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Numerical Solution of the Fokker- Planck Approximation of - DiVA PDF

104 Pages·2005·3.27 MB·English
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ITLicentiatetheses 2005-010 P a RReecceennttlliicceennttiiaatteetthheesseessffrroommtthheeDDeeppaarrtmtmeennttooffInInfoformrmataitoinonTeTcehcnhonloolgoygy u l 22000043--000153 EYrnikgvBeeSrge:le´Mne:tMhooddsefloSreRleucntiToinmeAnalysisofDataLocality Sjö b 22000044--000014 NMioclhaasmSmaenddgrEeln:ShPoabraakmie:tricOnM-CethhiopdsMfoonritForreinqguenfocyr-SNeloenc-tIivnetruMsiRveSpHeacr-d- er g twroasrceo/SpoyftwareObservability N Numerical Solution of the Fokker- 22000044--000025 MHeanrkriuksLNo¨ofr:dPe´anr:aPlalerlaizllienlgPtDheEMSeotlhveordsoofnCcocn-NjuUgaMteAGSryastdeimenstsforSharedMem- um 2004-003 YornygAvercSheiltee´nct:uMreosdelSelection eric Planck Approximation of the a 22000044--000046 MStoefhaanmJmoehdansEslonS:hHobigakhi:OrdOenr-CDhififpereMnoceniAtoprpinrgoxifmoratiNoonns-fIonrtrtuhseivLeinHeaarrdiz-ed l S wEaurleer/SEoqftuwaatrieonOsbservability olu Chemical Master Equation 22000054--000051 HJeesnpreikr LWo¨ifl:hPealmraslsleolniz:ingEftfihceiMenetthModemofoCryonMjuagnaategGemraednitenfotsrfoMreSshsaargeed-MPaesmsi-ng tion oCroynAcrucrhrietnecctyu—respartI:Single-threadedexecution of th 22000054--000062 SHtae˚kfaannJZoehfafnesrs:oHn:arHdiwgahreO-rSdoefrtwDairfefeTrernacdeeoAfpfsprinoxSimhaarteiodn-Ms feomrothryeILminpelaermizeendta- e F PAUL SJO¨BERG EtiuolnesrEquations o k k 22000055--000013 JMesapgenrusWA˚ilghreelnm:sHsoing:h-LEefvfieclieMntodMelelminogryanMdaLnoacgaelmSeenatrcfhor Message-Passing er - P 2005-004 COosnkcaurrWreinbclyin—g:pAadrtHI:ocSiRngoluet-itnhgrePardoetdoceoxlecVuatliiodnation la n 2005-002 Ha˚kanZeffer: Hardware-SoftwareTradeoffsinShared-MemoryImplementa- ck 2005-005 PeterNaucle´r:ModelingandControlofVibrationinMechanicaStructures A tions p 22000055--000036 MEraikgnBuas¨nA˚ggtsrseonn::HRigohb-uLsetvPelreMcoonddeliltiinogneadndIteLroactaivleSeSaorlcuhtionMethodsforLarge- pro ScaleNonsymmetricProblems x im 2005-004 OskarWibling:AdHocRoutingProtocolValidation a 2005-007 MeiHong:OnTwoMethodsforIdentifyingDynamicErrors-in-VariablesSys- tio 2005-005 PteemtesrNaucle´r:ModelingandControlofVibrationinMechanicalStructures n o 22000055--000068 ENriikklaBsa¨Jnoghtsasnosns:onR:oUbussatbPlereIcToSnydsittieomnsedfoIrteMraotibvieleSWolourtikonMethodsforLarge- f th 2005-009 SMcaaglenNusonEsvyemstmedettr:iPcaPrraombleetmersandStateEstimationusingAudioandVideoSig- e C h e nals m ic 2005-010 PaulSjo¨berg: NumericalSolutionoftheFokker-PlanckApproximationofthe al M ChemicalMasterEquation a s te r E q u a tio 05 n 0 2 a al s p p U et, eri k c y str et ersit UPPSALA UNIVERSITY v ni UDPepPaSrAtmLeAnt UoNf IInVfoErRmSaItTioYn Technology DepartmentofInformationTechnology,UppsalaUniversity,Sweden U DepartmentofInformationTechnology DepartmentofInformationTechnology,UppsalaUniversity,Sweden Numerical Solution of the Fokker-Planck Approximation of the Chemical Master Equation BY PAULSJO¨BERG December2005 DIVISIONOFSCIENTIFICCOMPUTING DEPARTMENTOFINFORMATIONTECHNOLOGY UPPSALAUNIVERSITY UPPSALA SWEDEN DissertationforthedegreeofLicentiateofTechnologyinScientificComputing atUppsalaUniversity2005 Numerical Solution of the Fokker-Planck Approximation of the Chemical Master Equation PaulSjo¨berg [email protected] DivisionofScientificComputing DepartmentofInformationTechnology UppsalaUniversity Box337 SE-75105 Uppsala Sweden http://www.it.uu.se/ (cid:176)c PaulSjo¨berg2005 ISSN1404-5117 PrintedbytheDepartmentofInformationTechnology,UppsalaUniversity,Sweden Abstract The chemical master equation (CME) describes the probability for the discrete molecular copy numbers that define the state of a chemical sys- tem. Each molecular species in the chemical model adds a dimension to the state space. The CME is a difference-differential equation which can be solved numerically if the state space is truncated at an upper limit of the copy number in each dimension. The size of the truncated CME suffers from an exponential growth for an increasing number of chemical species. In this thesis the CME is approximated by a continuous Fokker- Planck equation (FPE) which makes it possible to use sparser computa- tional grids than for CME. FPE on conservative form is used to compute steady state solutions by computation of an extremal eigenvalue and the corresponding eigenvector as well as time-dependent solutions by an im- plicit time-stepping scheme. The performance of the numerical solution is compared to a standard Monte Carlo algorithm. The computational work for solutions with the sameestimatederroriscomparedforthetwomethods. Dependingonthe problem, FPE or the Monte Carlo algorithm will be more efficient. FPE is well suited for problems in low dimensions, especially if high accuracy is desirable. ii List of Papers A J. Elf, P. L¨otstedt and P. Sj¨oberg. Problems of high dimension in molecular biology. Proceedings of the 19th GAMM-Seminar, Leipzig 2003. B L. Ferm, P. L¨otstedt and P. Sj¨oberg. Conservative solution of the Fokker-Planck equation for stochastic chemical equations. Techni- cal Report 2004-054, Department of Information Technology, Up- psala University, 2005, revised version. C P. Sj¨oberg, P. L¨otstedt and J. Elf. Fokker-Planck approximation of the master equation in molecular biology. Technical Report, Department of Information Technology, Uppsala University, 2005, to appear, will be available at: http://www.it.uu.se/research/publications/reports/. iii iv Acknowledgments I would like to thank my advisor Per L¨otstedt for his support and advice and also Johan Elf and M˚ans Ehrenberg for discussions and guidance in the microcosmos of molecular biology. This work has been supported by the National Graduate School in ScientificComputing,theSwedishFoundationforStrategicResearchand the Swedish Research Council. v vi

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The CME is a difference-differential equation which can be solved numerically if the state . Let us review the prerequisites of ordinary test tube chemistry. Con-.
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