{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:34:42Z","timestamp":1787236482692,"version":"build-2736575974"},"reference-count":49,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Multiscale Model. Simul."],"published-print":{"date-parts":[[2013,1]]},"abstract":"<jats:p>The slow processes of metastable stochastic dynamical systems are difficult to access by direct numerical simulation due to the sampling problems. Here, we suggest an approach for modeling the slow parts of Markov processes by approximating the dominant eigenfunctions and eigenvalues of the propagator. To this end, a variational principle is derived that is based on the maximization of a Rayleigh coefficient. It is shown that this Rayleigh coefficient can be estimated from statistical observables that can be obtained from short distributed simulations starting from different parts of state space. The approach forms a basis for the development of adaptive and efficient computational algorithms for simulating and analyzing metastable Markov processes while avoiding the sampling problem. Since any stochastic process with finite memory can be transformed into a Markov process, the approach is applicable to a wide range of processes relevant for modeling complex real-world phenomena.<\/jats:p>","DOI":"10.1137\/110858616","type":"journal-article","created":{"date-parts":[[2013,6,26]],"date-time":"2013-06-26T10:25:58Z","timestamp":1372242358000},"page":"635-655","source":"Crossref","is-referenced-by-count":292,"title":["A Variational Approach to Modeling Slow Processes in Stochastic Dynamical Systems"],"prefix":"10.1137","volume":"11","author":[{"given":"Frank","family":"No\u00e9","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Feliks","family":"N\u00fcske","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2013,6,26]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s00285-008-0197-8"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0910390107"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.physchem.53.082301.113146"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein,\n                      Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times\n                      , J. Eur. Math. Soc. (JEMS), 6 (2004), pp. 399-424.","DOI":"10.4171\/jems\/14"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein,\n                      Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues\n                      , J. Eur. Math. Soc. 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Kolkiewicz,\n                      Fitting diffusion models in finance\n                      , in Selected Proceedings of the Symposium on Estimating Functions (Athens, GA, 1996), IMS Lecture Notes Monogr. Ser. 32, Institute of Mathematical Statistics, Hayward, CA, 1997, pp. 327-350.","DOI":"10.1214\/lnms\/1215455054"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1063\/1.2335447"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1004646108"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1063\/1.2714539"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0905466106"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1063\/1.2959573"},{"key":"atypb30","unstructured":"J. D. 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