{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:16:46Z","timestamp":1787329006486,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100009193","name":"Marsden Fund","doi-asserted-by":"publisher","award":["UOO1015"],"award-info":[{"award-number":["UOO1015"]}],"id":[{"id":"10.13039\/501100009193","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We develop a finite-dimensional approximation of the Frobenius--Perron operator using the finite volume method applied to the continuity equation for the evolution of probability over the state of a dynamical system. A CFL condition ensures that the approximation satisfies the Markov property, while existing convergence theory for the finite volume method guarantees convergence of the discrete operator to the continuous operator and convergence of probability distributions as mesh size tends to zero. Properties of the approximation are demonstrated in two computed examples of sequential inference for the state of a low-dimensional mechanical system, with the second example also demonstrating parameter estimation when incomplete and uncertain observations give rise to multimodal distributions.<\/jats:p>","DOI":"10.1137\/16m1108698","type":"journal-article","created":{"date-parts":[[2018,2,20]],"date-time":"2018-02-20T13:14:20Z","timestamp":1519132460000},"page":"570-589","source":"Crossref","is-referenced-by-count":3,"title":["Numerical Approximation of the Frobenius--Perron Operator using the Finite Volume Method"],"prefix":"10.1137","volume":"56","author":[{"given":"Richard A.","family":"Norton","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Colin","family":"Fox","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Malcolm E.","family":"Morrison","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,2,20]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1109\/78.978374"},{"key":"atypb2","series-title":"Wiley Series in Probability and Mathematical Statistics","volume-title":"Probability and Measure","author":"Billingsley P.","year":"1995","edition":"3"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2001.7.219"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2011.07.005"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/0-387-28982-8"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-010-0322-x"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/0362-546X(91)90211-I"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/0362-546X(94)00152-8"},{"key":"atypb9","doi-asserted-by":"crossref","DOI":"10.1142\/9789812839183_0011","volume-title":"Positive Operators, and Applications","author":"Ding J.","year":"2009"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3437-9"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2004.04.003"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"R. Eymard, T. Gallou\u00ebt, and R. Herbin,\n                      Finite volume methods\n                      , in Solution of Equation in ${R}^n$ (Part 3), Techniques of Scientific Computing (Part 3), Handbook of Numerical Analysis 7, Elsevier, Amsterdam, 2000, pp. 713-1018,https:\/\/doi.org\/10.1016\/S1570-8659(00)07005-8.","DOI":"10.1016\/S1570-8659(00)07005-8"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/110819986"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-006-9392-x"},{"key":"atypb15","first-page":"107","volume":"140","author":"Gordon N.","year":"1993","journal-title":"IEEE Proc-F"},{"key":"atypb16","volume-title":"Stochastic Processes and Filtering Theory","author":"Jazwinski A. H.","year":"1970"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1117\/12.280797"},{"key":"atypb18","doi-asserted-by":"crossref","first-page":"51","DOI":"10.3934\/jcd.2016003","volume":"3","author":"Klus S.","year":"2016","journal-title":"J. Comput. Dyn."},{"key":"atypb19","unstructured":"P. Koltai,\n                      Efficient Approximation Methods for the Global Long-Term Behavior of Dynamical Systems-Theory, Algorithms and Examples\n                      , Ph.D. thesis, Technische Universit\u00e4t M\u00fcnchen, M\u00fcnchen, 2010."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4286-4"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/140984415"},{"key":"atypb22","volume-title":"Finite","volume":"2002","author":"LeVeque R. 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