Estimating the uncertainty in underresolved nonlinear dynamics

The Mori–Zwanzig formalism of statistical mechanics is used to estimate the uncertainty caused by underresolution in the solution of a nonlinear dynamical system. A general approach is outlined and applied to a simple example. The noise term that describes the uncertainty turns out to be neither Markovian nor Gaussian. It is argued that this is the general situation.

[1]  Eric Darve,et al.  Computing generalized Langevin equations and generalized Fokker–Planck equations , 2009, Proceedings of the National Academy of Sciences.

[2]  A. Yaglom,et al.  An Introduction to the Theory of Stationary Random Functions , 1963 .

[3]  D. Wilks.  Effects of stochastic parametrizations in the Lorenz '96 system , 2005 .

[4]  Andrew M. Stuart,et al.  Inverse problems: A Bayesian perspective , 2010, Acta Numerica.

[5]  Timothy J. Robinson,et al.  Sequential Monte Carlo Methods in Practice , 2003 .

[6]  R. Zwanzig.  Nonlinear generalized Langevin equations , 1973 .

[7]  Gary P. Morriss,et al.  Statistical Mechanics of Nonequilibrium Liquids , 2008 .

[8]  Matthias Morzfeld,et al.  A random map implementation of implicit filters , 2011, J. Comput. Phys..

[9]  Matthias Morzfeld,et al.  Conditions for successful data assimilation , 2013, 1303.2714.

[10]  Panagiotis Stinis.  Mori-Zwanzig reduced models for uncertainty quantification II: Initial condition uncertainty , 2012, 1212.6360.

[11]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[13]  Armen Shirikyan,et al.  Mathematics of Two-Dimensional Turbulence , 2012 .

[14]  J. Whitaker,et al.  Accounting for the Error due to Unresolved Scales in Ensemble Data Assimilation: A Comparison of Different Approaches , 2005 .

[15]  Christopher K. Wikle,et al.  Atmospheric Modeling, Data Assimilation, and Predictability , 2005, Technometrics.

[16]  Alain Bensoussan,et al.  Stochastic Navier-Stokes Equations , 1995 .

[17]  Alexandre J. Chorin,et al.  Optimal prediction with memory , 2002 .

[18]  Eugen Fick,et al.  The quantum statistics of dynamic processes , 1990 .

[19]  Panagiotis Stinis,et al.  Mori-Zwanzig reduced models for uncertainty quantification I: Parametric uncertainty , 2012, 1211.4285.

[20]  A. Chorin,et al.  Implicit sampling for particle filters , 2009, Proceedings of the National Academy of Sciences.

[21]  Jinqiao Duan,et al.  Stochastic parameterization for large eddy simulation of geophysical flows , 2006, math/0607214.

[22]  G. W. Ford,et al.  Statistical Mechanics of Assemblies of Coupled Oscillators , 1965 .

[23]  J. Klauder,et al.  Numerical Integration of Multiplicative-Noise Stochastic Differential Equations , 1985 .

[24]  A. Chorin,et al.  Stochastic Tools in Mathematics and Science , 2005 .

[25]  Andrew J. Majda,et al.  Filtering Complex Turbulent Systems , 2012 .

[26]  R. Zwanzig.  Nonequilibrium statistical mechanics , 2001, Physics Subject Headings (PhySH).