×

zbMATH — the first resource for mathematics

Solving chemical master equations by adaptive wavelet compression. (English) Zbl 1203.65104
After the introduction of the chemical master equation (CME) the approximation of the solution of the CME by means of wavelets is discussed. The solution is represented in a sparse wavelet basis which is adapted in each time step. At first, a wavelet method which is only adaptive in space is considered. Hereby, a 2-stage Gauss-Runge-Kutta method and higher order Daubechies wavelets are used. Then, a strategy for choosing the step-size in time adaptively is proposed. Four numerical examples are presented to demonstrate the performance of the adaptive wavelet method.

MSC:
65L05 Numerical methods for initial value problems
65T60 Numerical methods for wavelets
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
92B20 Neural networks for/in biological studies, artificial life and related topics
92E20 Classical flows, reactions, etc. in chemistry
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
Software:
RODAS
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Burrage, K.; Hegland, M.; MacNamara, S.; Sidje, R.B., A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems, (), 21-38
[2] Cohen, A., Numerical analysis of wavelet methods, Studies in mathematics and its applications, vol. 32, (2003), Elsevier · Zbl 1038.65151
[3] Cohen, A.; Dahmen, W.; DeVore, R., Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. comput., 70, 233, 27-75, (2001) · Zbl 0980.65130
[4] Cohen, A.; Dahmen, W.; DeVore, R., Adaptive wavelet methods. II: beyond the elliptic case, Found. comput. math., 2, 3, 203-245, (2002) · Zbl 1025.65056
[5] Dahmen, W., Wavelets and multiscale methods for operator equations, Acta numer., 6, 55-228, (1997) · Zbl 0884.65106
[6] Dahmen, W., Wavelet methods for PDEs – some recent developments, J. comput. appl. math., 128, 133-185, (2001) · Zbl 0974.65101
[7] Daubechies, I., Ten lectures on wavelets, CBMS-NSF regional conference series in applied mathematics, vol. 61, (1992), SIAM, Society for Industrial and Applied Mathematics Philadelphia, PA · Zbl 0776.42018
[8] Deuflhard, P.; Huisinga, W.; Jahnke, T.; Wulkow, M., Adaptive discrete Galerkin methods applied to the chemical master equation, SIAM J. sci. comput., 30, 6, 2990-3011, (2008) · Zbl 1178.41003
[9] Engblom, S., Galerkin spectral method applied to the chemical master equation, Commun. comput. phys., 5, 5, 871-896, (2009) · Zbl 1365.92151
[10] Engblom, S., Spectral approximation of solutions to the chemical master equation, J. comput. appl. math., 229, 1, 208-221, (2009) · Zbl 1168.65006
[11] Ferm, L.; Lötstedt, P., Adaptive solution of the master equation in low dimensions, Appl. numer. math., 59, 1, 187-204, (2009) · Zbl 1155.65008
[12] Ferm, L.; Lötstedt, P.; Hellander, A., A hierarchy of approximations of the master equation scaled by a size parameter, SIAM J. sci. comput., 34, 127-151, (2008) · Zbl 1136.65007
[13] Gillespie, D.T., A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. comput. phys., 22, 403-434, (1976)
[14] Gillespie, D.T., A rigorous derivation of the chemical master equation, Physica A, 188, 404-425, (1992)
[15] Hairer, E.; Wanner, G., Solving ordinary differential equations II: stiff and differential – algebraic problems, Springer series in computational mathematics, vol. 14, (1996), Springer Berlin · Zbl 0859.65067
[16] Hegland, M.; Burden, C.; Santoso, L.; MacNamara, S.; Booth, H., A solver for the stochastic master equation applied to gene regulatory networks, J. comput. appl. math., 205, 708-724, (2007) · Zbl 1121.65009
[17] Hegland, M.; Hellander, A.; Lötstedt, P., Sparse grids and hybrid methods for the chemical master equation, Bit, 48, 265-284, (2008) · Zbl 1155.65304
[18] Hellander, A.; Lötstedt, P., Hybrid method for the chemical master equation, J. comput. phys., 227, 100-122, (2007) · Zbl 1126.80010
[19] Hethcote, H.W., The mathematics of infectious diseases, SIAM rev., 42, 4, 599-653, (2000) · Zbl 0993.92033
[20] Jahnke, T., An adaptive wavelet method for the chemical master equation, SIAM J. scient. comput., 31, 6, 4373-4394, (2010) · Zbl 1205.65022
[21] T. Jahnke, S. Galan, Solving chemical master equations by an adaptive wavelet method, in: T.E. Simos, G. Psihoyios, C. Tsitouras (Eds.), Numerical Analysis and Applied Mathematics: International Conference on Numerical Analysis and Applied Mathematics 2008, Psalidi, Kos, Greece, 16-20 September 2008, vol. 1048 of AIP Conference Proceedings, pp. 290-293, 2008. · Zbl 1167.65441
[22] Jahnke, T.; Huisinga, W., Solving the chemical master equation for monomolecular reaction systems analytically, J. math. biol., 54, 1, 1-26, (2007) · Zbl 1113.92032
[23] Jahnke, T.; Huisinga, W., A dynamical low-rank approach to the chemical master equation, Bull. math. biol., 70, 8, 2283-2302, (2008) · Zbl 1169.92021
[24] MacNamara, S.; Bersani, A.M.; Burrage, K.; Sidje, R.B., Stochastic chemical kinetics and the total quasi-steady-state assumption: application to the stochastic simulation algorithm and chemical master equation, J. chem. phys., 129, 9, 95-105, (2008)
[25] MacNamara, S.; Burrage, K.; Sidje, R.B., Multiscale modeling of chemical kinetics via the master equation, Multiscale model. simul., 6, 4, 1146-1168, (2008) · Zbl 1153.60370
[26] Mallat, S., A wavelet tour of signal processing, (2009), Elsevier Amsterdam · Zbl 1170.94003
[27] Munsky, B.; Khammash, M., The finite state projection algorithm for the solution of the chemical master equation, J. chem. phys., (2006)
[28] von Petersdorff, T.; Schwab, Ch., Numerical solution of parabolic equations in high dimensions, Esaim: m2an, 38, 1, 93-127, (2004) · Zbl 1083.65095
[29] Schwab, Ch.; Stevenson, R., Space – time adaptive wavelet methods for parabolic evolution problems, Math. comput., 78, 1293-1318, (2009) · Zbl 1198.65249
[30] Sjöberg, P.; Lötstedt, P.; Elf, J., Fokker – planck approximation of the master equation in molecular biology, Comput. visual. sci., 12, 1, 37-50, (2009)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.