-asserted-by":"crossref","unstructured":"Rump, S.: INTLAB\u2014INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77\u2013104. Kluwer Academic Publishers, Dordrecht (1999)","DOI":"10.1007\/978-94-017-1247-7_7"},{"issue":"2","key":"2097_CR19","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/s10543-010-0294-0","volume":"51","author":"SM Rump","year":"2011","unstructured":"Rump, S.M.: Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse. BIT 51(2), 367\u2013384 (2011). https:\/\/doi.org\/10.1007\/s10543-010-0294-0","journal-title":"BIT"},{"key":"2097_CR20","doi-asserted-by":"publisher","first-page":"907","DOI":"10.1007\/s00211-020-01155-7","volume":"146","author":"K Sekine","year":"2020","unstructured":"Sekine, K., Nakao, M.T., Oishi, S.: A new formulation using the Schur complement for the numerical existence proof of solutions to elliptic problems: without direct estimation for an inverse of the linearized operator. Numer. Math. 146, 907\u2013926 (2020)","journal-title":"Numer. Math."},{"key":"2097_CR21","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1016\/j.apnum.2020.03.016","volume":"154","author":"Y Watanabe","year":"2020","unstructured":"Watanabe, Y., Kinoshita, T., Nakao, M.T.: Some improvements of invertibility verifications for second-order linear elliptic operators. Appl. Numer. Math. 154, 36\u201346 (2020)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"2097_CR22","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1587\/nolta.2.123","volume":"2","author":"Y Watanabe","year":"2011","unstructured":"Watanabe, Y., Nagatou, K., Plum, M., Nakao, M.: A computer-assisted stability proof for the Orr-Sommerfeld problem with poiseuille flow. Nonlinear Theory Appl. IEICE 2(1), 123\u2013127 (2011)","journal-title":"Nonlinear Theory Appl. IEICE"},{"issue":"2","key":"2097_CR23","doi-asserted-by":"publisher","first-page":"975","DOI":"10.1137\/120894683","volume":"52","author":"Y Watanabe","year":"2014","unstructured":"Watanabe, Y., Nagatou, K., Plum, M., Nakao, M.T.: Verified computations of eigenvalue exclosures for eigenvalue problems in Hilbert spaces. SIAM J. Numer. Anal. 52(2), 975\u2013992 (2014). https:\/\/doi.org\/10.1137\/120894683","journal-title":"SIAM J. Numer. Anal."},{"issue":"7","key":"2097_CR24","doi-asserted-by":"publisher","first-page":"6363","DOI":"10.1016\/j.jde.2015.12.041","volume":"260","author":"Y Watanabe","year":"2016","unstructured":"Watanabe, Y., Nagatou, K., Plum, M., Nakao, M.T.: Norm bound computation for inverses of linear operators in Hilbert spaces. J. Differ. Equ. 260(7), 6363\u20136374 (2016). https:\/\/doi.org\/10.1016\/j.jde.2015.12.041","journal-title":"J. Differ. Equ."},{"key":"2097_CR25","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/j.amc.2015.12.021","volume":"276","author":"Y Watanabe","year":"2016","unstructured":"Watanabe, Y., Nakao, M.T.: A numerical verification method for nonlinear functional equations based on infinite-dimensional Newton-like iteration. Appl. Math. Comput. 276, 239\u2013251 (2016). https:\/\/doi.org\/10.1016\/j.amc.2015.12.021","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"2097_CR26","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1002\/zamm.200700158","volume":"89","author":"Y Watanabe","year":"2009","unstructured":"Watanabe, Y., Plum, M., Nakao, M.T.: A computer-assisted instability proof for the Orr\u2013Sommerfeld problem with Poiseuille flow. ZAMM Z. Angew. Math. Mech. 89(1), 5\u201318 (2009). https:\/\/doi.org\/10.1002\/zamm.200700158","journal-title":"ZAMM Z. Angew. Math. Mech."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02097-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-023-02097-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02097-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,31]],"date-time":"2023-01-31T06:36:50Z","timestamp":1675147010000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-023-02097-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,11]]},"references-count":26,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["2097"],"URL":"https:\/\/doi.org\/10.1007\/s10915-023-02097-6","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2023,1,11]]},"assertion":[{"value":"8 May 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 November 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 December 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 January 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"43"}}