{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T05:08:14Z","timestamp":1775365694385,"version":"3.50.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2019,2,1]],"date-time":"2019-02-01T00:00:00Z","timestamp":1548979200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput. Visual Sci."],"published-print":{"date-parts":[[2019,2]]},"DOI":"10.1007\/s00791-018-00306-6","type":"journal-article","created":{"date-parts":[[2019,2,8]],"date-time":"2019-02-08T17:19:30Z","timestamp":1549646370000},"page":"29-46","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Exploiting multilevel Toeplitz structures in high dimensional nonlocal diffusion"],"prefix":"10.1007","volume":"20","author":[{"given":"Christian","family":"Vollmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Volker","family":"Schulz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,2,8]]},"reference":[{"key":"306_CR1","doi-asserted-by":"publisher","unstructured":"Acosta, G., Bersetche, F.M., Borthagaray, J.P.: A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian. Comput. Math. Appl. (2017). \n                    https:\/\/doi.org\/10.1016\/j.camwa.2017.05.026\n                    \n                  . \n                    http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0898122117303310","DOI":"10.1016\/j.camwa.2017.05.026"},{"issue":"2","key":"306_CR2","doi-asserted-by":"publisher","first-page":"472","DOI":"10.1137\/15M1033952","volume":"55","author":"G Acosta","year":"2017","unstructured":"Acosta, G., Borthagaray, J.P.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55(2), 472\u2013495 (2017). \n                    https:\/\/doi.org\/10.1137\/15M1033952","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"306_CR3","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1214\/08-AIHP201","volume":"46","author":"RF Bass","year":"2010","unstructured":"Bass, R.F., Kassmann, M., Kumagai, T.: Symmetric jump processes: localization, heat kernels and convergence. Ann. Inst. H. Poincar\u00e9 Probab. Stat. 46(1), 59\u201371 (2010). \n                    https:\/\/doi.org\/10.1214\/08-AIHP201","journal-title":"Ann. Inst. H. Poincar\u00e9 Probab. Stat."},{"key":"306_CR4","doi-asserted-by":"crossref","unstructured":"Burkovska, O., Gunzburger, M.: Regularity and approximation analyses of nonlocal variational equality and inequality problems. arXiv preprint. \n                    arXiv:1804.10282\n                    \n                   (2018)","DOI":"10.1016\/j.jmaa.2019.05.064"},{"key":"306_CR5","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1137\/S0895479897331941","volume":"21","author":"SS Capizzano","year":"2000","unstructured":"Capizzano, S.S., Tyrtyshnikov, E.E.: Any circulant-like preconditioner for multilevel matrices is not superlinear. SIAM J. Matrix Anal. Appl. 21, 431\u2013439 (2000)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"4","key":"306_CR6","doi-asserted-by":"publisher","first-page":"766","DOI":"10.1137\/0909051","volume":"9","author":"TF Chan","year":"1988","unstructured":"Chan, T.F.: An optimal circulant preconditioner for Toeplitz systems. SIAM J. Sci. Stat. Comput. 9(4), 766\u2013771 (1988)","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"306_CR7","doi-asserted-by":"publisher","unstructured":"Chen, J., Li, T.L., Anitescu, M.: A parallel linear solver for multilevel Toeplitz systems with possibly several right-hand sides. Parallel Comput. 40(8), 408\u2013424 (2014). \n                    https:\/\/doi.org\/10.1016\/j.parco.2014.06.004\n                    \n                  . \n                    http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0167819114000672","DOI":"10.1016\/j.parco.2014.06.004"},{"key":"306_CR8","unstructured":"Chen, M., Deng, W.: Convergence proof for the multigrid method of the nonlocal model. arXiv preprint. \n                    arXiv:1605.05481\n                    \n                   (2016)"},{"issue":"4","key":"306_CR9","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1137\/110833294","volume":"54","author":"Q Du","year":"2012","unstructured":"Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54(4), 667\u2013696 (2012). \n                    https:\/\/doi.org\/10.1137\/110833294","journal-title":"SIAM Rev."},{"issue":"03","key":"306_CR10","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1142\/S0218202512500546","volume":"23","author":"Q Du","year":"2013","unstructured":"Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23(03), 493\u2013540 (2013). \n                    https:\/\/doi.org\/10.1142\/S0218202512500546","journal-title":"Math. Models Methods Appl. Sci."},{"key":"306_CR11","doi-asserted-by":"publisher","unstructured":"Du, Q., Zhou, K.: Mathematical analysis for the peridynamic nonlocal continuum theory. ESAIM: M2AN 45(2), 217\u2013234 (2011). \n                    https:\/\/doi.org\/10.1051\/m2an\/2010040","DOI":"10.1051\/m2an\/2010040"},{"key":"306_CR12","doi-asserted-by":"publisher","unstructured":"D\u2019Elia, M., Gunzburger, M.: The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math. Appl. 66(7), 1245\u20131260 (2013). \n                    https:\/\/doi.org\/10.1016\/j.camwa.2013.07.022\n                    \n                  . \n                    http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0898122113004707","DOI":"10.1016\/j.camwa.2013.07.022"},{"issue":"3","key":"306_CR13","doi-asserted-by":"publisher","first-page":"1005","DOI":"10.1137\/070698592","volume":"7","author":"G Gilboa","year":"2009","unstructured":"Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005\u20131028 (2009). \n                    https:\/\/doi.org\/10.1137\/070698592","journal-title":"Multiscale Model. Simul."},{"key":"306_CR14","volume-title":"Finite Elements and Fast Iterative Solvers","author":"Howard Elman","year":"2005","unstructured":"Elman, Howard, Silvester, David, Wathen, Andy: Finite Elements and Fast Iterative Solvers. Oxford University Press, New York (2005)"},{"issue":"4","key":"306_CR15","doi-asserted-by":"publisher","first-page":"1963","DOI":"10.1090\/S0002-9947-08-04544-3","volume":"361","author":"Martin T Barlow","year":"2009","unstructured":"Barlow, Martin T., Bass, Richard F., Chen, Zhen-Qing, Krassmann, Moritz: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. 361(4), 1963\u20131999 (2009). \n                    https:\/\/doi.org\/10.1090\/S0002-9947-08-04544-3","journal-title":"Trans. Am. Math. Soc."},{"key":"306_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/b98874","volume-title":"Numerical Optimization","author":"J Nocedale","year":"1999","unstructured":"Nocedale, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999). \n                    https:\/\/doi.org\/10.1007\/978-0-387-40065-5"},{"key":"306_CR17","doi-asserted-by":"crossref","unstructured":"Olshanskii, M.A., Tyrtyshnikov, E.E.: Iterative methods for linear systems: theory and applications. SIAM (2014)","DOI":"10.1137\/1.9781611973464"},{"key":"306_CR18","doi-asserted-by":"publisher","unstructured":"Olshevsky, V., Oseledets, I., Tyrtyshnikov, E.: Tensor properties of multilevel Toeplitz and related matrices. Linear Algebra Appl. 412(1), 1\u201321 (2006). \n                    https:\/\/doi.org\/10.1016\/j.laa.2005.03.040\n                    \n                  . \n                    http:\/\/www.sciencedirect.com\/science\/article\/pii\/S002437950500296X","DOI":"10.1016\/j.laa.2005.03.040"},{"key":"306_CR19","doi-asserted-by":"publisher","unstructured":"Pang, H.K., Sun, H.W.: Multigrid method for fractional diffusion equations. J. Comput. Phys. 231(2), 693\u2013703 (2012). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2011.10.005\n                    \n                  . \n                    http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0021999111005985","DOI":"10.1016\/j.jcp.2011.10.005"},{"key":"306_CR20","doi-asserted-by":"publisher","unstructured":"Peyr\u00e9, G., Bougleux, S., Cohen, L.: Non-local regularization of inverse problems, pp. 57\u201368. Springer Berlin Heidelberg, Berlin, Heidelberg (2008). \n                    https:\/\/doi.org\/10.1007\/978-3-540-88690-7_5","DOI":"10.1007\/978-3-540-88690-7_5"},{"key":"306_CR21","unstructured":"Rosasco, L., Belkin, M., Vito, E.D.: On learning with integral operators. J. Mach. Learn. Res. 11, 905\u2013934 (2010). \n                    http:\/\/dl.acm.org\/citation.cfm?id=1756006.1756036"},{"key":"306_CR22","unstructured":"Serra-Capizzano, S.: Toeplitz matrices: spectral properties and preconditioning in the cg method (2007)"},{"issue":"5","key":"306_CR23","doi-asserted-by":"publisher","first-page":"A2444","DOI":"10.1137\/12086491X","volume":"34","author":"H Wang","year":"2012","unstructured":"Wang, H., Basu, T.S.: A fast finite difference method for two-dimensional space-fractional diffusion equations. SIAM J. Sci. Comput. 34(5), A2444\u2013A2458 (2012). \n                    https:\/\/doi.org\/10.1137\/12086491X","journal-title":"SIAM J. Sci. Comput."},{"key":"306_CR24","doi-asserted-by":"publisher","unstructured":"Wang, H., Du, N.: A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation. J. Comput. Phys. 253, 50\u201363 (2013). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2013.06.040\n                    \n                  . \n                    http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0021999113004701","DOI":"10.1016\/j.jcp.2013.06.040"},{"key":"306_CR25","doi-asserted-by":"publisher","unstructured":"Wang, H., Wang, K., Sircar, T.: A direct o(nlog2n) finite difference method for fractional diffusion equations. J. Comput. Phys. 229(21), 8095\u20138104 (2010). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2010.07.011\n                    \n                  . \n                    http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0021999110003943","DOI":"10.1016\/j.jcp.2010.07.011"},{"key":"306_CR26","doi-asserted-by":"publisher","unstructured":"Ye, K., Lim, L.H.: Algorithms for structured matrix-vector product of optimal bilinear complexity. In: 2016 IEEE Information Theory Workshop (ITW), pp. 310\u2013314 (2016). \n                    https:\/\/doi.org\/10.1109\/ITW.2016.7606846","DOI":"10.1109\/ITW.2016.7606846"},{"issue":"5","key":"306_CR27","doi-asserted-by":"publisher","first-page":"1759","DOI":"10.1137\/090781267","volume":"48","author":"K Zhou","year":"2010","unstructured":"Zhou, K., Du, Q.: Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48(5), 1759\u20131780 (2010). \n                    https:\/\/doi.org\/10.1137\/090781267","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Computing and Visualization in Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00791-018-00306-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00791-018-00306-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00791-018-00306-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,2,7]],"date-time":"2020-02-07T19:13:08Z","timestamp":1581102788000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00791-018-00306-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2]]},"references-count":27,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2019,2]]}},"alternative-id":["306"],"URL":"https:\/\/doi.org\/10.1007\/s00791-018-00306-6","relation":{},"ISSN":["1432-9360","1433-0369"],"issn-type":[{"value":"1432-9360","type":"print"},{"value":"1433-0369","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,2]]},"assertion":[{"value":"8 September 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 May 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 December 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 February 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}