TY - GEN A1 - Deuflhard, Peter A1 - Huisinga, Wilhelm A1 - Jahnke, Tobias A1 - Wulkow, Michael T1 - Adaptive Discrete Galerkin Methods Applied to the Chemical Master Equation N2 - \begin{abstract} In systems biology, the stochastic description of biochemical reaction kinetics is increasingly being employed to model gene regulatory networks and signalling pathways. Mathematically speaking, such models require the numerical solution of the underlying evolution equat ion, also known as the chemical master equation (CME). Up to now, the CME has almost exclusively been treated by Monte-Carlo techniques, the most prominent of which is the simulation algorithm suggest ed by Gillespie in 1976. Since this algorithm requires an update for each single reaction event, realizations can be computationally very costly. As an alternative, we here propose a novel approach, which focuses on the discrete partial differential equation (PDE) structure of the CME and thus allows to adopt ideas from adaptive discrete Galerkin methods (as designed by two of the present authors in 1989), which have proven to be highly efficient in the mathematical modelling of polyreaction kinetics. Among the two different options of discretizing the CME as a discrete PDE, the method of lines approach (first space, then time) and the Rothe method (first time, then space), we select the latter one for clear theoretical and algorithmic reasons. First numeric al experiments at a challenging model problem illustrate the promising features of the proposed method and, at the same time, indicate lines of necessary further research. \end{abstract} T3 - ZIB-Report - 07-04 KW - adaptive discrete Galerkin methods KW - adaptive Rothe method KW - discrete Chebyshev polynomials KW - stochastic reactions kinetics KW - chemical master equation Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9471 ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Best, Christoph A1 - Cordes, Frank A1 - Roitzsch, Rainer A1 - Schütte, Christof T1 - Identification of Molecular Conformations via Statistical Analysis of Simulation Data JF - Comp. Chem. Y1 - 1999 VL - 20 SP - 1760 EP - 1774 ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Telgmann, R. A1 - Wulkow, Michael T1 - The virtual lab approach to pharmacokinetic JF - Drug Discov Today Y1 - 2006 VL - 11 SP - 800 EP - 805 ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Best, Christoph A1 - Roitzsch, Rainer A1 - Schütte, Christof A1 - Cordes, Frank T1 - From Simulation Data to Conformational Ensembles JF - J. Comp. Chem. Y1 - 1999 VL - 20 IS - 16 SP - 1760 EP - 1774 ER - TY - CHAP A1 - Galliat, Tobias A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter ED - H. Bothe, R. T1 - Self-Organizing Maps Combined with Eigenmode Analysis for Automated Cluster Identification T2 - Neural Computation, ICSC Academic Press Y1 - 2000 SP - 227 EP - 232 ER - TY - JOUR A1 - Schütte, Christof A1 - Fischer, Alexander A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter T1 - A Direct Approach to Conformational Dynamics Based on Hybrid Monte Carlo JF - J. Comput. Phys. Y1 - 1999 UR - http://www.zib.de/PaperWeb/abstracts/SC-98-45 VL - 151 SP - 146 EP - 168 ER - TY - CHAP A1 - Schütte, Christof A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter ED - Fiedler, Bernold T1 - Transfer operator approach to conformational dynamics in biomolecular systems T2 - Ergodic theory, analysis, and efficient simulation of dynamical systems Y1 - 2001 SP - 191 EP - 223 PB - Berlin: Springer ER - TY - JOUR A1 - von Kleist, Max A1 - Menz, Stephan A1 - Stocker, Hartmut A1 - Arasteh, Keikawus A1 - Huisinga, Wilhelm A1 - Schütte, Christof T1 - HIV Quasispecies Dynamics during Pro-active Treatment Switching JF - Plos One Y1 - 2011 UR - http://publications.imp.fu-berlin.de/1051/ U6 - https://doi.org/10.1371/journal.pone.0018204 VL - 6 IS - 3 SP - e18204 PB - Public Library of Science ER - TY - JOUR A1 - Menz, Stephan A1 - Latorre, J. A1 - Schütte, Christof A1 - Huisinga, Wilhelm T1 - Hybrid Stochastic--Deterministic Solution of the Chemical Master Equation JF - SIAM Interdisciplinary Journal Multiscale Modeling and Simulation (MMS) Y1 - 2012 UR - http://publications.imp.fu-berlin.de/1039/ U6 - https://doi.org/10.1137/110825716 VL - 10 IS - 4 SP - 1232 EP - 1262 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Huisinga, Wilhelm A1 - Jahnke, Tobias A1 - Wulkow, Michael T1 - Adaptive Discrete Galerkin Methods Applied to the Chemical Master Equation JF - SIAM J. Sci. Comput Y1 - 2008 VL - 30 IS - 6 SP - 2990 EP - 3011 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Huisinga, Wilhelm A1 - Fischer, Alexander A1 - Schütte, Christof T1 - Identification of Almost Invariant Aggregates in Reversible Nearly Uncoupled Markov Chains JF - Lin. Alg. Appl. Y1 - 2000 UR - http://publications.imp.fu-berlin.de/84/ U6 - https://doi.org/10.1016/S0024-3795(00)00095-1 VL - 315 IS - 1-3 SP - 39 EP - 59 ER - TY - CHAP A1 - Schütte, Christof A1 - Huisinga, Wilhelm ED - Fiedler, B. ED - Grger, K. ED - Sprekels, Jürgen T1 - On Conformational Dynamics induced by Langevin Processes T2 - Equadiff 99 Y1 - 2000 UR - http://publications.imp.fu-berlin.de/131/ U6 - https://doi.org/10.1142/9789812792617_0234 VL - 2 SP - 1247 EP - 1262 PB - World Scientific ER - TY - JOUR A1 - Schütte, Christof A1 - Fischer, Alexander A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter T1 - A Direct Approach to Conformational Dynamics based on Hybrid Monte Carlo JF - J. Comp. Phys Y1 - 1999 UR - http://publications.imp.fu-berlin.de/87/ U6 - https://doi.org/10.1006/jcph.1999.6231 VL - 151 IS - 1 SP - 146 EP - 168 ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Best, Christoph A1 - Roitzsch, Rainer A1 - Schütte, Christof A1 - Cordes, Frank T1 - From Simulation Data to Conformational Ensembles JF - J. Comp. Chem. Y1 - 1999 UR - http://publications.imp.fu-berlin.de/92/ U6 - https://doi.org/10.1002/(SICI)1096-987X(199912)20:16<1760::AID-JCC8>3.0.CO;2-2 VL - 20 IS - 16 SP - 1760 EP - 1774 ER - TY - JOUR A1 - Horenko, Illia A1 - Lorenz, S. A1 - Schütte, Christof A1 - Huisinga, Wilhelm T1 - Adaptive Approach for Non-Linear Sensitivity Analysis of Reaction Kinetics JF - J. Comp. Chem. Y1 - 2005 UR - http://publications.imp.fu-berlin.de/49/ U6 - https://doi.org/10.1002/jcc.20234 VL - 26 IS - 9 SP - 941 EP - 948 ER - TY - JOUR A1 - Schütte, Christof A1 - Walter, J. A1 - Hartmann, Carsten A1 - Huisinga, Wilhelm T1 - An Averaging Principle for Fast Degrees of Freedom Exhibiting Long-Term Correlations JF - Multiscale Model. Simul. Y1 - 2004 UR - http://publications.imp.fu-berlin.de/62/ U6 - https://doi.org/10.1137/030600308 VL - 2 IS - 3 SP - 501 EP - 526 ER - TY - JOUR A1 - Nettesheim, Peter A1 - Huisinga, Wilhelm A1 - Schütte, Christof T1 - Chebyshev-Approximation for Wavepacket-Dynamics JF - preprint Y1 - 1996 UR - http://publications.imp.fu-berlin.de/102/ ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Meyn, S. A1 - Schütte, Christof T1 - Phase Transitions and Metastability in Markovian and Molecular Systems JF - Ann. Appl. Prob. Y1 - 2004 UR - http://publications.imp.fu-berlin.de/67/ U6 - https://doi.org/10.1214/aoap/1075828057 VL - 14 IS - 1 SP - 419 EP - 458 ER - TY - CHAP A1 - Schütte, Christof A1 - Huisinga, Wilhelm ED - Ciarlet, P. ED - Le Bris, Claude T1 - Biomolecular Conformations can be Identified as Metastable Sets of Molecular Dynamics T2 - Special Volume Y1 - 2003 UR - http://publications.imp.fu-berlin.de/126/ VL - X SP - 699 EP - 744 PB - Elsevier ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Schütte, Christof A1 - Stuart, Andrew T1 - Extracting Macroscopic Stochastic Dynamics JF - Comm. Pure Appl. Math. Y1 - 2003 UR - http://publications.imp.fu-berlin.de/72/ U6 - https://doi.org/10.1002/cpa.10057 VL - 56 IS - 2 SP - 234 EP - 269 ER -