Stochastic modeling of unresolved scales in complex systems

doi:10.1007/s11464-009-0027-3

Front Math Chin

2009, Vol. 4

Issue (3) : 425-436 https://doi.org/10.1007/s11464-009-0027-3

RESEARCH ARTICLE

Stochastic modeling of unresolved scales in complex systems

Jinqiao DUAN^1,2(

)

1. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA; 2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

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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)

Corresponding Author(s): DUAN Jinqiao,Email:duan@iit.edu

Issue Date: 05 September 2009

Cite this article:

Jinqiao DUAN. Stochastic modeling of unresolved scales in complex systems[J]. Front Math Chin, 2009, 4(3): 425-436.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-009-0027-3
https://academic.hep.com.cn/fmc/EN/Y2009/V4/I3/425

1	Arnold L. Random Dynamical Systems. New York: Springer-Verlag, 1998
2	Arnold L. Hasselmann’s program visited: The analysis of stochasticity in deterministic climate models. In: von Storch J-S, Imkeller P, eds. Stochastic Climate Models . Boston: Birkh?user, 2001, 141-158
3	Berloff P S. Random-forcing model of the mesoscale oceanic eddies. J Fluid Mech , 2005, 529: 71-95 doi: 10.1017/S0022112005003393
4	Berselli L C, Iliescu T, Layton W J. Mathematics of Large Eddy Simulation of Turbulent Flows. Berlin: Springer-Verlag , 2005
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 doi: 10.1111/1467-9892.00318
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 ?ows. Proc American Math Soc , 2007, 135: 1187-1196 doi: 10.1090/S0002-9939-06-08631-X
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 doi: 10.1137/S036301299834171X
9	Francfort G A, Suquet P M. Homogenization and mechanical dissipation in thermoviscoelasticity. Arch Ratinal Mech Anal , 1986, 96: 879-895 doi: 10.1007/BF00251909
10	Garcia-Ojalvo J, Sancho J M. Noise in Spatially Extended Systems. Berlin: Springer, 1999
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 doi: 10.1007/s000300050049
12	Hasselmann K. Stochastic climate models: Part I. Theory. Tellus , 1976, 28: 473-485
13	Horsthemke W, Lefever R. Noise-Induced Transitions. Berlin: Springer-Verlag, 1984
14	Huisinga W, Schutte C, Stuart A M. Extracting macroscopic stochastic dynamics: Model problems. Comm. Pure Appl Math , 2003, 56: 234-269 doi: 10.1002/cpa.10057
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 doi: 10.1088/0305-4470/34/15/302
16	Leith C E. Stochastic backscatter in a subgrid-scale model: Plane shear mixing layer. Phys Fluids A , 1990, 2: 297-299 doi: 10.1063/1.857779
17	Lin J W-B, Neelin J D. Considerations for stochastic convective parameterization. J Atmos Sci , 2002, 59(5): 959-975 doi: 10.1175/1520-0469(2002)059<0959:CFSCP>2.0.CO;2
18	Majda A J, Timofeyev I, Vanden-Eijnden E. Models for stochastic climate prediction. PNAS , 1999, 96: 14687-14691 doi: 10.1073/pnas.96.26.14687
19	Marchenko V A, Khruslov E Y. Homogenization of Partial Differential Equations. Boston: Birkh?user , 2006
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 doi: 10.1017/S0022112092002271
22	Mehrabi A R, Rassamdana H, Sahimi M. Characterization of long-range correlation in complex distributions and pro?les. Physical Review E , 1997, 56: 712 doi: 10.1103/PhysRevE.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 doi: 10.1016/S0167-7152(00)00157-7
24	Meneveau C, Katz J. Scale-invariance and turbulence models for large-eddy simulation. Annu Rev Fluid Mech , 2000, 32: 1-32 doi: 10.1146/annurev.fluid.32.1.1
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 doi: 10.1146/annurev.earth.33.092203.122552
27	Pasquero C, Tziperman E. Statistical parameterization of heterogeneous oceanic convection. J Phys Oceanography , 2007, 37: 214-229 doi: 10.1175/JPO3008.1
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 June2005
29	Peters H, Jones W E. Bottom layer turbulence in the red sea out?ow plume. J Phys Oceanography , 2006, 36: 1763-1785 doi: 10.1175/JPO2939.1
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 doi: 10.1142/S0218348X00000408
32	Rozovskii B L. Stochastic Evolution Equations. Boston: Kluwer Acad Publishers, 1990
33	Sagaut P. Large Eddy Simulation for Incompressible Flows. 3rd Ed. New York: Springer, 2005
34	Sardeshmukh P. Issues in stochastic parameterisation. In: Proc ECMWFWorkshop on Representation of Subscale Processes using Stochastic-Dynamic Models. Reading , England, 6-8 June2005
35	Schumann U. Stochastic backscatter of turbulent energy and scalar variance by random subgrid-scale ?uxes. Proc R Soc Lond A , 1995, 451: 293-318 doi: 10.1098/rspa.1995.0126
36	Tarantola A. Inverse Problem Theory and Methods for Model Parameter Estimation. Philadelphia: SIAM , 2004
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. Philadelphia: SIAM , 2000
39	Waymire E, Duan J, eds. Probability and Partial Differential Equations in Modern Applied Mathematics. New York: Springer-Verlag , 2005
40	Wilks D S. Effects of stochastic parameterizations in the Lorenz ’96 system. Q J R Meteorol Soc , 2005, 131: 389-407 doi: 10.1256/qj.04.03
41	Williams P D. Modelling climate change: the role of unresolved processes. Phil Trans R Soc A , 2005363: 2931-2946
42	Wu J. Theory and Applications of Partial Functional Differential Equations. New York: Springer, 1996

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