{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T01:26:53Z","timestamp":1773365213118,"version":"3.50.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,2,28]],"date-time":"2023-02-28T00:00:00Z","timestamp":1677542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,2,28]],"date-time":"2023-02-28T00:00:00Z","timestamp":1677542400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100018934","name":"Universit\u00e4t Greifswald","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100018934","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We apply the charge simulation method (CSM) in order to compute the logarithmic capacity of compact sets consisting of (infinitely) many \u201csmall\u201d components. This application allows to use just a single charge point for each component. The resulting method therefore is significantly more efficient than methods based on discretizations of the boundaries (for example, our own method presented in\u00a0Liesen et al. (<jats:italic>Comput. Methods Funct. Theory<\/jats:italic><jats:bold>17<\/jats:bold>, 689\u2013713, 2017)), while maintaining a very high level of accuracy. We study properties of the linear algebraic systems that arise in the CSM, and show how these systems can be solved efficiently using preconditioned iterative methods, where the matrix-vector products are computed using the fast multipole method. We illustrate the use of the method on generalized Cantor sets and the Cantor dust.<\/jats:p>","DOI":"10.1007\/s11075-022-01428-2","type":"journal-article","created":{"date-parts":[[2023,2,28]],"date-time":"2023-02-28T03:02:32Z","timestamp":1677553352000},"page":"581-614","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Computing the logarithmic capacity of compact sets having (infinitely) many components with the charge simulation method"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3677-373X","authenticated-orcid":false,"given":"J\u00f6rg","family":"Liesen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2561-0978","authenticated-orcid":false,"given":"Mohamed M. S.","family":"Nasser","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3107-3053","authenticated-orcid":false,"given":"Olivier","family":"S\u00e8te","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,2,28]]},"reference":[{"key":"1428_CR1","doi-asserted-by":"publisher","first-page":"1169","DOI":"10.1137\/S1064827595294307","volume":"19","author":"K Amano","year":"1998","unstructured":"Amano, K.: A charge simulation method for numerical conformal mapping onto circular and radial slit domains. SIAM J. Sci. Comput. 19, 1169\u20131187 (1998)","journal-title":"SIAM J. Sci. Comput."},{"key":"1428_CR2","doi-asserted-by":"publisher","first-page":"53","DOI":"10.14495\/jsiaml.2.53","volume":"2","author":"K Amano","year":"2010","unstructured":"Amano, K., Okano, D.: A circular and radial slit mapping of unbounded multiply connected domains. JSIAM Lett. 2, 53\u201356 (2010)","journal-title":"JSIAM Lett."},{"key":"1428_CR3","unstructured":"Augustin, M.A.: A Method of Fundamental Solutions in Poroelasticity to Model the Stress Field in Geothermal Reservoirs, Lecture Notes in Geosystems Mathematics and Computing, Birkh\u00e4user Basel (1995)"},{"issue":"14124","key":"1428_CR4","first-page":"1","volume":"1","author":"P Baddoo","year":"2021","unstructured":"Baddoo, P., Trefethen, L.N.: Log-lightning computation of capacity and Green\u2019s function. Maple Trans. 1(14124), 1\u201313 (2021)","journal-title":"Maple Trans."},{"key":"1428_CR5","unstructured":"Dijkstra, W., Hochstenbach, M.E.: Numerical approximation of the logarithmic capacity, CASA report, 0809 (2009)"},{"key":"1428_CR6","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1016\/S0024-3795(02)00567-0","volume":"364","author":"H Fassbender","year":"2003","unstructured":"Fassbender, H., Ikramov, K.D.: Computing matrix-vector products with centrosymmetric and centrohermitian matrices. Linear Algebra Appl. 364, 235\u2013241 (2003)","journal-title":"Linear Algebra Appl."},{"key":"1428_CR7","unstructured":"Garnett, J.B., Marshall, D.E.: Harmonic measure, vol. 2 of New Mathematical Monographs. Cambridge University Press, Cambridge (2005)"},{"key":"1428_CR8","unstructured":"Greengard, L., Gimbutas, Z.: FMMLIB2D: a MATLAB toolbox for fast multipole method in two dimensions, version 1.2.\u00a0ed., 2012. http:\/\/www.cims.nyu.edu\/cmcl\/fmm2dlib\/fmm2dlib.html. Accessed 1 Jan 2018"},{"key":"1428_CR9","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1016\/0021-9991(87)90140-9","volume":"73","author":"L Greengard","year":"1987","unstructured":"Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73, 325\u2013348 (1987)","journal-title":"J. Comput. Phys."},{"key":"1428_CR10","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1016\/j.jcp.2004.06.024","volume":"202","author":"J Helsing","year":"2005","unstructured":"Helsing, J., Wadbro, E.: Laplace\u2019s equation and the Dirichlet-Neumann map: a new mode for Mikhlin\u2019s method. J. Comput. Phys. 202, 391\u2013410 (2005)","journal-title":"J. Comput. Phys."},{"key":"1428_CR11","doi-asserted-by":"publisher","first-page":"409","DOI":"10.6028\/jres.049.044","volume":"49","author":"MR Hestenes","year":"1952","unstructured":"Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Research Nat. Bur. Standards 49, 409\u2013436 (1952)","journal-title":"J. Research Nat. Bur. Standards"},{"key":"1428_CR12","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1007\/s10044-022-01074-3","volume":"25","author":"E Kalmoun","year":"2022","unstructured":"Kalmoun, E., Nasser, M.: Harmonic image inpainting using the charge simulation method. Pattern Anal. Applic 25, 795\u2013806 (2022)","journal-title":"Pattern Anal. Applic"},{"key":"1428_CR13","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/j.jat.2014.04.003","volume":"191","author":"H Kr\u00fcger","year":"2015","unstructured":"Kr\u00fcger, H., Simon, B.: Cantor polynomials and some related classes of OPRL. J. Approx. Theory 191, 71\u201393 (2015)","journal-title":"J. Approx. Theory"},{"key":"1428_CR14","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1007\/s40315-017-0207-1","volume":"17","author":"J Liesen","year":"2017","unstructured":"Liesen, J., S\u00e8te, O., Nasser, M.M.S.: Fast and accurate computation of the logarithmic capacity of compact sets. Comput. Methods Funct. Theory 17, 689\u2013713 (2017)","journal-title":"Comput. Methods Funct. Theory"},{"key":"1428_CR15","doi-asserted-by":"crossref","unstructured":"Liesen, J., Strako\u0161, Z.: Krylov subspace methods. Principles and analysis, Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)","DOI":"10.1093\/acprof:oso\/9780199655410.003.0002"},{"key":"1428_CR16","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1017\/S096249290626001X","volume":"15","author":"G Meurant","year":"2006","unstructured":"Meurant, G., Strako\u0161, Z.: The Lanczos and conjugate gradient algorithms in finite precision arithmetic. Acta Numer. 15, 471\u2013542 (2006)","journal-title":"Acta Numer."},{"key":"1428_CR17","doi-asserted-by":"publisher","first-page":"1169","DOI":"10.4153\/CJM-1974-109-1","volume":"26","author":"CD Minda","year":"1974","unstructured":"Minda, C.D.: Capacity estimates for planar cantor-like sets. Can. J. Math. 26, 1169\u20131172 (1974)","journal-title":"Can. J. Math."},{"key":"1428_CR18","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1007\/s40315-016-0159-x","volume":"16","author":"MMS Nasser","year":"2016","unstructured":"Nasser, M.M.S., Liesen, J., S\u00e8te, O.: Numerical computation of the conformal map onto lemniscatic domains. Comput. Methods Funct. Theory 16, 609\u2013635 (2016)","journal-title":"Comput. Methods Funct. Theory"},{"key":"1428_CR19","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/BF03167460","volume":"20","author":"H Ogata","year":"2003","unstructured":"Ogata, H., Okano, D., Sugihara, M., Amano, K.: Unique solvability of the linear system appearing in the invariant scheme of the charge simulation method. Japan J. Indust. Appl. Math. 20, 17\u201335 (2003)","journal-title":"Japan J. Indust. Appl. Math."},{"key":"1428_CR20","doi-asserted-by":"crossref","unstructured":"Okano, D., Ogata, H., Amano, K.: A method of numerical conformal mapping of curved slit domains by the charge simulation method. In: Proceedings of the International Conference on Recent Advances in Computational Mathematics (ICRACM 2001) (Matsuyama), vol.\u00a0152, pp.\u00a0441\u2013450 (2003)","DOI":"10.1016\/S0377-0427(02)00722-7"},{"key":"1428_CR21","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1137\/0712047","volume":"12","author":"CC Paige","year":"1975","unstructured":"Paige, C.C., Saunders, M.A.: Solutions of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617\u2013629 (1975)","journal-title":"SIAM J. Numer. Anal."},{"key":"1428_CR22","unstructured":"Ransford, T.: Potential theory in the complex plane, vol.\u00a028 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1995)"},{"key":"1428_CR23","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1007\/BF03321780","volume":"10","author":"T Ransford","year":"2010","unstructured":"Ransford, T.: Computation of logarithmic capacity. Comput. Methods Funct. Theory 10, 555\u2013578 (2010)","journal-title":"Comput. Methods Funct. Theory"},{"key":"1428_CR24","doi-asserted-by":"publisher","first-page":"1499","DOI":"10.1090\/S0025-5718-07-01941-2","volume":"76","author":"T Ransford","year":"2007","unstructured":"Ransford, T., Rostand, J.: Computation of capacity. Math. Comp. 76, 1499\u20131520 (2007)","journal-title":"Math. Comp."},{"key":"1428_CR25","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1080\/10586458.1997.10504611","volume":"6","author":"J Rostand","year":"1997","unstructured":"Rostand, J.: Computing logarithmic capacity with linear programming. Experiment. Math. 6, 221\u2013238 (1997)","journal-title":"Experiment. Math."},{"key":"1428_CR26","doi-asserted-by":"publisher","first-page":"856","DOI":"10.1137\/0907058","volume":"7","author":"Y Saad","year":"1986","unstructured":"Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7, 856\u2013869 (1986)","journal-title":"SIAM J. Sci. Statist. Comput."},{"key":"1428_CR27","first-page":"1","volume":"45","author":"O S\u00e8te","year":"2016","unstructured":"S\u00e8te, O., Liesen, J.: On conformal maps from multiply connected domains onto lemniscatic domains. Electron. Trans. Numer. Anal. 45, 1\u201315 (2016)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"1428_CR28","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1007\/BF01187465","volume":"21","author":"G Szeg\u00f6","year":"1924","unstructured":"Szeg\u00f6, G.: Bemerkungen zu einer Arbeit von Herrn M. Fekete: \u00dcber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 21, 203\u2013208 (1924)","journal-title":"Math. Z."},{"key":"1428_CR29","doi-asserted-by":"publisher","first-page":"128","DOI":"10.1090\/S0002-9947-1956-0080727-2","volume":"82","author":"JL Walsh","year":"1956","unstructured":"Walsh, J.L.: On the conformal mapping of multiply connected regions. Trans. Amer. Math. Soc. 82, 128\u2013146 (1956)","journal-title":"Trans. Amer. Math. Soc."},{"key":"1428_CR30","volume-title":"Methods of Fundamental Solutions in Solid Mechanics","author":"H Wang","year":"2020","unstructured":"Wang, H., Qin, Q.-H.: Methods of Fundamental Solutions in Solid Mechanics. Elsevier, New York (2020)"},{"key":"1428_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-0180-5","volume-title":"Visual Complex Functions","author":"E Wegert","year":"2012","unstructured":"Wegert, E.: Visual Complex Functions. Birkh\u00e4user\/Springer Basel AG, Basel (2012)"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01428-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-022-01428-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01428-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,8]],"date-time":"2023-05-08T05:11:44Z","timestamp":1683522704000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-022-01428-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,28]]},"references-count":31,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["1428"],"URL":"https:\/\/doi.org\/10.1007\/s11075-022-01428-2","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,28]]},"assertion":[{"value":"8 February 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}