Lessons in uncertainty quantification for turbulent dynamical systems

Andrew J. Majda; Michal Branicki

doi:10.3934/dcds.2012.32.3133

Article Contents

2012, Volume 32, Issue 9: 3133-3221. Doi: 10.3934/dcds.2012.32.3133

This issue Previous Article Dynamics of a three species competition model Next Article Adaptation of an ecological territorial model to street gang spatial patterns in Los Angeles

Lessons in uncertainty quantification for turbulent dynamical systems

Andrew J. Majda^1, and
Michal Branicki^1,

1.
Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute of Mathematical Sciences, New York University

Received: December 31, 2011

Revised: February 29, 2012

Published: August 2012

Abstract Related Papers Cited by

Abstract

The modus operandi of modern applied mathematics in developing very recent mathematical strategies for uncertainty quantification in partially observed high-dimensional turbulent dynamical systems is emphasized here. The approach involves the synergy of rigorous mathematical guidelines with a suite of physically relevant and progressively more complex test models which are mathematically tractable while possessing such important features as the two-way coupling between the resolved dynamics and the turbulent fluxes, intermittency and positive Lyapunov exponents, eddy diffusivity parameterization and turbulent spectra. A large number of new theoretical and computational phenomena which arise in the emerging statistical-stochastic framework for quantifying and mitigating model error in imperfect predictions, such as the existence of information barriers to model improvement, are developed and reviewed here with the intention to introduce mathematicians, applied mathematicians, and scientists to these remarkable emerging topics with increasing practical importance.

Keywords:

Mathematics Subject Classification: Primary: 94A15, 60H30, 35Q86, 35Q93; Secondary: 35Q94, 62B10, 35Q84, 60H15.

Citation:

Related Papers

Cited by

References

[1]	R. V. Abramov and A. J. Majda, Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems, Nonlinearity, 20 (2007), 2793-2821.doi: 10.1088/0951-7715/20/12/004.
[2]	R. V. Abramov, A. J. Majda and R. Kleeman, Information theory and predictability for low-frequency variability, J. Atmos. Sci., 62 (2005), 65-87.doi: 10.1175/JAS-3373.1.
[3]	A. Alexanderian, O. Le Maitre, H. Najm, M. Iskandarani and O. Knio, Multiscale Stochastic Preconditioners in Non-intrusive Spectral Projection, Journal of Scientific Computing, (2011), 1-35.
[4]	L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
[5]	M. Avellaneda and A. J. Majda, Approximate and exact renormalization theories for a model for turbulent transport, Phys. Fluids A, 4 (1992), 41-57.doi: 10.1063/1.858499.
[6]	M. Avellaneda and A. J. Majda, Renormalization theory for eddy diffusivity in turbulent transport, Phys. Rev. Lett., 68 (1992), 3028-3031.doi: 10.1103/PhysRevLett.68.3028.
[7]	M. Avellaneda and A. J. Majda, Simple examples with features of renormalization for turbulent transport, Phil. Trans. R. Soc. Lond. A, 346 (1994), 205-233.doi: 10.1098/rsta.1994.0019.
[8]	A. Bourlioux and A. J. Majda, An elementary model for the validation of flamelet approximations in non-premixed turbulent combustion, Combust. Theory Model., 4 (2000), 189-210.doi: 10.1088/1364-7830/4/2/307.
[9]	M. Branicki, B. Gershgorin and A. J. Majda, Filtering skill for turbulent signals for a suite of nonlinear and linear kalman filters, J. Comp. Phys, 231 (2012), 1462-1498.doi: 10.1016/j.jcp.2011.10.029.
[10]	M. Branicki and A. J. Majda, Quantifying uncertainty for long range forecasting scenarios with model errors in non-Guassian models with intermittency, Nonlinearity, submitted, 2012.
[11]	G. Branstator and H. Teng, Two limits of initial-value decadal predictability in a CGCM, J. Climate, 23 (2010), 6292-6311.doi: 10.1175/2010JCLI3678.1.
[12]	R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Ann. Math. (2), 48 (1947), 385-392.doi: 10.2307/1969178.
[13]	M. D. Chekroun, D. Kondrashov and M. Ghil, Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation, PNAS, 108 (2011), 11766-11771.doi: 10.1073/pnas.1015753108.
[14]	A. J. Chorin, Gaussian fields and random flow, J. Fluid Mech., 63 (1974), 21-32.doi: 10.1017/S0022112074000991.
[15]	S. C. Crow and G. H. Canavan, Relationship between a Wiener-Hermite expansion and an energy cascade, J. Fluid Mech., 41 (1970), 387-403.doi: 10.1017/S0022112070000654.
[16]	T. DelSole, Predictability and information theory. I: Measures of predictability, J. Atmos. Sci., 61 (2004), 2425-2440.doi: 10.1175/1520-0469(2004)061<2425:PAITPI>2.0.CO;2.
[17]	T. DelSole, Stochastic models of quasigeostrophic turbulence, Surveys in Geophys., 25 (2004), 107-149.
[18]	T. DelSole, Predictability and information theory. II: Imperfect models, J. Atmos. Sci., 62 (2005), 3368-3381.doi: 10.1175/JAS3522.1.
[19]	T. DelSole and J. Shukla, Model fidelity versus skill in seasonal forecasting, J. Climate, 23 (2010), 4794-4806.doi: 10.1175/2010JCLI3164.1.
[20]	K. A. Emanuel, J. C. Wyngaard, J. C. McWilliams, D. A. Randall and Y. L. Yung, "Improving the Scientific Foundation for Atmosphere-Land Ocean Simulations," Natl. Acad. Press, Washington DC, 2005.
[21]	E. S. Epstein, Stochastic dynamic predictions, Tellus, 21 (1969), 739-759.doi: 10.1111/j.2153-3490.1969.tb00483.x.
[22]	C. Franzke, D. Crommelin, A. Fischer and A. J. Majda, A hidden Markov model perspective on regimes and metastability in atmospheric flows, J. Climate, 21 (2008), 1740-1757.doi: 10.1175/2007JCLI1751.1.
[23]	C. Franzke, I. Horenko, A. J. Majda and R. Klein, Systematic metastable atmospheric regime identification in an AGCM, J. Atmos. Sci., 66 (2009), 1997-2012.doi: 10.1175/2009JAS2939.1.
[24]	P. Frauenfelder, C. Schwab and R. A. Todor, Finite elements for elliptic problems with stochastic coefficients, Comput. Methods Appl. Mech. Eng., 194 (2005), 205-228.doi: 10.1016/j.cma.2004.04.008.
[25]	D. Frierson, Robust increases in midlatitude static stability in global warming simulations, Geophys. Res. Lett., 33 (2006), L24816.doi: 10.1029/2006GL027504.
[26]	D. Frierson, Midlatitude static stability in simple and comprehensive general circulation models, J. Atmos. Sci., 65 (2008), 1049-1062.doi: 10.1175/2007JAS2373.1.
[27]	C. Gardiner, "Stochastic Methods: A Handbook for the Natural and Social Sciences," Fourth edition, Springer Series in Synergetics, Springer-Verlag, Berlin, 2009.
[28]	B. Gershgorin, J. Harlim and A. J. Majda, Improving filtering and prediction of spatially extended turbulent systems with model errors through stochastic parameter estimation, J. Comp. Phys., 229 (2010), 32-57.doi: 10.1016/j.jcp.2009.09.022.
[29]	B. Gershgorin, J. Harlim and A. J. Majda, Test models for improving filtering with model errors through stochastic parameter estimation, J. Comp. Phys., 229 (2010), 1-31.doi: 10.1016/j.jcp.2009.08.019.
[30]	B. Gershgorin and A. J. Majda, A test model for fluctuation-dissipation theorems with time-periodic statistics, Physica D, 239 (2010), 1741-1757.doi: 10.1016/j.physd.2010.05.009.
[31]	B. Gershgorin and A. J. Majda, Quantifying uncertainty for climate change and long range forecasting scenarios with model errors. I: Gaussian models, J. Climate, in press, 2012.
[32]	B. Gershgorin and A. J. Majda, A nonlinear test model for filtering slow-fast systems, Comm. Math. Sci., 6 (2008), 611-650.
[33]	B. Gershgorin and A. J. Majda, Filtering a nonlinear slow-fast system with strong fast forcing, Comm. Math. Sci., 8 (2010), 67-92.
[34]	B. Gershgorin and A. J. Majda, Filtering a statistically exactly solvable test model for turbulent tracers from partial observations, J. Comp. Phys., 230 (2011), 1602-1638.doi: 10.1016/j.jcp.2010.11.024.
[35]	R. G. Ghanem and P. D. Spanos, "Stochastic Finite Elements: A Spectral Approach," Springer-Verlag, New York, 1991.doi: 10.1007/978-1-4612-3094-6.
[36]	D. Giannakis and A. J. Majda, Quantifying the predictive skill in long-range forecasting. I: Coarse-grained predictions in a simple ocean model, J. Climate, in press, 2011.doi: 10.1175/2011JCLI4143.1.
[37]	D. Giannakis and A. J. Majda, Quantifying the predictive skill in long-range forecasting. II: Model error in coarse-grained Markov models with application to ocean-circulation regimes, J. Climate, in press, 2011.doi: 10.1175/JCLI-D-11-00110.1.
[38]	D. Giannakis, A. J. Majda and I. Horenko, Information theory, model error, and predictive skill of stochastic models for complex nonlinear systems, Physica D, submitted, 2011.
[39]	J. P. Gollub, J. Clarke, M. Gharib, B. Lane and O. N. Mesquita, Fluctuations and transport in a stirred fluid with a mean gradient, Phys. Rev. Lett., 67 (1991), 3507-3510.doi: 10.1103/PhysRevLett.67.3507.
[40]	M. Hairer and A. J. Majda, A simple framework to justify linear response theory, Nonlinearity, 23 (2010), 909-922.doi: 10.1088/0951-7715/23/4/008.
[41]	J. Harlim and A. J. Majda, Filtering turbulent sparsely observed geophysical flows, Mon. Wea. Rev., 138 (2010), 1050-1083.doi: 10.1175/2009MWR3113.1.
[42]	H. G. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Comput. Methods Appl. Mech. Eng., 194 (2005), 1295-1331.doi: 10.1016/j.cma.2004.05.027.
[43]	I. Horenko, On clustering of non-stationary meteorological time series, Dyn. Atmos. Oceans, 49 (2010), 164-187.doi: 10.1016/j.dynatmoce.2009.04.003.
[44]	I. Horenko, On robust estimation of low-frequency variability trends in discrete Markovian sequences of atmospheric circulation patterns, J. Atmos. Sci., 66 (2009), 2059-2072.doi: 10.1175/2008JAS2959.1.
[45]	I. Horenko, Finite element approach to clustering of multidimensional time series, SIAM J. Sci. Comput., 32 (2010), 62-83.doi: 10.1137/080715962.
[46]	I. Horenko, On the identification of nonstationary factor models and their application to atmospheric data analysis, J. Atmos. Sci., 67 (2010), 1559-1574.doi: 10.1175/2010JAS3271.1.
[47]	I. Horenko, Nonstationarity in multifactor models of discrete jump processes, memory and application to cloud modeling, J. Atmos. Sci., 2011.doi: 10.1175/2011JAS3692.1.
[48]	T. Hou, W. Luo, B. Rozovskii and H.-M. Zhou, Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics, J. Comp. Phys., 216 (2006), 687-706.doi: 10.1016/j.jcp.2006.01.008.
[49]	Jayesh and Z. Warhaft, Probability distribution of a passive scalar in grid-generated turbulence, Phys. Rev. Lett., 67 (1991), 3503-3506.doi: 10.1103/PhysRevLett.67.3503.
[50]	Jayesh and Z. Warhaft, Probability distribution, conditional dissipation, and transport of passive temperature fluctuations in grid-generated turbulence, Phys. Fluids A, 4 (1992), 2292.doi: 10.1063/1.858469.
[51]	A. H. Jazwinski, "Stochastic Processes and Filtering Theory," Academic Press, New York, 1970.
[52]	M. I. Jordan, Graphical models, Statistical Science, 19 (2004), 140-155.doi: 10.1214/088342304000000026.
[53]	M. A. Katsoulakis, A. J. Majda and A. Sopasakis, Intermittency, metastability and coarse graining for coupled deterministic-stochastic lattice systems, Nonlinearity, 19 (2006), 1021-1047.doi: 10.1088/0951-7715/19/5/002.
[54]	S. R. Keating, A. J. Majda and K. S. Smith, New methods for estimating poleward eddy heat transport using satellite altimetry, Mon. Wea. Rev., in press, 2011.
[55]	R. Kleeman, Measuring dynamical prediction utility using relative entropy, J. Atmos. Sci., 59 (2002), 2057-2072.doi: 10.1175/1520-0469(2002)059<2057:MDPUUR>2.0.CO;2.
[56]	P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992.
[57]	R. H. Kraichnan, Small-scale structure of a scalar field convected by turbulence, Phys. Fluids, 11 (1968), 945-953.doi: 10.1063/1.1692063.
[58]	R. H. Kraichnan, Eddy viscosity and diffusivity: Exact formulas and approximations, Complex Systems, 1 (1987), 805-820.
[59]	R. H. Kraichnan, Anomalous scaling of a randomly advected passive scalar, Phys. Rev. Lett., 72 (1994), 1016-1019.doi: 10.1103/PhysRevLett.72.1016.
[60]	R. H. Kraichnan, V. Yakhot and S. Chen, Scaling relations for a randomly advected passive scalar field, Phys. Rev. Lett., 75 (1995), 240-243.doi: 10.1103/PhysRevLett.75.240.
[61]	S. Kullback and R. Leibler, On information and sufficiency, Ann. Math. Stat., 22 (1951), 79-86.doi: 10.1214/aoms/1177729694.
[62]	O. P. Le Maître, O. Knio, H. Najm and R. Ghanem, A stochastic projection method for fluid flow. I. Basic formulation, J. Comput. Phys., 173 (2001), 481-511.doi: 10.1006/jcph.2001.6889.
[63]	O. P. Le Maître, L. Mathelin, O. M. Knio and M. Y. Hussaini, Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics, DCDS, 28 (2010), 199-226.doi: 10.3934/dcds.2010.28.199.
[64]	C. E. Leith, Climate response and fluctuation dissipation, J. Atmospheric Sci., 32 (1975), 2022-2025.doi: 10.1175/1520-0469(1975)032<2022:CRAFD>2.0.CO;2.
[65]	E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130-141.doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
[66]	E. N. Lorenz, A study of predictability of a 28-variable atmospheric model, Tellus, 17 (1968), 321-333.doi: 10.1111/j.2153-3490.1965.tb01424.x.
[67]	E. N. Lorenz, The predictability of a flow which possesses many scales of motion, Tellus, 21 (1969), 289-307.doi: 10.1111/j.2153-3490.1969.tb00444.x.
[68]	W. Luo, "Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations," Ph.D thesis, Caltech, 2006.
[69]	W. Luo, "Wiener Chaos Expansion and Numerical Solutions of Stochastic PDE," VDM Verlag, 2010.
[70]	A. Majda and P. Kramer, Simplified models for turbulent diffusion: Theory, numerical modeling, and physical phenomena, Phys. Reports, 314 (1999), 237-574.doi: 10.1016/S0370-1573(98)00083-0.
[71]	A. J. Majda, Real world turbulence and modern applied mathematics, in "Mathematics: Frontiers and Perspectives," American Math. Society, Providence, RI, (2000), 137-151.
[72]	A. J. Majda, Challenges in Climate Science and Contemporary Applied Mathematics, Comm. Pure Appl. Math, in press, 2011.
[73]	A. J. Majda, R. Abramov and B. Gershgorin, High skill in low frequency climate response through fluctuation dissipation theorems despite structural instability, Proc. Natl. Acad. Sci. USA, 107 (2010), 581-586.doi: 10.1073/pnas.0912997107.
[74]	A. J. Majda, R. V. Abramov and M. J. Grote, "Information Theory and Stochastics for Multiscale Nonlinear Systems," CRM Monograph Series, 25, Americal Mathematical Society, Providence, RI, 2005.
[75]	A. J. Majda, C. Franzke and D. Crommelin, Normal forms for reduced stochastic climate models, Proc. Natl. Acad. Sci. USA, 106 (2009), 3649-3653.doi: 10.1073/pnas.0900173106.
[76]	A. J. Majda, C. Franzke, A. Fischer and D. T. Crommelin, Distinct metastable atmospheric regimes despite nearly Gaussian statistics: A paradigm model, Proc. Natl. Acad. Sci. USA, 103 (2006), 8309-8314.doi: 10.1073/pnas.0602641103.
[77]	A. J. Majda, C. Franzke and B. Khouider, An applied mathematics perspective on stochastic modelling for climate, Phil. Trans. R Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), 2429-2455.
[78]	A. J. Majda and B. Gershgorin, Quantifying uncertainty in climage change science through empirical information theory, Proc. Natl. Acad. Sci., 107 (2010), 14958-14963.doi: 10.1073/pnas.1007009107.
[79]	A. J. Majda and B. Gershgorin, Improving model fidelity and sensitivity for complex systems through empirical information theory, Proc. Natl. Acad. Sci. USA, 108 (2011), 10044-10049.doi: 10.1073/pnas.1105174108.
[80]	A. J. Majda and B. Gershgorin, Link between statistical equilibrium fidelity and forecasting skill for complex systems with model error, Proc. Natl. Acad. Sci. USA, 108 (2011), 12599-12604.doi: 10.1073/pnas.1108132108.
[81]	A. J. Majda, B. Gershgorin and Y. Yuan, Low-frequency climate response and fluctuation-Dissipation theorems: Theory and practice, J. Atmos. Sci., 67 (2010), 1186-1201.doi: 10.1175/2009JAS3264.1.
[82]	A. J. Majda and J. Harlim, Physics constrained nonlinear regression models for time series, Nonlinearity, submitted, 2012.
[83]	A. J. Majda, R. Kleeman and D. Cai, A mathematical framework for predictability through relative entropy, Methods Appl. Anal., 9 (2002), 425-444.
[84]	A. J. Majda, I. I. Timofeyev and E. Vanden Eijnden, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math., 54 (2001), 891-974.doi: 10.1002/cpa.1014.
[85]	A. J. Majda, I. I. Timofeyev and E. Vanden Eijnden, Systematic strategies for stochastic mode reduction in climate, J. Atmos. Sci., 60 (2003), 1705-1722.doi: 10.1175/1520-0469(2003)060<1705:SSFSMR>2.0.CO;2.
[86]	A. J. Majda and X. Wang, "Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows," Cambridge University Press, Cambridge, 2006.doi: 10.1017/CBO9780511616778.
[87]	A. J. Majda and X. Wang, Linear response theory for statistical ensembles in complex systems with time-periodic forcing, Comm. Math. Sci., 8 (2010), 145-172.
[88]	A. J. Majda and Y. Yuan, Fundamental limitations of Ad hoc linear and quadratic multi-level regression models for physical systems, Discrete Cont. Dyn. Systems, 17 (2010), 1333-1363.doi: 10.3934/dcdsb.2012.17.1333.
[89]	A. J. Majda, Explicit inertial range renormalization theory in a model for turbulent diffusion, J. Statist. Phys., 73 (1993), 515-542.doi: 10.1007/BF01054338.
[90]	A. J. Majda, Random shearing direction models for isotropic turbulent diffusion, J. Statist. Phys., 25 (1994), 1153-1165.doi: 10.1007/BF02186761.
[91]	A. J. Majda and B. Gershgorin, Elementary models for turbulent diffusion with complex physical features: Eddy diffusivity, spectrum, and intermittency, Phil. Trans. Roy. Soc., in press, 2011.
[92]	A. J. Majda and J. Harlim, "Filtering Complex Turbulent Systems," Cambridge University Press, 2012.doi: 10.1017/CBO9781139061308.
[93]	A. J. Majda, J. Harlim and B. Gershgorin, Mathematical strategies for filtering turbulent dynamical systems, DCDS, 27 (2010), 441-486.doi: 10.3934/dcds.2010.27.441.
[94]	R. Mikulevicius and B. Rozovskii, Stochastic Navier-Stokes equations for turbulence flow, SIAM J. Math. Anal., 35 (2004), 1250-1310.doi: 10.1137/S0036141002409167.
[95]	N. Wiener, The Homogeneous Chaos, Am. J. Math., 60 (1938), 897-936.doi: 10.2307/2371268.
[96]	N. Wiener, The use of statistical theory in the study of turbulence, in "Proc. Fifth Int. Cong. Appl. Mech.," Wiley, New York, (1939), 356.
[97]	H. N. Najm, Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics, in "Annu. Rev. Fluid Mech.," 41, Annual Reviews, Palo Alto, CA, (2009), 35-52.
[98]	J. Neelin, B. Lintner, B. Tian, Q. Li, L. Zhang, P. Patra, M. Chahine and S. Stechmann, Long tails in deep columns of natural and anthropogenic tropospheric tracers, Geophysical Res. Lett., 37 (2010), L05804.doi: 10.1029/2009GL041726.
[99]	J. D. Neelin, M. Munnich, H. Su, J. E. Meyerson and C. E. Holloway, Tropical drying trends in global warming models and observations, Proc Natl Acad Sci USA, 103 (2006), 6110-6115.doi: 10.1073/pnas.0601798103.
[100]	J. Von Neumann, Some remarks on the problem of forecasting climatic fluctuations, in "Dynamics of Climate," (ed. R. L. Preffer), Pergamon Press, New York, (1960), 9-11.
[101]	B. K. Oksendal, "Stochastic Differential Equations: An Introduction with Applications," Springer, 2010.
[102]	S. A. Orszag and L. R. Bissonnette, Dynamical properties of truncated Wiener-Hermite expansions, Phys. Fluids, 10 (1967), 2603-2613.doi: 10.1063/1.1762082.
[103]	T. Palmer, A nonlinear dynamical perspective on model error: A proposal for nonlocal stochastic dynamic parameterizations in weather and climate prediction models, Quart. J. Roy. Meteor. Soc., 127 (2001), 279-303.
[104]	N. Peters, "Turbulent Combustion," Cambridge Monographs on Mechanics, Cambridge University Press, 2000.
[105]	C. L. Pettit and P. S. Beran, Spectral and multiresolution Wiener expansions of oscillatory stochastic processes, J. Sound Vib., 294 (2006), 752-779.doi: 10.1016/j.jsv.2005.12.043.
[106]	S. B. Pope, The probability approach to the modelling of turbulent reacting flows, Combust. Flame, 27 (1976), 299-312.doi: 10.1016/0010-2180(76)90035-3.
[107]	C. Prévôt and M. Röckner, "A Concise Course on Stochastic Partial Differential Equations," Lecture Notes in Mathematics, 1905, Springer, Berlin, 2007.
[108]	T. Sapsis and P. Lermusiaux, Dynamically orthogonal field equations for continuous stochastic dynamical systems, Physica D, 238 (2009), 2347-2360.doi: 10.1016/j.physd.2009.09.017.
[109]	T. Sapsis and P. Lermusiaux, Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty, Physica D, 241 (2012), 60-76.doi: 10.1016/j.physd.2011.10.001.
[110]	T. Sapsis and A. J. Majda, Blended dynamic reduced subspace algorithms for uncertainty quantification in turbulent dynamical systems, Nonlinearity, in preparation, 2012.
[111]	T. Sapsis and A. J. Majda, Dynamic reduced subspace and quasilinear gaussian methods for uncertainty quantification in turbulent dynamical systems, Comm. Math. Sci., in preparation, 2012.
[112]	T. Sapsis and A. J. Majda, Efficient blended dynamic reduced subspace algorithms for uncertainty quantification in high-dimensional turbulent dynamical systems, J. Comp. Phys., in preparation, 2012.
[113]	R.-A. Todor and C. Schwab, Sparse finite elements for elliptic problems with stochastic loading, Numer. Math., 95 (2003), 707-734.doi: 10.1007/s00211-003-0455-z.
[114]	L. M. Smith and S. L. Woodruff, Renormalization-group analysis of turbulence, in "Annu. Rev. Fluid Mech.," Vol. 30, Annual Reviews, Palo Alto, CA, (1998), 275-310.
[115]	K. R. Sreenivasan, The passive scalar spectrum and the Obukhov-Corrsin constant, Phys. Fluids, 8 (1996), 189-196.doi: 10.1063/1.868826.
[116]	H. Teng and G. Branstator, Initial-value predictability of prominent modes of North Pacific subsurface temperature in a CGCM, Climate Dyn., 36 (2010), 1813-1834.doi: 10.1007/s00382-010-0749-7.
[117]	D. M. Titterington, Bayesian methods for neural networks and related models, Statistical Science, 19 (2004), 128-139.doi: 10.1214/088342304000000099.
[118]	X. Wang, Stationary statistical properties of Rayleigh-Bénard convection at large Prandtl number, Comm. Pur. Appl. Math., 61 (2008), 789-815.doi: 10.1002/cpa.20214.
[119]	X. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization, Math. Comp., 79 (2010), 259-280.doi: 10.1090/S0025-5718-09-02256-X.
[120]	C. K. Wikle and M. B. Hooten, A general science-based framework for dynamical spatio-temporal models, TEST, 19 (2010), 417-451.doi: 10.1007/s11749-010-0209-z.
[121]	D. Xiu and G. E. Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Comput. Methods Appl. Mech. Engrg., 191 (2002), 4927-4948.doi: 10.1016/S0045-7825(02)00421-8.
[122]	D. Xiu and G. E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys., 187 (2003), 137-167.doi: 10.1016/S0021-9991(03)00092-5.
[123]	D. Xiu, D. Lucor, C. H. Su and G. E. Karniadakis, Stochastic modeling of flow-structure interactions using generalized polynomial chaos, J. Fluid Engrg., 124 (2002), 51-59.doi: 10.1115/1.1436089.
[124]	G. E. Karnidakis and D. Xiu, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619-644.doi: 10.1137/S1064827501387826.
[125]	Y. Yuan and A. J. Majda, Invariant measures and asymptotic bounds for normal forms of stochastic climate models, Chin. Ann. Math. B, 32 (2010), 343-368.doi: 10.1007/s11401-011-0647-2.