{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T22:20:52Z","timestamp":1778538052267,"version":"3.51.4"},"reference-count":17,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["14-11-00083"],"award-info":[{"award-number":["14-11-00083"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The well-known random walk on spheres method (RWS) for the Laplace equation is here extended to drift-diffusion problems. First we derive a generalized spherical mean value relation which is an extension of the\nclassical integral mean value relation for the Laplace equation. Next we give a probabilistic interpretation\nof the kernel. The distribution on the sphere generated by this kernel is then related to the von Mises\u2013Fisher distribution\non the sphere which can be efficiently simulated. The rigorous expressions are given for the case of constant velocity drift,\nbut the algorithm is then extended to solve drift-diffusion problems with arbitrary varying drift velocity vector.\nApplications to cathodoluminescence and EBIC imaging of defects and dislocations in semiconductors are discussed.<\/jats:p>","DOI":"10.1515\/mcma-2016-0118","type":"journal-article","created":{"date-parts":[[2016,11,17]],"date-time":"2016-11-17T10:02:10Z","timestamp":1479376930000},"page":"265-275","source":"Crossref","is-referenced-by-count":42,"title":["Random walk on spheres method for solving drift-diffusion problems"],"prefix":"10.1515","volume":"22","author":[{"given":"Karl K.","family":"Sabelfeld","sequence":"first","affiliation":[{"name":"Institute of Computational Mathematics and Mathematical Geophysics, Russian Academy of Sciences, Novosibirsk, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2016,11,17]]},"reference":[{"key":"2023040102000306943_j_mcma-2016-0118_ref_001_w2aab2b8e1324b1b7b1ab2ab1Aa","unstructured":"Brown G. W.,\nMonte Carlo Methods,\nMcGraw\u2013Hill, New York, 1956."},{"key":"2023040102000306943_j_mcma-2016-0118_ref_002_w2aab2b8e1324b1b7b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"Ermakov S. M., Nekrutkin V. V. and Sipin A. S.,\nRandom Processes for Classical Equations of Mathematical Physics,\nKluwer Academic Publishers, Dodrecht, 1989.","DOI":"10.1007\/978-94-009-2243-3"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_003_w2aab2b8e1324b1b7b1ab2ab3Aa","unstructured":"Fischer N. I.,\nStatistical Analysis of Circular Data,\nCambridge University Press, Cambridge, 1995."},{"key":"2023040102000306943_j_mcma-2016-0118_ref_004_w2aab2b8e1324b1b7b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"Forbes C., Evans M., Hastings N. and Peacock B.,\nStatistical Distributions, 4th ed.,\nJohn Wiley & Sons, Hoboken, 2011.","DOI":"10.1002\/9780470627242"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_005_w2aab2b8e1324b1b7b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"Golyandina N.,\nConvergence rate for spherical processes with shifted centres,\nMonte Carlo Methods Appl. 10 (2004), no. 3\u20134, 287\u2013296.","DOI":"10.1515\/mcma.2004.10.3-4.287"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_006_w2aab2b8e1324b1b7b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"Haji-Sheikh A. and Sparrow E. M.,\nThe floating random walk and its application to Monte Carlo solutions of heat equations,\nSIAM J. Appl. Math. 14 (1966), no. 2, 570\u2013589.","DOI":"10.1137\/0114031"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_007_w2aab2b8e1324b1b7b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"Mascagni M. and Simonov N. A.,\nMonte Carlo methods for calculating some physical properties of large molecules,\nSIAM J. Sci. Comput. 26 (2004), no. 1, 339\u2013357.","DOI":"10.1137\/S1064827503422221"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_008_w2aab2b8e1324b1b7b1ab2ab8Aa","unstructured":"Mikhailov G. A.,\nWeighted Monte Carlo Methods,\nSB RAS, Novosibirsk, 2000."},{"key":"2023040102000306943_j_mcma-2016-0118_ref_009_w2aab2b8e1324b1b7b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"Motoo M.,\nSome evaluations for continuous Monte Carlo method by using Brownian hitting process,\nAnn. Inst. Statist. Math. 11 (1959), 49\u201355.","DOI":"10.1007\/BF01831723"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_010_w2aab2b8e1324b1b7b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"M\u00fcller M. E.,\nSome continuous Monte Carlo methods for the Dirichlet problem,\nAnn. Math. Statist. 27 (1956), no. 3, 569\u2013589.","DOI":"10.1214\/aoms\/1177728169"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_011_w2aab2b8e1324b1b7b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"\u00d8ksendal B.,\nStochastic Differential Equations: An Introduction with Applications,\nSpringer, Berlin, 2003.","DOI":"10.1007\/978-3-642-14394-6"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_012_w2aab2b8e1324b1b7b1ab2ac12Aa","unstructured":"Prudnikov A. P., Brychkov J. F. and Marichev O. I.,\nIntegrals and Series,\nNauka, Moscow, 1981."},{"key":"2023040102000306943_j_mcma-2016-0118_ref_013_w2aab2b8e1324b1b7b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"Sabelfeld K. K.,\nMonte Carlo Methods in Boundary Value Problems,\nSpringer, Berlin, 1991.","DOI":"10.1007\/978-3-642-75977-2"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_014_w2aab2b8e1324b1b7b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"Sabelfeld K. K.,\nA mesh free floating random walk method for solving diffusion imaging problems,\nStatist. Probab. Lett. 121 (2017), 6\u201311.","DOI":"10.1016\/j.spl.2016.10.006"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_015_w2aab2b8e1324b1b7b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"Sabelfeld K. K. and Shalimova I. A.,\nSpherical and Plane Integral Operators for PDEs: Construction, Analysis, and Applications,\nDe Gruyter, Berlin, 2013.","DOI":"10.1515\/9783110315332"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_016_w2aab2b8e1324b1b7b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"Sabelfeld K. K. and Simonov N. A.,\nStochastic Methods for Boundary Value Problems. Numerics for High-Dimensional PDEs and Applications,\nDe Gruyter, Berlin, 2016.","DOI":"10.1515\/9783110479454"},{"key":"2023040102000306943_j_mcma-2016-0118_ref_017_w2aab2b8e1324b1b7b1ab2ac17Aa","doi-asserted-by":"crossref","unstructured":"Simonov N. 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