Random walk algorithms for solving nonlinear chemotaxis problems

References

[1] H. Amann, Eine Monte-Carlo-Methode mit Informationsspeicherung zur Lösung von elliptischen Randwertproblemen, Z. Wahrscheinlichkeitstheorie Verw. Geb. 8 (1967), 117–130. 10.1007/BF00536914Suche in Google Scholar

[2] G. Arumugam and J. Tyagi, Keller–Segel chemotaxis models: A review, Acta Appl. Math. 171 (2021), Paper No. 6. 10.1007/s10440-020-00374-2Suche in Google Scholar

[3] A. F. Cheshkova, Global estimate of the solution of the Dirichlet problem for the Helmholtz n-dimensional equation by the Monte Carlo method, Russian J. Numer. Anal. Math. Modelling 10 (1995), no. 6, 495–510. 10.1515/rnam.1995.10.6.495Suche in Google Scholar

[4] A. Codutti, K. Bente, D. Faivre and S. Klumpp, Chemotaxis in external fields: Simulations for active magnetic biological matter, PLoS Comput. Biol. 15 (2019), no. 12, Article ID e1007548. 10.1371/journal.pcbi.1007548Suche in Google Scholar PubMed PubMed Central

[5] J. Haškovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller–Segel system, J. Stat. Phys. 135 (2009), no. 1, 133–151. 10.1007/s10955-009-9717-1Suche in Google Scholar

[6] T. Hillen and K. J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), no. 1–2, 183–217. 10.1007/s00285-008-0201-3Suche in Google Scholar PubMed

[7] D. Horstmann, From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. I, Jahresber. Dtsch. Math.-Ver. 105 (2003), no. 3, 103–165. Suche in Google Scholar

[8] S. Kawaguchi, Chemotaxis-growth under the influence of lateral inhibition in a three-component reaction-diffusion system, Nonlinearity 24 (2011), no. 4, 1011–1031. 10.1088/0951-7715/24/4/002Suche in Google Scholar

[9] E. F. Keller and L. A. Segel, Models of chemotaxis, J. Theoret. Biol. 30 (1971), 225–234. 10.1016/0022-5193(71)90050-6Suche in Google Scholar PubMed

[10] K. J. Painter, Mathematical models for chemotaxis and their applications in self-organisation phenomena, J. Theoret. Biol. 481 (2019), 162–182. 10.1016/j.jtbi.2018.06.019Suche in Google Scholar PubMed

[11] K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems, Springer Ser. Comput. Phys., Springer, Berlin, 1991. 10.1007/978-3-642-75977-2Suche in Google Scholar

[12] K. K. Sabelfeld, Vector Monte Carlo stochastic matrix-based algorithms for large linear systems, Monte Carlo Methods Appl. 22 (2016), no. 3, 259–264. 10.1515/mcma-2016-0112Suche in Google Scholar

[13] K. K. Sabelfeld, A new randomized vector algorithm for iterative solution of large linear systems, Appl. Math. Lett. 126 (2022), Article ID 107830. 10.1016/j.aml.2021.107830Suche in Google Scholar

[14] K. K. Sabelfeld and O. Bukhasheev, Global random walk on grid algorithm for solving Navier–Stokes and Burgers equations, Monte Carlo Methods Appl. 28 (2022), no. 4, 293–305. 10.1515/mcma-2022-2126Suche in Google Scholar

[15] K. K. Sabelfeld and A. E. Kireeva, Discrete stochastic modeling of electron and hole recombination in 2D and 3D inhomogeneous semiconductors, J. Comput. Electron 16 (2017), no. 2, 325–339. 10.1007/s10825-017-0961-3Suche in Google Scholar

[16] K. K. Sabelfeld and A. Kireeva, Randomized vector iterative linear solvers of high precision for large dense system, Monte Carlo Methods Appl. 29 (2023), no. 4, 323–332. 10.1515/mcma-2023-2013Suche in Google Scholar

[17] K. K. Sabelfeld and N. Loshchina, Stochastic iterative projection methods for large linear systems, Monte Carlo Methods Appl. 16 (2010), no. 3–4, 343–359. 10.1515/mcma.2010.020Suche in Google Scholar

[18] K. K. Sabelfeld and N. S. Mozartova, Sparsified randomization algorithms for low rank approximations and applications to integral equations and inhomogeneous random field simulation, Math. Comput. Simulation 82 (2011), no. 2, 295–317. 10.1016/j.matcom.2011.08.002Suche in Google Scholar

[19] K. K. Sabelfeld and D. Smirnov, A global random walk on grid algorithm for second order elliptic equations, Monte Carlo Methods Appl. 27 (2021), no. 3, 211–225. 10.1515/mcma-2021-2092Suche in Google Scholar

[20] T. Strohmer and R. Vershynin, A randomized Kaczmarz algorithm with exponential convergence, J. Fourier Anal. Appl. 15 (2009), no. 2, 262–278. 10.1007/s00041-008-9030-4Suche in Google Scholar

[21] A. F. Voter, Introduction to the kinetic Monte Carlo method, Radiation Effects in Solids, NATO Sci. Ser., Springer, Berlin (2007), 1–23. 10.1007/978-1-4020-5295-8_1Suche in Google Scholar

[22] A. J. Walker, New fast method for generating discrete random numbers with arbitrary frequency distributions, Electr. Lett. 10 (1974), no. 8, 127–128. 10.1049/el:19740097Suche in Google Scholar

[23] C. C. N. Wang, K.-L. Ng, Y.-C. Chen, P. C. Y. Sheu and J. J. P. Tsai, Simulation of bacterial chemotaxis by the random run and tumble model, 11th IEEE International Conference on Bioinformatics and Bioengineering, IEEE Press, Piscataway (2011), 228–233. 10.1109/BIBE.2011.41Suche in Google Scholar

[24] X. Xiao, X. Feng and Y. He, Numerical simulations for the chemotaxis models on surfaces via a novel characteristic finite element method, Comput. Math. Appl. 78 (2019), no. 1, 20–34. 10.1016/j.camwa.2019.02.004Suche in Google Scholar