Stochastic modeling of unresolved scales in complex systems

Abstract

Model uncertainties or simulation uncertainties occur in mathematical modeling of multiscale complex systems, since some mechanisms or scales are not represented (i.e., ‘unresolved’) due to a lack in our understanding of these mechanisms or limitations in computational power. The impact of these unresolved scales on the resolved scales needs to be parameterized or taken into account. A stochastic scheme is devised to take the effects of unresolved scales into account, in the context of solving nonlinear partial differential equations. An example is presented to demonstrate this strategy.

Keywords

Stochastic partial differential equation (SPDE) / stochastic modeling / impact of unresolved scales on resolved scales / model error / large eddy simulation (LES) / fractional Brownian motion (fBM)

References

[1.] Arnold L. Random Dynamical Systems, 1998, New York: Springer-Verlag.

[2.] Arnold L. von Storch J. -S., Imkeller P. Hasselmann’s program visited: The analysis of stochasticity in deterministic climate models. Stochastic Climate Models, 2001, Boston: Birkhäuser, 141-158.

[3.] Berloff P. S. Random-forcing model of the mesoscale oceanic eddies. J Fluid Mech, 2005, 529: 71-95.

[4.] Berselli L. C., Iliescu T., Layton W. J. Mathematics of Large Eddy Simulation of Turbulent Flows, 2005, Berlin: Springer-Verlag.

[5.] Craigmile P. F. Simulating a class of stationary Gaussian processes using the Davies-Harte algorithm, with application to long memory processes. J Time Series Anal, 2003, 24: 505-511.

[6.] Du A, Duan J. A stochastic approach for parameterizing unresolved scales in a system with memory. Journal of Algorithms & Computational Technology (to appear)

[7.] Duan J., Nadiga B. Stochastic parameterization of large eddy simulation of geophysical flows. Proc American Math Soc, 2007, 135: 1187-1196.

[8.] Duncan T. E., Hu Y. Z., Pasik-Duncan B. Stochastic calculus for fractional Brownian motion. I: Theory. SIAM Journal on Control and Optimization, 2000, 38: 582-612.

[9.] Francfort G. A., Suquet P. M. Homogenization and mechanical dissipation in thermoviscoelasticity. Arch Ratinal Mech Anal, 1986, 96: 879-895.

[10.] Garcia-Ojalvo J., Sancho J. M. Noise in Spatially Extended Systems, 1999, Berlin: Springer.

[11.] Giorgi C., Marzocchi A., Pata V. Asymptotic behavior of a similinear problem in heat conduction with memory. NoDEA Nonl Diff Equa Appl, 1998, 5: 333-354.

[12.] Hasselmann K. Stochastic climate models: Part I. Theory. Tellus, 1976, 28: 473-485.

[13.] Horsthemke W., Lefever R. Noise-Induced Transitions, 1984, Berlin: Springer-Verlag.

[14.] Huisinga W., Schutte C., Stuart A. M. Extracting macroscopic stochastic dynamics: Model problems. Comm. Pure Appl Math, 2003, 56: 234-269.

[15.] Just W., Kantz H., Rodenbeck C., Helm M. Stochastic modelling: replacing fast degrees of freedom by noise. J Phys, A: Math Gen, 2001, 34: 3199-3213.

[16.] Leith C. E. Stochastic backscatter in a subgrid-scale model: Plane shear mixing layer. Phys Fluids A, 1990, 2: 297-299.

[17.] Lin J. W. -B., Neelin J. D. Considerations for stochastic convective parameterization. J Atmos Sci, 2002, 59(5): 959-975.

[18.] Majda A. J., Timofeyev I., Vanden-Eijnden E. Models for stochastic climate prediction. PNAS, 1999, 96: 14687-14691.

[19.] Marchenko V. A., Khruslov E. Y. Homogenization of Partial Differential Equations, 2006, Boston: Birkhäuser.

[20.] Maslowski B, Schmalfuss B. Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion. Stoch Anal Appl (to appear)

[21.] Mason P. J., Thomson D. J. Stochastic backscatter in large-eddy simulations of boundary layers. J Fluid Mech, 1992, 242: 51-78.

[22.] Mehrabi A. R., Rassamdana H., Sahimi M. Characterization of long-range correlation in complex distributions and profiles. Physical Review E, 1997, 56: 712

[23.] Memin J., Mishura Y., Valkeila E. Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Stat & Prob Lett, 2001, 51: 197-206.

[24.] Meneveau C., Katz J. Scale-invariance and turbulence models for large-eddy simulation. Annu Rev Fluid Mech, 2000, 32: 1-32.

[25.] Nualart D. Stochastic calculus with respect to the fractional Brownian motion and applications. Contemporary Mathematics, 2003, 336: 3-39.

[26.] Palmer T. N., Shutts G. J., Hagedorn R., Doblas-Reyes F. J., Jung T., Leutbecher M. Representing model uncertainty in weather and climate prediction. Annu Rev Earth Planet Sci, 2005, 33: 163-193.

[27.] Pasquero C., Tziperman E. Statistical parameterization of heterogeneous oceanic convection. J Phys Oceanography, 2007, 37: 214-229.

[28.] Penland C, Sura P. Sensitivity of an ocean model to “details” of stochastic forcing. In: Proc ECMWF Workshop on Representation of Subscale Processes using Stochastic-Dynamic Models. Reading, England, 6–8 June 2005

[29.] Peters H., Jones W. E. Bottom layer turbulence in the red sea outflow plume. J Phys Oceanography, 2006, 36: 1763-1785.

[30.] Peters H, Lee C M, Orlic M, Dorman C E. Turbulence in the wintertime northern Adriatic sea under strong atmospheric forcing. J Geophys Res (to appear)

[31.] Pipiras V., Taqqu M. S. Convergence of the Weierstrass-Mandelbrot process to fractional Brownian motion. Fractals, 2000, 8(4): 369-384.

[32.] Rozovskii B. L. Stochastic Evolution Equations, 1990, Boston: Kluwer Acad Publishers.

[33.] Sagaut P. Large Eddy Simulation for Incompressible Flows, 2005 3 New York: Springer.

[34.] Sardeshmukh P. Issues in stochastic parameterisation. In: Proc ECMWF Workshop on Representation of Subscale Processes using Stochastic-Dynamic Models. Reading, England, 6–8 June 2005

[35.] Schumann U. Stochastic backscatter of turbulent energy and scalar variance by random subgrid-scale fluxes. Proc R Soc Lond A, 1995, 451: 293-318.

[36.] Tarantola A. Inverse Problem Theory and Methods for Model Parameter Estimation, 2004, Philadelphia: SIAM.

[37.] Tindel S., Tudor C. A., Viens F. Stochastic evolution equations with fractional Brownian motion. Probability Theory and Related Fields, 2003, 127(2): 186-204.

[38.] Trefethen L. N. Spectral Methods in Matlab, 2000, Philadelphia: SIAM.

[39.] Waymire E., Duan J. Probability and Partial Differential Equations in Modern Applied Mathematics, 2005, New York: Springer-Verlag.

[40.] Wilks D. S. Effects of stochastic parameterizations in the Lorenz’ 96 system. Q J R Meteorol Soc, 2005, 131: 389-407.

[41.] Williams P. D. Modelling climate change: the role of unresolved processes. Phil Trans R Soc A, 2005, 363: 2931-2946.

[42.] Wu J. Theory and Applications of Partial Functional Differential Equations, 1996, New York: Springer.