An adaptive wavelet method for the chemical master equation

SIAM Journal on Scientific Computing

Volumn 31, Issue 6, 2009, Pages 4373-4394

  Jahnke, Tobias ^a  

^a UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

Adaptive methods; Chemical master equation; Error bounds; Haar wavelets; High dimensional problems; Numerical integration; Stochastic reaction kinetics

Indexed keywords

DEGREES OF FREEDOM (MECHANICS); ERROR ANALYSIS; ITERATIVE METHODS; NUMERICAL METHODS; REACTION KINETICS;

ADAPTIVE METHODS; ADAPTIVE WAVELETS; CHEMICAL MASTER EQUATION; ERROR BOUND; HAAR-WAVELETS; HIGH-DIMENSIONAL PROBLEMS; HIGHER-DIMENSIONAL PROBLEMS; NUMERICAL INTEGRATIONS; STOCHASTIC REACTION KINETICS; WAVELET METHODS;

STOCHASTIC SYSTEMS;

EID: 77953020635     PISSN: 10648275     EISSN: 10957200     Source Type: Journal    
DOI: 10.1137/080742324     Document Type: Article

References (32)

1. F. Bornemann, An adaptive multilevel approach to parabolic equations: I. General theory and 1d implementation, IMPACT Comput. Sci. Eng., 2 (1990), pp. 279-317.
2. F. Bornemann, An adaptive multilevel approach to parabolic equations: II. Variable-order time discretization based on a multiplicative error correction, IMPACT Comput. Sci. Eng., 3 (1991), pp. 93-122.
3. K. Burrage, M. Hegland, S. MacNamara, and R. B. Sidje, A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems, in Markov Anniversary Meeting: An international conference to celebrate the 150th anniversary of the birth of A. A. Markov. A. N. Langville and W. J. Stewart, eds., Boson Books, Raleigh, NC, 2006, pp. 21-38.
4. A. Cohen, Numerical Analysis of wavelet methods, Volume 32 of Studies in Mathematics and its applications, Elsevier, New York, 2003.
5. W. Dahmen, Wavelets and multiscale methods for operator equations, Acta Numer., 6 (1997), pp. 55-228.
6. W. Dahmen, Wavelet methods for PDEs-some recent developments, J. Comput. Appl. Math., 128 (2001), pp. 133-185.
7. P. Deuflhard, W. Huisinga, T. Jahnke, and M. Wulkow, Adaptive discrete Galerkin methods applied to the chemical master equation, SIAM J. Sci. Comput., 30 (2008), pp. 2990-3011.
8. P. Deuflhard and M. Wulkow, Computational treatment of polyreaction kinetics by orthogonal polynomials of a discrete variable, IMPACT Comput. Sci. Eng., 1 (1989), pp. 269-301.
9. J. Elf, O. Berg, and M. Ehrenberg, Comparison of repressor and transcriptional attenuator systems for control of amino acid biosynthetic operons, J. Mol. Biol., 313 (2001), pp. 941-954.
10. S. Engblom, Spectral approximation of solutions to the chemical master equation, J. Comput. Appl. Math., 229 (2009), pp. 208-221.
11. S. Engblom, Galerkin spectral method applied to the chemical master equation, Commun. Comput. Phys., 5 (2009), pp. 871-896.
12. K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Number 194 in Grad. Texts in Math., Springer, Berlin, 2000.
13. L. Ferm and P. LÖtstedt, Adaptive solution of the master equation in low dimensions, Appl. Numer. Math., 59 (2009), pp. 187-204.
14. L. Ferm, P. LÖtstedt, and A. Hellander, A hierarchy of approximations of the master equation scaled by a size parameter, SIAM J. Sci. Comput., 34 (2008), pp. 127-151.
15. C. W. Gardiner, Handbook of Stochastic Methods, 3rd ed., Springer, New York, 2004.
16. D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), pp. 403-434.
17. D. T. Gillespie, A rigorous derivation of the chemical master equation, Physica A, 188 (1992), pp. 404-425.
18. M. Hegland, C. Burden, L. Santoso, S. MacNamara, and H. Booth, A solver for the stochastic master equation applied to gene regulatory networks, J. Comput. Appl. Math., 205 (2007), pp. 708-724.
19. M. Hegland, A. Hellander, and P. LÖtstedt, Sparse grids and hybrid methods for the chemical master equation, BIT, 48 (2008), pp. 265-284.
20. A. Hellander and P. LÖtstedt, Hybrid method for the chemical master equation, J. Comput. Phys., 227 (2007), pp. 100-122.
21. H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), pp. 599-653.
22. T. Jahnke and W. Huisinga, Solving the chemical master equation for monomolecular reaction systems analytically, J. Math. Biol., 54 (2007), pp. 1-26.
23. T. Jahnke and W. Huisinga, A dynamical low-rank approach to the chemical master equation, Bull. Math. Biol., 70 (2008), pp. 2283-2302.
24. T. Jahnke and S. Galan, Solving chemical master equations by an adaptive wavelet method, in Numerical analysis and applied mathematics: International Conference on Numerical Analysis and Applied Mathematics 2008, Psalidi, Kos, Greece, 2008, volume 1048 of AIP Conference Proceedings, T. E. Simos, G. Psihoyios, and C. Tsitouras, eds., 2008, AIP, New York, pp. 290-293.
25. P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Number 23 in Applications of Mathematics, Springer, New York, 1992.
26. S. MacNamara, K. Burrage, and R. B. Sidje, Multiscale modeling of chemical kinetics via the master equation, Multiscale Model. Simul., 6 (2008), pp. 1146-1168.
27. B. Munsky and M. Khammash, The finite state projection algorithm for the solution of the chemical master equation, J. Chem. Phys., 124 (2006), p. 044104.
28. Y. Nievergelt, Wavelets made easy, Birkhäuser, Boston, MA, 1999.
29. P. SjÖberg, P. LÖtstedt, and J. Elf, Fokker-Planck approximation of the master equation in molecular biology, Comput. Vis. Sci., 12 (2009), pp. 37-50.
30. Ch. Schwab and R. Stevenson, Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comp., 78 (2009), pp. 1293-1318.
31. T. von Petersdorff and Ch. Schwab, Numerical solution of parabolic equations in high dimensions, M2AN Math. Model. Numer. Anal., 38 (2004), pp. 93-127.
32. M. Wulkow, The simulation of molecular weight distribution in polyreaction kinetics by discrete Galerkin methods, Macromol. Theory Simul., 5 (1996), pp. 393-416.