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

Yves Frederix; Giovanni Samaey; Dirk Roose

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- Volume: 45, Issue: 3, page 541-561
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topFrederix, Yves, Samaey, Giovanni, and Roose, Dirk. "An analysis of noise propagation in the multiscale simulation of coarse Fokker-Planck equations." ESAIM: Mathematical Modelling and Numerical Analysis 45.3 (2011): 541-561. <http://eudml.org/doc/197481>.

@article{Frederix2011,

abstract = {
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.
},

author = {Frederix, Yves, Samaey, Giovanni, Roose, Dirk},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Multiscale computing; stochastic systems; Fokker-Planck
equation; uncertainty propagation; multiscale computing; Fokker-Planck equation},

language = {eng},

month = {1},

number = {3},

pages = {541-561},

publisher = {EDP Sciences},

title = {An analysis of noise propagation in the multiscale simulation of coarse Fokker-Planck equations},

url = {http://eudml.org/doc/197481},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Frederix, Yves

AU - Samaey, Giovanni

AU - Roose, Dirk

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2011/1//

PB - EDP Sciences

VL - 45

IS - 3

SP - 541

EP - 561

AB -
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.

LA - eng

KW - Multiscale computing; stochastic systems; Fokker-Planck
equation; uncertainty propagation; multiscale computing; Fokker-Planck equation

UR - http://eudml.org/doc/197481

ER -

## References

top- Y. Ait-Sahalia, Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica70 (2002) 223–262. Zbl1104.62323
- 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. E73 (2006) 051901.
- W. E and B. Engquist, The heterogeneous multi-scale methods. Commun. Math. Sci.1 (2003) 87–132. Zbl1093.35012
- W. E, D. Liu and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations. Commun. Pure Appl. Math.58 (2005) 1544–1585. Zbl1080.60060
- W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden, Heterogeneous multiscale methods: A review. Commun. Comput. Phys.2 (2007) 367–450. Zbl1164.65496
- 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. Zbl1073.35205
- 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. Zbl1058.65065
- 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 Engineering75, Springer-Verlag (2010).
- 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. B11 (2009) 855–874. Zbl05574125
- C. Gear, Projective integration methods for distributions. Technical report, NEC Research Institute (2001).
- 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. Zbl1170.34343
- D. Givon, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity17 (2004) R55–R127. Zbl1073.82038
- R.M. Gray, Toeplitz and circulant matrices: A review. Found. Trends Commun. Inf. Theory2 (2005) 155–239.
- 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. Zbl1143.76333
- I.G. Kevrekidis and G. Samaey, Equation-free multiscale computation: Algorithms and applications. Ann. Rev. Phys. Chem.60 (2009) 321–344.
- 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. Zbl1086.65066
- 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.
- G. Pavliotis and A. Stuart, Multiscale Methods: Averaging and Homogenization, Texts in Applied Mathematics53. Springer, New York (2007). Zbl1160.35006
- G.A. Pavliotis and A.M. Stuart, Parameter estimation for multiscale diffusions. J. Stat. Phys.127 (2007) 741–781. Zbl1137.82016
- Y. Pokern, A.M. Stuart and E. Vanden-Eijnden, Remarks on drift estimation for diffusion processes. SIAM Multiscale Model. Simul.8 (2009) 69–95. Zbl1183.62145
- H. Risken, The Fokker-Planck Equation: Methods of Solutions and Applications. Springer Series in Synergetics, Second Edition, Springer (1989). Zbl0665.60084
- M. Rousset and G. Samaey, Simulating individual-based models of bacterial chemotaxis with asymptotic variance reduction. INRIA, inria-00425065, available at (2009). Zbl1291.35417URIhttp://hal.inria.fr/inria-00425065/fr/
- A. Skorokhod, Asymptotic methods in the theory of stochastic differential equations, Translations of mathematical monographs78. AMS, Providence (1999).
- N. Van Kampen, Elimination of fast variables. Phys. Rep.124 (1985) 69–160.
- 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. Zbl1248.76121
- E. Vanden-Eijnden, Numerical techniques for multi-scale dynamical systems with stochastic effects. Commun. Math. Sci.1 (2003) 385–391. Zbl1088.60060

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.