Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics

Abstract

Many physical systems are described by nonlinear differential equations that are too complicated to solve in full. A natural way to proceed is to divide the variables into those that are of direct interest and those that are not, formulate solvable approximate equations for the variables of greater interest, and use data and statistical methods to account for the impact of the other variables. In the present paper we consider time-dependent problems and introduce a fully discrete solution method, which simplifies both the analysis of the data and the numerical algorithms. The resulting time series are identified by a NARMAX (nonlinear autoregression moving average with exogenous input) representation familiar from engineering practice. The connections with the MoriZwanzig formalism of statistical physics are discussed, as well as an application to the Lorenz 96 system.T here are many time-dependent problems in science, where the equations of motion are too complex for full solution, either because the equations are not certain or because the computational cost is too high, but one is interested only in the dynamics of a subset of the variables. Such problems appear, for example, in weather and climate modeling, e.g., refs. 1, 2; in economics, e.g., ref. 3; in statistical mechanics, e.g., refs. 4, 5; and in the mechanics of turbulent flow, e.g., refs. 6, 7. In these settings it is natural to look for simpler equations that involve only the variables of interest, and then use data to assess the effect of the remaining variables on the variables of interest in some approximate way. In the present paper we focus on stochastic methods for doing this, with data coming either from experimental observations or from prior numerical calculations.Consider a set of differential equations of the formwhere t is the time, x = ðx 1 , x 2 , . . . , x m Þ is the vector of resolved variables, and y = ðy 1 , y 2 , . . . , y ℓ Þ is the vector of unresolved variables, where ℓ is not necessarily finite, with initial data xð0Þ = α, yð0Þ = β. Consider a situation where this system is too complicated to solve, but where data are available, usually as sequences of measured values of x, assumed here to be observed with negligible observation errors. One can write Rðx, yÞ in the form Rðx, yÞ = R 0 ðxÞ + zðx, yÞ,where R 0 approximates Rðx, yÞ in some sense and is such that one is able to solve the equationThe remainder zðx, yÞ = Rðx, yÞ − R 0 ðxÞ is the contribution of the unresolved variables to the equation for x and must be taken into account. It has been variously called "unresolved tendency" (8, 9), "model error" (10, 11), or "model noise" (12); we call it simply the "noise." In the present paper we do not discuss general methods for constructing R 0 ; they depend on practical considerations which differ from case to case. A usual approach to the problem of computing x is to identify z from data (8, 13, 14), i.e., find a concise approximate representationẑ of z that can be evaluated on the computer, and then solve the equationWhenẑ is a...

Accounting for Model Error from Unresolved Scales in Ensemble Kalman Filters by Stochastic Parameterization

“…It has been shown to have a mean close to zero (see e.g. (Arnold et al 2013;Chorin and Lu 2015;Mitchell and Carrassi 2015)), hence the biases in the means of forecast ensembles are negligible. This removes a concern about ensemble bias in the use of covariance inflation and localization to account for such model error (Dee and Da Silva 1998;Li et al 2009).…”

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