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

Abstract

top
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.

How to cite

top

Frederix, 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
  1. Y. Ait-Sahalia, Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica70 (2002) 223–262.  
  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. E73 (2006) 051901.  
  3. W. E and B. Engquist, The heterogeneous multi-scale methods. Commun. Math. Sci.1 (2003) 87–132.  
  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.  
  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.  
  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.  
  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.  
  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 Engineering75, Springer-Verlag (2010).  
  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. B11 (2009) 855–874.  
  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.  
  12. D. Givon, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity17 (2004) R55–R127.  
  13. R.M. Gray, Toeplitz and circulant matrices: A review. Found. Trends Commun. Inf. Theory2 (2005) 155–239.  
  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.  
  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.  
  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 Mathematics53. Springer, New York (2007).  
  19. G.A. Pavliotis and A.M. Stuart, Parameter estimation for multiscale diffusions. J. Stat. Phys.127 (2007) 741–781.  
  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.  
  21. H. Risken, The Fokker-Planck Equation: Methods of Solutions and Applications. Springer Series in Synergetics, Second Edition, Springer (1989).  
  22. M. Rousset and G. Samaey, Simulating individual-based models of bacterial chemotaxis with asymptotic variance reduction. INRIA, inria-00425065, available at (2009).  URIhttp://hal.inria.fr/inria-00425065/fr/
  23. A. Skorokhod, Asymptotic methods in the theory of stochastic differential equations, Translations of mathematical monographs78. AMS, Providence (1999).  
  24. N. Van Kampen, Elimination of fast variables. Phys. Rep.124 (1985) 69–160.  
  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.  
  26. E. Vanden-Eijnden, Numerical techniques for multi-scale dynamical systems with stochastic effects. Commun. Math. Sci.1 (2003) 385–391.  

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.