An analysis of noise propagation in the multiscale simulation of coarse Fokker-Planck equations

Frederix, Yves; Samaey, Giovanni; Roose, Dirk

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 3, pp. 541-561.

We consider multiscale systems for which only a fine-scale model describing the evolution of individuals (atoms, molecules, bacteria, agents) is given, while we are interested in the evolution of the population density on coarse space and time scales. Typically, this evolution is described by a coarse Fokker-Planck equation. In this paper, we consider a numerical procedure to compute the solution of this Fokker-Planck equation directly on the coarse level, based on the estimation of the unknown parameters (drift and diffusion) using only appropriately chosen realizations of the fine-scale, individual-based system. As these parameters might be space- and time-dependent, the estimation is performed in every spatial discretization point and at every time step. If the fine-scale model is stochastic, the estimation procedure introduces noise on the coarse level. We investigate stability conditions for this procedure in the presence of this noise and present an analysis of the propagation of the estimation error in the numerical solution of the coarse Fokker-Planck equation.

[1] Y. Ait-Sahalia, Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70 (2002) 223-262. | MR | Zbl

[2] M. Alber, N. Chen, T. Glimm and P.M. Lushnikov, Multiscale dynamics of biological cells with chemotactic interactions: From a discrete stochastic model to a continuous description. Phys. Rev. E 73 (2006) 051901. | MR

[3] W. E and B. Engquist, The heterogeneous multi-scale methods. Commun. Math. Sci. 1 (2003) 87-132. | Zbl

[4] W. E, D. Liu and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations. Commun. Pure Appl. Math. 58 (2005) 1544-1585. | MR | Zbl

[5] W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden, Heterogeneous multiscale methods: A review. Commun. Comput. Phys. 2 (2007) 367-450. | MR | Zbl

[6] R. Erban and H.G. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology. SIAM Multiscale Model. Simul. 3 (2005) 362-394. | MR | Zbl

[7] I. Fatkullin and E. Vanden-Eijnden, A computational strategy for multiscale systems with applications to Lorenz 96 model. J. Comput. Phys. 200 (2004) 605-638. | MR | Zbl

[8] Y. Frederix and D. Roose, A drift-filtered approach to diffusion estimation for multiscale processes, in Coping with complexity: model reduction and data analysis, Lecture Notes in Computational Science and Engineering 75, Springer-Verlag (2010). | MR

[9] Y. Frederix, G. Samaey, C. Vandekerckhove, T. Li, E. Nies and D. Roose, Lifting in equation-free methods for molecular dynamics simulations of dense fluids. Discrete Continuous Dyn. Syst. Ser. B 11 (2009) 855-874. | MR

[10] C. Gear, Projective integration methods for distributions. Technical report, NEC Research Institute (2001).

[11] C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes. SIAM J. Appl. Dyn. Syst. 4 (2005) 711-732. | MR | Zbl

[12] D. Givon, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17 (2004) R55-R127. | MR | Zbl

[13] R.M. Gray, Toeplitz and circulant matrices: A review. Found. Trends Commun. Inf. Theory 2 (2005) 155-239. | Zbl

[14] B. Jourdain, C.L. Bris and T. Lelièvre, On a variance reduction technique for micro-macro simulations of polymeric fluids. J. Non-Newton. Fluid Mech. 122 (2004) 91-106. | Zbl

[15] I.G. Kevrekidis and G. Samaey, Equation-free multiscale computation: Algorithms and applications. Ann. Rev. Phys. Chem. 60 (2009) 321-344.

[16] I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis. Commun. Math. Sci. 1 (2003) 715-762. | MR | Zbl

[17] H.C. Öttinger, B.H.A.A. Van Den Brule and M.A. Hulsen, Brownian configuration fields and variance reduced CONNFFESSIT. J. Non-Newton. Fluid Mech. 70 (1997) 255-261.

[18] G. Pavliotis and A. Stuart, Multiscale Methods: Averaging and Homogenization, Texts in Applied Mathematics 53. Springer, New York (2007). | MR | Zbl

[19] G.A. Pavliotis and A.M. Stuart, Parameter estimation for multiscale diffusions. J. Stat. Phys. 127 (2007) 741-781. | MR | Zbl

[20] Y. Pokern, A.M. Stuart and E. Vanden-Eijnden, Remarks on drift estimation for diffusion processes. SIAM Multiscale Model. Simul. 8 (2009) 69-95. | MR | Zbl

[21] H. Risken, The Fokker-Planck Equation: Methods of Solutions and Applications. Springer Series in Synergetics, Second Edition, Springer (1989). | MR | Zbl

[22] M. Rousset and G. Samaey, Simulating individual-based models of bacterial chemotaxis with asymptotic variance reduction. INRIA, inria-00425065, available at http://hal.inria.fr/inria-00425065/fr/ (2009). | Zbl

[23] A. Skorokhod, Asymptotic methods in the theory of stochastic differential equations, Translations of mathematical monographs 78. AMS, Providence (1999). | Zbl

[24] N. Van Kampen, Elimination of fast variables. Phys. Rep. 124 (1985) 69-160. | MR

[25] P. Van Leemput, W. Vanroose and D. Roose, Mesoscale analysis of the equation-free constrained runs initialization scheme. SIAM Multiscale Model. Simul. 6 (2007) 1234-1255. | MR | Zbl

[26] E. Vanden-Eijnden, Numerical techniques for multi-scale dynamical systems with stochastic effects. Commun. Math. Sci. 1 (2003) 385-391. | MR | Zbl