{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T21:38:39Z","timestamp":1772314719791,"version":"3.50.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,11,29]],"date-time":"2024-11-29T00:00:00Z","timestamp":1732838400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,11,29]],"date-time":"2024-11-29T00:00:00Z","timestamp":1732838400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["12301522"],"award-info":[{"award-number":["12301522"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["12101524"],"award-info":[{"award-number":["12101524"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2025,1]]},"DOI":"10.1007\/s10915-024-02726-8","type":"journal-article","created":{"date-parts":[[2024,11,28]],"date-time":"2024-11-28T23:54:07Z","timestamp":1732838047000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Computing Optimal Partition Problems via Lagrange Multiplier approach"],"prefix":"10.1007","volume":"102","author":[{"given":"Qing","family":"Cheng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jing","family":"Guo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dong","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,11,29]]},"reference":[{"issue":"3","key":"2726_CR1","doi-asserted-by":"publisher","first-page":"1639","DOI":"10.1137\/20M1335431","volume":"59","author":"R An","year":"2021","unstructured":"An, R., Gao, H., Sun, W.: Optimal error analysis of Euler and Crank-Nicolson projection finite difference schemes for Landau-Lifshitz equation. SIAM J. Numer. Anal. 59(3), 1639\u20131662 (2021)","journal-title":"SIAM J. Numer. Anal."},{"key":"2726_CR2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110328","volume":"437","author":"X Antoine","year":"2021","unstructured":"Antoine, X., Shen, J., Tang, Q.: Scalar auxiliary variable\/lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear Schr\u00f6dinger\/Gross-Pitaevskii equations. J. Comput. Phys. 437, 110328 (2021)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"2726_CR3","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1137\/030600209","volume":"2","author":"W Bao","year":"2004","unstructured":"Bao, W.: Ground states and dynamics of multicomponent Bose-Einstein condensates. Multiscale Model. Simul. 2(2), 210\u2013236 (2004). https:\/\/doi.org\/10.1137\/030600209","journal-title":"Multiscale Model. Simul."},{"issue":"5","key":"2726_CR4","doi-asserted-by":"publisher","first-page":"1674","DOI":"10.1137\/s1064827503422956","volume":"25","author":"W Bao","year":"2004","unstructured":"Bao, W., Du, Q.: Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25(5), 1674\u20131697 (2004). https:\/\/doi.org\/10.1137\/s1064827503422956","journal-title":"SIAM J. Sci. Comput."},{"key":"2726_CR5","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/j.amc.2018.03.087","volume":"333","author":"B Bogosel","year":"2018","unstructured":"Bogosel, B.: Efficient algorithm for optimizing spectral partitions. Appl. Math. Comput. 333, 61\u201375 (2018). https:\/\/doi.org\/10.1016\/j.amc.2018.03.087","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"2726_CR6","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1137\/140976406","volume":"54","author":"B Bogosel","year":"2016","unstructured":"Bogosel, B., Velichkov, B.: A multiphase shape optimization problem for eigenvalues: qualitative study and numerical results. SIAM J. Numer. Anal. 54(1), 210\u2013241 (2016). https:\/\/doi.org\/10.1137\/140976406","journal-title":"SIAM J. Numer. Anal."},{"key":"2726_CR7","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1051\/cocv:2008074","volume":"16","author":"V Bonnaillie-No\u00ebl","year":"2010","unstructured":"Bonnaillie-No\u00ebl, V., Helffer, B., Vial, G.: Numerical simulations for nodal domains and spectral minimal partitions. ESAIM Control Optim. Calc. Var. 16, 221\u2013246 (2010). https:\/\/doi.org\/10.1051\/cocv:2008074","journal-title":"ESAIM Control Optim. Calc. Var."},{"issue":"6","key":"2726_CR8","doi-asserted-by":"publisher","first-page":"4100","DOI":"10.1137\/090747087","volume":"31","author":"B Bourdin","year":"2010","unstructured":"Bourdin, B., Bucur, D., Oudet, E.: Optimal Partitions for Eigenvalues. SIAM J. Sci. Comput. 31(6), 4100\u20134114 (2010). https:\/\/doi.org\/10.1137\/090747087","journal-title":"SIAM J. Sci. Comput."},{"key":"2726_CR9","first-page":"571","volume":"8","author":"D Bucur","year":"1998","unstructured":"Bucur, D., Butazzo, G., Henrot, A.: Existence results for some optimal partition problems. Adv. Math. Sci. Appl. 8, 571\u2013579 (1998)","journal-title":"Adv. Math. Sci. Appl."},{"issue":"1 &2","key":"2726_CR10","first-page":"49","volume":"23","author":"L Caffarelli","year":"2008","unstructured":"Caffarelli, L., Lin, F.: Nonlocal heat flows preserving the L2 energy. Discrete Contin. Dynam. Syst. 23(1 &2), 49\u201364 (2008)","journal-title":"Discrete Contin. Dynam. Syst."},{"issue":"1\u20132","key":"2726_CR11","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/s10915-006-9114-8","volume":"31","author":"LA Cafferelli","year":"2007","unstructured":"Cafferelli, L.A., Lin, F.H.: An Optimal Partition Problem for Eigenvalues. J. Sci. Comp. 31(1\u20132), 5\u201318 (2007). https:\/\/doi.org\/10.1007\/s10915-006-9114-8","journal-title":"J. Sci. Comp."},{"issue":"3","key":"2726_CR12","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1016\/j.physd.2004.06.002","volume":"196","author":"SM Chang","year":"2004","unstructured":"Chang, S.M., Lin, C.S., Lin, T.C., Lin, W.W.: Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates. Physica D 196(3), 341\u2013361 (2004). https:\/\/doi.org\/10.1016\/j.physd.2004.06.002","journal-title":"Physica D"},{"issue":"6","key":"2726_CR13","doi-asserted-by":"publisher","first-page":"A3982","DOI":"10.1137\/18M1166961","volume":"40","author":"Q Cheng","year":"2018","unstructured":"Cheng, Q., Shen, J.: Multiple scalar auxiliary variable (MSAV) approach and its application to the phase-field vesicle membrane model. SIAM J. Sci. Comput. 40(6), A3982\u2013A4006 (2018)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"2726_CR14","doi-asserted-by":"publisher","first-page":"A2489","DOI":"10.1137\/19M1306221","volume":"42","author":"Q Cheng","year":"2020","unstructured":"Cheng, Q., Shen, J.: Global constraints preserving scalar auxiliary variable schemes for gradient flows. SIAM J. Sci. Comput. 42(4), A2489\u2013A2513 (2020)","journal-title":"SIAM J. Sci. Comput."},{"key":"2726_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.114585","volume":"391","author":"Q Cheng","year":"2022","unstructured":"Cheng, Q., Shen, J.: .A new Lagrange multiplier approach for constructing structure preserving schemes, I. Positivity preserving. Comput. Methods Appl. Mech. Eng. 391, 114585 (2022)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"3","key":"2726_CR16","doi-asserted-by":"publisher","first-page":"970","DOI":"10.1137\/21M144877X","volume":"60","author":"Q Cheng","year":"2022","unstructured":"Cheng, Q., Shen, J.: A new Lagrange multiplier approach for constructing structure preserving schemes, II. Bound preserving. SIAM J. Num. Anal. 60(3), 970\u2013998 (2022)","journal-title":"SIAM J. Num. Anal."},{"issue":"2","key":"2726_CR17","doi-asserted-by":"publisher","first-page":"A530","DOI":"10.1137\/22M1501143","volume":"45","author":"Q Cheng","year":"2023","unstructured":"Cheng, Q., Shen, J.: Length Preserving Numerical Schemes for Landau-Lifshitz Equation Based on Lagrange Multiplier Approaches. SIAM J. Sci. Comput. 45(2), A530\u2013A553 (2023)","journal-title":"SIAM J. Sci. Comput."},{"key":"2726_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-020-01368-w","author":"K Chu","year":"2021","unstructured":"Chu, K., Leung, S.: A level set method for the Dirichlet $$k$$-partition problem. J. Sci. Comput. (2021). https:\/\/doi.org\/10.1007\/s10915-020-01368-w","journal-title":"J. Sci. Comput."},{"issue":"6","key":"2726_CR19","doi-asserted-by":"publisher","first-page":"871","DOI":"10.1016\/s0294-1449(02)00104-x","volume":"19","author":"M Conti","year":"2002","unstructured":"Conti, M., Terracini, S., Verzini, G.: Nehari\u2019s problem and competing species systems. Ann. l\u2019IHP Anal. Nonlin\u00e9aire 19(6), 871\u2013888 (2002). https:\/\/doi.org\/10.1016\/s0294-1449(02)00104-x","journal-title":"Ann. l\u2019IHP Anal. Nonlin\u00e9aire"},{"issue":"1","key":"2726_CR20","doi-asserted-by":"publisher","first-page":"160","DOI":"10.1016\/s0022-1236(02)00105-2","volume":"198","author":"M Conti","year":"2003","unstructured":"Conti, M., Terracini, S., Verzini, G.: An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198(1), 160\u2013196 (2003). https:\/\/doi.org\/10.1016\/s0022-1236(02)00105-2","journal-title":"J. Funct. Anal."},{"issue":"4","key":"2726_CR21","doi-asserted-by":"publisher","first-page":"46130","DOI":"10.1103\/physreve.71.046130","volume":"71","author":"O Cybulski","year":"2005","unstructured":"Cybulski, O., Babin, V., Holyst, R.: Minimization of the Renyi entropy production in the space-partitioning process. Phys. Rev. E 71(4), 46130 (2005). https:\/\/doi.org\/10.1103\/physreve.71.046130","journal-title":"Phys. Rev. E"},{"issue":"5","key":"2726_CR22","doi-asserted-by":"publisher","first-page":"56101","DOI":"10.1103\/physreve.77.056101","volume":"77","author":"O Cybulski","year":"2008","unstructured":"Cybulski, O., Holyst, R.: Three-dimensional space partition based on the first Laplacian eigenvalues in cells. Phys. Rev. E 77(5), 56101 (2008). https:\/\/doi.org\/10.1103\/physreve.77.056101","journal-title":"Phys. Rev. E"},{"issue":"1","key":"2726_CR23","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1088\/0951-7715\/22\/1\/005","volume":"22","author":"Q Du","year":"2008","unstructured":"Du, Q., Lin, F.: Numerical approximations of a norm-preserving gradient flow and applications to an optimal partition problem. Nonlinearity 22(1), 67\u201383 (2008). https:\/\/doi.org\/10.1088\/0951-7715\/22\/1\/005","journal-title":"Nonlinearity"},{"key":"2726_CR24","first-page":"101","volume":"26","author":"B Helffer","year":"2009","unstructured":"Helffer, B., Hoffmann-Ostenhof, T., Terracini, S.: Nodal domains and spectral minimal partitions. Ann. l\u2019IHP Anal. lin\u00e9aire 26, 101\u2013138 (2009)","journal-title":"Ann. l\u2019IHP Anal. lin\u00e9aire"},{"issue":"3","key":"2726_CR25","doi-asserted-by":"publisher","first-page":"764","DOI":"10.1016\/j.camwa.2019.07.030","volume":"79","author":"X Jing","year":"2020","unstructured":"Jing, X., Wang, Q.: Linear second order energy stable schemes for phase field crystal growth models with nonlocal constraints. Comput. Math. Appl. 79(3), 764\u2013788 (2020)","journal-title":"Comput. Math. Appl."},{"issue":"1","key":"2726_CR26","doi-asserted-by":"publisher","first-page":"B219","DOI":"10.1137\/20M1328002","volume":"43","author":"W Liu","year":"2021","unstructured":"Liu, W., Cai, Y.: Normalized gradient flow with lagrange multiplier for computing ground states of Bose-Einstein Condensates. SIAM J. Sci. Comput. 43(1), B219\u2013B242 (2021)","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"2726_CR27","doi-asserted-by":"publisher","first-page":"4251","DOI":"10.1137\/16m1098309","volume":"49","author":"B Osting","year":"2017","unstructured":"Osting, B., Reeb, T.H.: Consistency of Dirichlet partitions. SIAM J. Math. Anal. 49(5), 4251\u20134274 (2017). https:\/\/doi.org\/10.1137\/16m1098309","journal-title":"SIAM J. Math. Anal."},{"issue":"3","key":"2726_CR28","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1017\/S095679250800778X","volume":"20","author":"S Rosencrans","year":"2009","unstructured":"Rosencrans, S., Wang, X., Winter, W., Zhao, S.: Measuring the insulating ability of anisotropic thermal conductors via principal Dirichlet eigenvalue. Eur. J. Appl. Math. 20(3), 231\u2013246 (2009)","journal-title":"Eur. J. Appl. Math."},{"issue":"11 &12","key":"2726_CR29","doi-asserted-by":"publisher","first-page":"5441","DOI":"10.3934\/dcds.2013.33.5441","volume":"33","author":"S Rosencrans","year":"2013","unstructured":"Rosencrans, S., Wang, X., Zhao, S.: Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements. Discrete Contin. Dyn. Syst. 33(11 &12), 5441 (2013)","journal-title":"Discrete Contin. Dyn. Syst."},{"issue":"4","key":"2726_CR30","doi-asserted-by":"publisher","first-page":"A2061","DOI":"10.1137\/21M1443406","volume":"44","author":"D Wang","year":"2022","unstructured":"Wang, D.: An efficient unconditionally stable method for Dirichlet partitions in arbitrary domains. SIAM J. Sci. Comput. 44(4), A2061\u2013A2088 (2022)","journal-title":"SIAM J. Sci. Comput."},{"key":"2726_CR31","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1016\/j.cam.2018.11.015","volume":"351","author":"D Wang","year":"2019","unstructured":"Wang, D., Osting, B.: A diffusion generated method for computing Dirichlet partitions. J. Comput. Appl. Math. 351, 302\u2013316 (2019). https:\/\/doi.org\/10.1016\/j.cam.2018.11.015","journal-title":"J. Comput. Appl. Math."},{"key":"2726_CR32","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/j.jcp.2016.04.004","volume":"316","author":"X Wang","year":"2016","unstructured":"Wang, X., Ju, L., Du, Q.: Efficient and stable exponential time differencing Runge-Kutta methods for phase field elastic bending energy models. J. Comput. Phys. 316, 21\u201338 (2016)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"2726_CR33","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1006\/jcph.2001.6793","volume":"171","author":"XP Wang","year":"2001","unstructured":"Wang, X.P., Garc\u0131a-Cervera, C.J., Weinan, E.: A Gauss-Seidel projection method for micromagnetics simulations. J. Comput. Phys. 171(1), 357\u2013372 (2001)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"2726_CR34","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1007\/s10915-022-01838-3","volume":"91","author":"J Yin","year":"2022","unstructured":"Yin, J., Huang, Z., Zhang, L.: Constrained high-index saddle dynamics for the solution landscape with equality constraints. J. Sci. Comput. 91(2), 62 (2022)","journal-title":"J. Sci. Comput."},{"key":"2726_CR35","doi-asserted-by":"publisher","first-page":"1043","DOI":"10.1007\/s10915-015-0117-1","volume":"67","author":"L Zhu","year":"2016","unstructured":"Zhu, L., Ju, L., Zhao, W.: Fast high-order compact exponential time differencing Runge-Kutta methods for second-order semilinear parabolic equations. J. Sci. Comput. 67, 1043\u20131065 (2016)","journal-title":"J. Sci. Comput."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-024-02726-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-024-02726-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-024-02726-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,13]],"date-time":"2025-01-13T14:58:47Z","timestamp":1736780327000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-024-02726-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,29]]},"references-count":35,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["2726"],"URL":"https:\/\/doi.org\/10.1007\/s10915-024-02726-8","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,29]]},"assertion":[{"value":"27 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 October 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 October 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 November 2024","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 confirm that there are no competing interests to declare for any of the authors of this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"22"}}