{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T06:44:50Z","timestamp":1762325090391,"version":"3.37.3"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,8,25]],"date-time":"2021-08-25T00:00:00Z","timestamp":1629849600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,8,25]],"date-time":"2021-08-25T00:00:00Z","timestamp":1629849600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11901211","11871486"],"award-info":[{"award-number":["11901211","11871486"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Hong Kong RGC GRF","award":["11337216","11305318"],"award-info":[{"award-number":["11337216","11305318"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,10]]},"DOI":"10.1007\/s10915-021-01592-y","type":"journal-article","created":{"date-parts":[[2021,8,25]],"date-time":"2021-08-25T08:03:48Z","timestamp":1629878628000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Projection Method for Saddle Points of Energy Functional in $$H^{-1}$$ Metric"],"prefix":"10.1007","volume":"89","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4214-5545","authenticated-orcid":false,"given":"Shuting","family":"Gu","sequence":"first","affiliation":[]},{"given":"Ling","family":"Lin","sequence":"additional","affiliation":[]},{"given":"Xiang","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,8,25]]},"reference":[{"key":"1592_CR1","doi-asserted-by":"publisher","first-page":"1085","DOI":"10.1016\/0001-6160(79)90196-2","volume":"27","author":"SM Allen","year":"1979","unstructured":"Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica 27, 1085\u20131095 (1979)","journal-title":"Acta Metallurgica"},{"key":"1592_CR2","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/S0022-247X(02)00205-6","volume":"273","author":"PW Bates","year":"2002","unstructured":"Bates, P.W., Chen, F.: Spectral analysis and multidimensional stability of traveling waves for nonlocal allen-cahn equation. J. Math. Anal. Appl. 273, 45\u201357 (2002)","journal-title":"J. Math. Anal. Appl."},{"key":"1592_CR3","doi-asserted-by":"crossref","unstructured":"Brachet, M., Chehab, J.: Fast and Stable Schemes for Phase Fields Models (2020)","DOI":"10.1016\/j.camwa.2020.07.015"},{"key":"1592_CR4","doi-asserted-by":"crossref","unstructured":"Cahn, J. W., Hilliard, J. E.: Free energy of a nonuniform system. I. interfacial free energy. J. Chem. Phys. 28, 258\u2013267 (1958)","DOI":"10.1063\/1.1744102"},{"key":"1592_CR5","doi-asserted-by":"crossref","unstructured":"Crippen, G. M., Scheraga, H. A.: Minimization of polypeptide energy: XI. The method of gentlest ascent. Arch. Biochem. Biophys. 144, 462\u2013466 (1971)","DOI":"10.1016\/0003-9861(71)90349-3"},{"key":"1592_CR6","doi-asserted-by":"crossref","unstructured":"Dang, H., Fife, P.C., Peletier, L.A.: Saddle solutions of the bistable diffusion equation. Ztschrift F\u00fcr Angewandte Mathematik Und Physik Zamp 43, 984\u2013998 (1992)","DOI":"10.1007\/BF00916424"},{"key":"1592_CR7","doi-asserted-by":"crossref","unstructured":"Dawson,D. A., G\u00e4rtner, J.: Large Deviations, Free Energy Functional and Quasi-Potential for a Mean Field Model of Interacting Diffusions, vol. 78. Memoirs of American Mathematical Society (1989)","DOI":"10.1090\/memo\/0398"},{"key":"1592_CR8","doi-asserted-by":"publisher","first-page":"1039","DOI":"10.4310\/CMS.2009.v7.n4.a11","volume":"7","author":"Q Du","year":"2009","unstructured":"Du, Q., Zhang, L.: A constrained string method and its numerical analysis. Commun. Math. Sci. 7, 1039\u20131051 (2009)","journal-title":"Commun. Math. Sci."},{"key":"1592_CR9","doi-asserted-by":"crossref","unstructured":"Feng, J., Kurtz, T.G.: Large Deviations for Stochastic Processes. Mathematical Surveys and Monographs, vol. 131. American Mathematical Society, Prividence, RI (2006)","DOI":"10.1090\/surv\/131"},{"key":"1592_CR10","first-page":"1","volume":"2000","author":"PC Fife","year":"2000","unstructured":"Fife, P.C.: Models for phase separation and their mathematics. Electron. J. Differ. Equ. 2000, 1\u201326 (2000)","journal-title":"Electron. J. Differ. Equ."},{"key":"1592_CR11","doi-asserted-by":"publisher","first-page":"1786","DOI":"10.1137\/130930339","volume":"53","author":"W Gao","year":"2015","unstructured":"Gao, W., Leng, J., Zhou, X.: An iterative minimization formulation for saddle point search. SIAM J. Numer. Anal. 53, 1786\u20131805 (2015)","journal-title":"SIAM J. Numer. Anal."},{"key":"1592_CR12","doi-asserted-by":"crossref","unstructured":"Gao, W., Leng, J., Zhou, X.: Iterative minimization algorithm for efficient calculations of transition states. J. Comput. Phys. 309, 69 \u2013 87 (2016)","DOI":"10.1016\/j.jcp.2015.12.056"},{"key":"1592_CR13","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1016\/j.jcp.2017.10.028","volume":"353","author":"S Gu","year":"2018","unstructured":"Gu, S., Zhou, X.: Convex splitting method for the calculation of transition states of energy functional. J. Comput. Phys. 353, 417\u2013434 (2018)","journal-title":"J. Comput. Phys."},{"key":"1592_CR14","doi-asserted-by":"publisher","first-page":"7010","DOI":"10.1063\/1.480097","volume":"111","author":"G Henkelman","year":"1999","unstructured":"Henkelman, G., J\u00f3nsson, H.: A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J. Chem. Phys. 111, 7010\u20137022 (1999)","journal-title":"J. Chem. Phys."},{"key":"1592_CR15","doi-asserted-by":"crossref","unstructured":"J\u00f2nsson, H., Mills, G., Jacobsen, K. W.: Nudged elasic band method for finding minimum energy paths of transitions. In Berne, B.J., Ciccotti, G., Coker, D.F. (eds.) Classical and Quantum Dynamics in Condensed Phase Simulations, p. 385. New Jersey, 1998 (LERICI, Villa Marigola, Proceedings of the International School of Physics, World Scientific)","DOI":"10.1142\/9789812839664_0016"},{"key":"1592_CR16","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1137\/140957780","volume":"14","author":"T Li","year":"2016","unstructured":"Li, T., Li, X., Zhou, X.: Finding transition pathways on manifolds. Multiscale Model Simul. 14, 173\u2013206 (2016)","journal-title":"Multiscale Model Simul."},{"key":"1592_CR17","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1137\/120876307","volume":"11","author":"T Li","year":"2013","unstructured":"Li, T., Zhang, P., Zhang, W.: Nucleation rate calculation for the phase transition of diblock copolymers undr stochastic cahn-hilliard dynamics. Multiscale Model. Simul. 11, 385\u2013409 (2013)","journal-title":"Multiscale Model. Simul."},{"key":"1592_CR18","doi-asserted-by":"crossref","unstructured":"Lin, L., Cheng, X., E, W., Shi, A., Zhang, P.: A numerical method for the study of nucleation of ordered phases. J. Comput. Phys. 229, 1797\u20131809 (2010)","DOI":"10.1016\/j.jcp.2009.11.009"},{"key":"1592_CR19","doi-asserted-by":"publisher","first-page":"2419","DOI":"10.1103\/PhysRevE.57.2419","volume":"57","author":"N Mousseau","year":"1998","unstructured":"Mousseau, N., Barkema, G.: Traveling through potential energy surfaces of disordered materials: the activation-relaxation technique. Phys. Rev. E 57, 2419 (1998)","journal-title":"Phys. Rev. E"},{"key":"1592_CR20","doi-asserted-by":"crossref","unstructured":"Ren, W.E, Weiqing, R., Vanden-Eijnden, E.: String method for the study of rare events. Phys. Rev. B 66, 052301 (2002)","DOI":"10.1103\/PhysRevB.66.052301"},{"key":"1592_CR21","doi-asserted-by":"publisher","first-page":"134105","DOI":"10.1063\/1.4798344","volume":"138","author":"W Ren","year":"2013","unstructured":"Ren, W., Vanden-Eijnden, E.: A climbing string method for saddle point search. J. Chem. Phys. 138, 134105 (2013)","journal-title":"J. Chem. Phys."},{"key":"1592_CR22","doi-asserted-by":"publisher","first-page":"1669","DOI":"10.3934\/dcds.2010.28.1669","volume":"28","author":"J Shen","year":"2010","unstructured":"Shen, J., Yang, X.: Numerical approximations of allen-cahn and cahn-hilliard equations. Disc. Contin. Dyn. Syst. 28, 1669\u20131691 (2010)","journal-title":"Disc. Contin. Dyn. Syst."},{"key":"1592_CR23","unstructured":"Wales, D.J.: Energy Landscapes with Application to Clusters. Cambridge University Press, Biomolecules and Glasses (2003)"},{"key":"1592_CR24","doi-asserted-by":"crossref","unstructured":"Wei, J., Winter, M.: Stationary solutions for the cahn-hilliard equation. Annales de l\u2019Institut Henri Poincar\u00e9 C, Analyse non lin\u00e9aire 15, 459\u2013492 (1998)","DOI":"10.1016\/S0294-1449(98)80031-0"},{"key":"1592_CR25","doi-asserted-by":"crossref","unstructured":"Weinan E., Zhou, X.: The gentlest ascent dynamics. Nonlinearity 24, 1831 (2011)","DOI":"10.1088\/0951-7715\/24\/6\/008"},{"key":"1592_CR26","doi-asserted-by":"publisher","first-page":"2269","DOI":"10.1137\/080738143","volume":"47","author":"SM Wise","year":"2009","unstructured":"Wise, S.M., Wang, C., Lowengrub, J.: An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47, 2269\u20132288 (2009)","journal-title":"SIAM J. Numer. Anal."},{"key":"1592_CR27","doi-asserted-by":"publisher","first-page":"4745","DOI":"10.1016\/j.jcp.2012.03.006","volume":"231","author":"J Zhang","year":"2012","unstructured":"Zhang, J., Du, Q.: Constrained shrinking dimer dynamics for saddle point search with constraints. J. Comput. Phys. 231, 4745\u20134758 (2012)","journal-title":"J. Comput. Phys."},{"key":"1592_CR28","doi-asserted-by":"publisher","first-page":"1105","DOI":"10.4310\/CMS.2012.v10.n4.a4","volume":"10","author":"W Zhang","year":"2012","unstructured":"Zhang, W., Li, T., Zhang, P.: Numerical study for the nucleation of one-dimensional stochastic cahn-hilliard dynamics. Comun. Math. Sci. 10, 1105\u20131132 (2012)","journal-title":"Comun. Math. Sci."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01592-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-021-01592-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01592-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,23]],"date-time":"2021-10-23T04:11:35Z","timestamp":1634962295000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-021-01592-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,25]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,10]]}},"alternative-id":["1592"],"URL":"https:\/\/doi.org\/10.1007\/s10915-021-01592-y","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2021,8,25]]},"assertion":[{"value":"24 November 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 March 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 August 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"12"}}