{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T13:58:01Z","timestamp":1761919081469},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2007,12,20]],"date-time":"2007-12-20T00:00:00Z","timestamp":1198108800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2008,2]]},"DOI":"10.1007\/s10915-007-9179-z","type":"journal-article","created":{"date-parts":[[2007,12,19]],"date-time":"2007-12-19T20:28:07Z","timestamp":1198096087000},"page":"127-151","source":"Crossref","is-referenced-by-count":44,"title":["A Hierarchy of Approximations of the Master Equation Scaled by a Size Parameter"],"prefix":"10.1007","volume":"34","author":[{"given":"Lars","family":"Ferm","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Per","family":"L\u00f6tstedt","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andreas","family":"Hellander","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2007,12,20]]},"reference":[{"key":"9179_CR1","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1038\/35002258","volume":"403","author":"N. Barkai","year":"2000","unstructured":"Barkai, N., Leibler, S.: Circadian clocks limited by noise. Nature 403, 267\u2013268 (2000)","journal-title":"Nature"},{"key":"9179_CR2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0962492900002804","volume":"7","author":"R.E. Caflisch","year":"1998","unstructured":"Caflisch, R.E.: Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 7, 1\u201349 (1998)","journal-title":"Acta Numer."},{"key":"9179_CR3","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.jcp.2004.12.014","volume":"206","author":"Y. Cao","year":"2005","unstructured":"Cao, Y., Gillespie, D., Petzold, L.: Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems. J. Comput. Phys. 206, 395\u2013411 (2005)","journal-title":"J. Comput. Phys."},{"key":"9179_CR4","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1016\/j.jtbi.2004.11.033","volume":"234","author":"W.-Y. Chen","year":"2005","unstructured":"Chen, W.-Y., Bokka, A.: Stochastic modeling of nonlinear epidemiology. J. Theor. Biol. 234, 455\u2013470 (2005)","journal-title":"J. Theor. Biol."},{"key":"9179_CR5","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1006\/jtbi.1995.0179","volume":"176","author":"U. Dieckmann","year":"1995","unstructured":"Dieckmann, U., Marrow, P., Law, R.: Evolutionary cycling in predator-prey interactions: population dynamics and the Red Queen. J. Theor. Biol. 176, 91\u2013102 (1995)","journal-title":"J. Theor. Biol."},{"key":"9179_CR6","volume-title":"Foundations of Modern Analysis","author":"J. Dieudonn\u00e9","year":"1969","unstructured":"Dieudonn\u00e9, J.: Foundations of Modern Analysis. Academic Press, New York (1969)"},{"key":"9179_CR7","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.jcp.2006.06.019","volume":"221","author":"W. E","year":"2007","unstructured":"E, W., Liu, D., Vanden-Eijnden, E.: Nested stochastic simulation algorithm for chemical kinetic systems with multiple time scales. J. Comput. Phys. 221, 158\u2013180 (2007)","journal-title":"J. Comput. Phys."},{"key":"9179_CR8","doi-asserted-by":"crossref","first-page":"2475","DOI":"10.1101\/gr.1196503","volume":"13","author":"J. Elf","year":"2003","unstructured":"Elf, J., Ehrenberg, M.: Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Res. 13, 2475\u20132484 (2003)","journal-title":"Genome Res."},{"key":"9179_CR9","unstructured":"Elf, J., L\u00f6tstedt, P., Sj\u00f6berg, P.: Problems of high dimension in molecular biology. In: Hackbusch, W. (ed.) High-Dimensional Problems\u2014Numerical Treatment and Applications. Proceedings of the 19th GAMM-Seminar, Leipzig, 2003, pp. 21\u201330; available at http:\/\/www.mis.mpg.de\/conferences\/gamm\/2003\/ (2003)"},{"key":"9179_CR10","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/S0006-3495(03)74839-5","volume":"84","author":"J. Elf","year":"2003","unstructured":"Elf, J., Paulsson, J., Berg, O.G., Ehrenberg, M.: Near-critical phenomena in intracellular metabolite pools. Biophys. J. 84, 154\u2013170 (2003)","journal-title":"Biophys. J."},{"key":"9179_CR11","first-page":"498","volume":"180","author":"S. Engblom","year":"2006","unstructured":"Engblom, S.: Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput. 180, 498\u2013515 (2006)","journal-title":"Appl. Math. Comput."},{"key":"9179_CR12","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1017\/S0962492902000119","volume":"3","author":"B. Engquist","year":"2003","unstructured":"Engquist, B., Runborg, O.: Computational high frequency wave propagation. Acta Numer. 3, 181\u2013266 (2003)","journal-title":"Acta Numer."},{"key":"9179_CR13","doi-asserted-by":"crossref","first-page":"021908","DOI":"10.1103\/PhysRevE.69.021908","volume":"69","author":"C. Escudera","year":"2004","unstructured":"Escudera, C., Buceta, J., de la Rubia, F.J., Lindenberg, K.: Extinction in population dynamics. Phys. Rev. E 69, 021908 (2004)","journal-title":"Phys. Rev. E"},{"key":"9179_CR14","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316658","volume-title":"Markov Processes, Characterization and Convergence","author":"S.N. Ethier","year":"1986","unstructured":"Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (1986)"},{"key":"9179_CR15","doi-asserted-by":"crossref","first-page":"337","DOI":"10.4064\/aa-41-4-337-351","volume":"41","author":"H. Faure","year":"1982","unstructured":"Faure, H.: Discr\u00e9pance de suites associ\u00e9es \u00e0 un syst\u00e8me de num\u00e9ration (en dimension s). Acta Aritm. 41, 337\u2013351 (1982)","journal-title":"Acta Aritm."},{"key":"9179_CR16","unstructured":"Ferm, L., L\u00f6tstedt, P.: Adaptive solution of the master equation in low dimensions. Technical report 2007-023, Department of Information Technology, Uppsala University, Uppsala, Sweden; available at http:\/\/www.it.uu.se\/research\/publications\/reports\/2007-023\/ (2007)"},{"key":"9179_CR17","doi-asserted-by":"crossref","first-page":"S61","DOI":"10.1007\/s10543-006-0082-z","volume":"46","author":"L. Ferm","year":"2006","unstructured":"Ferm, L., L\u00f6tstedt, P., Sj\u00f6berg, P.: Conservative solution of the Fokker-Planck equation for stochastic chemical reactions. BIT 46, S61\u2013S83 (2006)","journal-title":"BIT"},{"key":"9179_CR18","doi-asserted-by":"crossref","first-page":"1709","DOI":"10.1103\/PhysRevA.43.1709","volume":"43","author":"R.F. Fox","year":"1991","unstructured":"Fox, R.F., Keizer, J.: Amplification of intrinsic fluctuations by chaotic dynamics in physical systems. Phys. Rev. A 43, 1709\u20131720 (1991)","journal-title":"Phys. Rev. A"},{"key":"9179_CR19","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-05389-8","volume-title":"Handbook of Stochastic Methods","author":"C.W. Gardiner","year":"2004","unstructured":"Gardiner, C.W.: Handbook of Stochastic Methods, 3rd edn. Springer, Berlin (2004)","edition":"3"},{"key":"9179_CR20","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1017\/S096249290200003X","volume":"11","author":"M.B. Giles","year":"2002","unstructured":"Giles, M.B., S\u00fcli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145\u2013236 (2002)","journal-title":"Acta Numer."},{"key":"9179_CR21","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/0021-9991(76)90041-3","volume":"22","author":"D.T. Gillespie","year":"1976","unstructured":"Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403\u2013434 (1976)","journal-title":"J. Comput. Phys."},{"key":"9179_CR22","doi-asserted-by":"crossref","first-page":"R55","DOI":"10.1088\/0951-7715\/17\/6\/R01","volume":"17","author":"D. Givon","year":"2004","unstructured":"Givon, D., Kupferman, R., Stewart, A.: Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17, R55\u2013R127 (2004)","journal-title":"Nonlinearity"},{"key":"9179_CR23","volume-title":"Solving Ordinary Differential Equations I. Nonstiff Problems","author":"E. Hairer","year":"1993","unstructured":"Hairer, E., N\u00f8rsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd edn. Springer, Berlin (1993)","edition":"2"},{"key":"9179_CR24","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1145\/779359.779360","volume":"29","author":"H.S. Hong","year":"2003","unstructured":"Hong, H.S., Hickernell, F.J.: Algorithm 823: implementing scrambled digital sequences. ACM Trans. Math. Softw. 29, 95\u2013109 (2003)","journal-title":"ACM Trans. Math. Softw."},{"key":"9179_CR25","doi-asserted-by":"crossref","first-page":"49","DOI":"10.2307\/3212147","volume":"7","author":"T.G. Kurtz","year":"1970","unstructured":"Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure jump Markov processes. J.\u00a0Appl. Probab. 7, 49\u201358 (1970)","journal-title":"J.\u00a0Appl. Probab."},{"key":"9179_CR26","doi-asserted-by":"crossref","first-page":"344","DOI":"10.2307\/3211904","volume":"7","author":"T.G. Kurtz","year":"1971","unstructured":"Kurtz, T.G.: Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab. 7, 344\u2013356 (1971)","journal-title":"J. Appl. Probab."},{"key":"9179_CR27","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1137\/050639120","volume":"5","author":"P. L\u00f6tstedt","year":"2006","unstructured":"L\u00f6tstedt, P., Ferm, L.: Dimensional reduction of the Fokker-Planck equation for stochastic chemical reactions. Multiscale Methods Simul. 5, 593\u2013614 (2006)","journal-title":"Multiscale Methods Simul."},{"key":"9179_CR28","doi-asserted-by":"crossref","first-page":"814","DOI":"10.1073\/pnas.94.3.814","volume":"94","author":"H.H. McAdams","year":"1997","unstructured":"McAdams, H.H., Arkin, A.: Stochastic mechanisms in gene expression. Proc. Nat. Acad. Sci. USA 94, 814\u2013819 (1997)","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"9179_CR29","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/S0168-9525(98)01659-X","volume":"15","author":"H.H. McAdams","year":"1999","unstructured":"McAdams, H.H., Arkin, A.: It\u2019s a noisy business. Genetic regulation at the nanomolar scale. Trends Genet. 15, 65\u201369 (1999)","journal-title":"Trends Genet."},{"key":"9179_CR30","doi-asserted-by":"crossref","first-page":"041902","DOI":"10.1103\/PhysRevE.70.041902","volume":"70","author":"A.J. McKane","year":"2004","unstructured":"McKane, A.J., Newman, T.J.: Stochastic models in population biology and their deterministic analogs. Phys. Rev. E 70, 041902 (2004)","journal-title":"Phys. Rev. E"},{"key":"9179_CR31","doi-asserted-by":"crossref","DOI":"10.1007\/b98868","volume-title":"Mathematical Biology I. An Introduction","author":"J.D. Murray","year":"2002","unstructured":"Murray, J.D.: Mathematical Biology I. An Introduction, 3rd edn. Springer, New York (2002)","edition":"3"},{"key":"9179_CR32","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1006\/jcph.2002.7183","volume":"182","author":"J.T. Oden","year":"2002","unstructured":"Oden, J.T., Prudhomme, S.: Estimation of modeling error in computational mechanics. J. Comput. Phys. 182, 496\u2013515 (2002)","journal-title":"J. Comput. Phys."},{"key":"9179_CR33","doi-asserted-by":"crossref","first-page":"1884","DOI":"10.1137\/S0036142994277468","volume":"34","author":"A.B. Owen","year":"1997","unstructured":"Owen, A.B.: Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34, 1884\u20131910 (1997)","journal-title":"SIAM J. Numer. Anal."},{"key":"9179_CR34","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61544-3","volume-title":"The Fokker-Planck Equation","author":"H. Risken","year":"1996","unstructured":"Risken, H.: The Fokker-Planck Equation, 2nd edn. Springer, Berlin (1996)","edition":"2"},{"key":"9179_CR35","author":"P. Sj\u00f6berg","year":"2007","unstructured":"Sj\u00f6berg, P., L\u00f6tstedt, P., Elf, J.: Fokker-Planck approximation of the master equation in molecular biology. Comput. Vis. Sci. (2007). doi: 10.1007\/s00791-006-0045-6","journal-title":"Comput. Vis. Sci."},{"key":"9179_CR36","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/S0022-5193(03)00041-9","volume":"222","author":"N. Stollenwerk","year":"2003","unstructured":"Stollenwerk, N., Jensen, V.A.A.: Meningitis, pathogenicity near criticality: the epidemiology of meningococcal disease as a model for accidental pathogens. J. Theor. Biol. 222, 347\u2013359 (2003)","journal-title":"J. Theor. Biol."},{"key":"9179_CR37","volume-title":"Nonlinear Dynamics and Chaos","author":"S.H. Strogatz","year":"1994","unstructured":"Strogatz, S.H.: Nonlinear Dynamics and Chaos. Perseus Books, Cambridge (1994)"},{"key":"9179_CR38","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198503989.001.0001","volume-title":"The Lattice Boltzmann Equation for Fluid Dynamics and Beyond","author":"S. Succi","year":"2001","unstructured":"Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Clarendon, Oxford (2001)"},{"key":"9179_CR39","first-page":"245","volume":"34","author":"N.G. Kampen van","year":"1976","unstructured":"van Kampen, N.G.: The expansion of the master equation. Adv. Chem. Phys. 34, 245\u2013309 (1976)","journal-title":"Adv. Chem. Phys."},{"key":"9179_CR40","volume-title":"Stochastic Processes in Physics and Chemistry","author":"N.G. Kampen van","year":"1992","unstructured":"van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1992)"},{"key":"9179_CR41","doi-asserted-by":"crossref","first-page":"5988","DOI":"10.1073\/pnas.092133899","volume":"99","author":"J.M.G. Vilar","year":"2002","unstructured":"Vilar, J.M.G., Kueh, H.Y., Barkai, N., Leibler, S.: Mechanisms of noise-resistance in genetic oscillators. Proc. Nat. Acad. Sci. USA 99, 5988\u20135992 (2002)","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"9179_CR42","volume-title":"Sociodynamics. A Systematic Approach to Mathematical Modelling in the Social Sciences","author":"W. Weidlich","year":"2000","unstructured":"Weidlich, W.: Sociodynamics. A Systematic Approach to Mathematical Modelling in the Social Sciences. Taylor and Francis, London (2000)"},{"key":"9179_CR43","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.chaos.2004.07.022","volume":"24","author":"W. Weidlich","year":"2005","unstructured":"Weidlich, W.: Thirty years of sociodynamics. An integrated strategy of modelling in the social sciences: applications to migration and urban evolution. Chaos Solit. Fract. 24, 45\u201356 (2005)","journal-title":"Chaos Solit. Fract."},{"key":"9179_CR44","unstructured":"Wilcox, D.C.: Turbulence modeling for CFD. DCW Industries, La Ca\u00f1ada, CA (1994)"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-007-9179-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-007-9179-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-007-9179-z","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,20]],"date-time":"2024-02-20T15:51:47Z","timestamp":1708444307000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-007-9179-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,12,20]]},"references-count":44,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2008,2]]}},"alternative-id":["9179"],"URL":"https:\/\/doi.org\/10.1007\/s10915-007-9179-z","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,12,20]]}}}