Ilya Timofeyev

Systematic Strategies for Stochastic Mode Reduction in Climate

A systematic strategy for stochastic mode reduction is applied here to three prototype ‘‘toy’’ models with nonlinear behavior mimicking several features of low-frequency variability in the extratropical atmosphere. Two of the models involve explicit stable periodic orbits and multiple equilibria in the projected nonlinear climate dynamics. The systematic strategy has two steps: stochastic consistency and stochastic mode elimination. Both aspects of the mode reduction strategy are tested in an a priori fashion in the paper. In all three models the stochastic mode elimination procedure applies in a quantitative fashion for moderately large values of « 0.5 or even « 1, where the parameter « roughly measures the ratio of correlation times of unresolved variables to resolved climate variables, even though the procedure is only justified mathematically for « K 1. The results developed here provide some new perspectives on both the role of stable nonlinear structures in projected nonlinear climate dynamics and the regression fitting strategies for stochastic climate modeling. In one example, a deterministic system with 102 degrees of freedom has an explicit stable periodic orbit for the projected climate dynamics in two variables; however, the complete deterministic system has instead a probability density function with two large isolated peaks on the ‘‘ghost’’ of this periodic orbit, and correlation functions that only weakly ‘‘shadow’’ this periodic orbit. Furthermore, all of these features are predicted in a quantitative fashion by the reduced stochastic model in two variables derived from the systematic theory; this reduced model has multiplicative noise and augmented nonlinearity. In a second deterministic model with 101 degrees of freedom, it is established that stable multiple equilibria in the projected climate dynamics can be either relevant or completely irrelevant in the actual dynamics for the climate variable depending on the strength of nonlinearity and the coupling to the unresolved variables. Furthermore, all this behavior is predicted in a quantitative fashion by a reduced nonlinear stochastic model for a single climate variable with additive noise, which is derived from the systematic mode reduction procedure. Finally, the systematic mode reduction strategy is applied in an idealized context to the stochastic modeling of the effect of mountain torque on the angular momentum budget. Surprisingly, the strategy yields a nonlinear stochastic equation for the large-scale fluctuations, and numerical simulations confirm significantly improved predicted correlation functions from this model compared with a standard linear model with damping and white noise forcing.

A priori tests of a stochastic mode reduction strategy

Several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors [Proc. Natl. Acad. Sci. 96 (1999) 14687; Commun. Pure Appl. Math. 54 (2001) 891] are developed here. In this procedure, reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables. While the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small, it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big. Typically, the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1/2. The examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated Burgers–Hopf equations as a nonlinear heat bath. Systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two-time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables. In an important illustrative example presented here, the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises. Even for large value of the correlation time ratio of the order of 1/2, the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non-Gaussian statistics of the original nonlinear systems with 102 modes extremely well. Furthermore, it is shown here that the standard regression fitting of the second-order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example.

Stochastic models for selected slow variables in large deterministic systems

A new stochastic mode-elimination procedure is introduced for a class of deterministic systems. Under assumptions of ergodicity and mixing, the procedure gives closed-form stochastic models for the slow variables in the limit of infinite separation of timescales. The procedure is applied to the truncated Burgers–Hopf (TBH) system as a test case where the separation of timescale is only approximate. It is shown that the stochastic models reproduce exactly the statistical behaviour of the slow modes in TBH when the fast modes are artificially accelerated to enforce the separation of timescales. It is shown that this operation of acceleration only has a moderate impact on the bulk statistical properties of the slow modes in TBH. As a result, the stochastic models are sound for the original TBH system.

Quantifying predictability in a model with statistical features of the atmosphere

The Galerkin truncated inviscid Burgers equation has recently been shown by the authors to be a simple model with many degrees of freedom, with many statistical properties similar to those occurring in dynamical systems relevant to the atmosphere. These properties include long time-correlated, large-scale modes of low frequency variability and short time-correlated “weather modes” at smaller scales. The correlation scaling in the model extends over several decades and may be explained by a simple theory. Here a thorough analysis of the nature of predictability in the idealized system is developed by using a theoretical framework developed by R.K. This analysis is based on a relative entropy functional that has been shown elsewhere by one of the authors to measure the utility of statistical predictions precisely. The analysis is facilitated by the fact that most relevant probability distributions are approximately Gaussian if the initial conditions are assumed to be so. Rather surprisingly this holds for both the equilibrium (climatological) and nonequilibrium (prediction) distributions. We find that in most cases the absolute difference in the first moments of these two distributions (the “signal” component) is the main determinant of predictive utility variations. Contrary to conventional belief in the ensemble prediction area, the dispersion of prediction ensembles is generally of secondary importance in accounting for variations in utility associated with different initial conditions. This conclusion has potentially important implications for practical weather prediction, where traditionally most attention has focused on dispersion and its variability.

Spatiotemporal Smoothing as a Basis for Facial Tissue Tracking in Thermal Imaging

Accurate tracking of facial tissue in thermal infrared imaging is challenging because it is affected not only by positional but also physiological (functional) changes. This paper presents a particle filter tracker driven by a probabilistic template function with both spatial and temporal smoothing components, which is capable of adapting to abrupt positional and physiological changes. The method was tested on tracking facial regions of subjects under varying physiological and environmental conditions in 25 thermal clips. It demonstrated robustness and accuracy, outperforming other strategies. This new method promises improved performance in a number of biomedical applications that involve physiological measurements on the face, such as unobtrusive sleep and stress studies.

PEDESTRIAN FLOW MODELS WITH SLOWDOWN INTERACTIONS

In this paper, we introduce and study one-dimensional models for the behavior of pedestrians in a narrow street or corridor. We begin at the microscopic level by formulating a stochastic cellular automata model with explicit rules for pedestrians moving in two opposite directions. Coarse-grained mesoscopic and macroscopic analogs are derived leading to the coupled system of PDEs for the density of the pedestrian traffic. The obtained first-order system of conservation laws is only conditionally hyperbolic. We also derive higher-order nonlinear diffusive corrections resulting in a parabolic macroscopic PDE model. Numerical experiments comparing and contrasting the behavior of the microscopic stochastic model and the resulting coarse-grained PDEs for various parameter settings and initial conditions are performed. These numerical experiments demonstrate that the nonlinear diffusion is essential for reproducing the behavior of the stochastic system in the nonhyperbolic regime.

Finite-dimensional dynamical system modeling thermal instabilities

We describe a three-dimensional dynamical system, which is obtained as a pseudo-spectral approximation to a free boundary problem modeling solid combustion and rapid solidification, and is capable of generating its major dynamical patterns. These patterns include a Hopf bifurcation followed by a sequence of secondary period doubling and a transition to chaos, reverse sequences, and sequences followed by Shilnikov type trajectories. A computer-assisted bifurcation analysis uncovers some novel mechanisms of stability exchange. The most striking of them is an infinite period bifurcation which resembles the classical Shilnikov bifurcation, but instead of a funnel-shaped spiral along which the period is continually increasing, the continuation produced a series of isolas. Each isola is a closed branch of solutions of roughly the same period, and with the same number of oscillations. The isolas corresponding to consecutive numbers of low amplitude oscillations about the equilibrium are adjacent to each other, and appear to accumulate on a saddle-focus homoclinic connection of Shilnikov type. ©2000 Elsevier Science B.V. All rights reserved.

On Cellular Automata Models of Traffic Flow with Look Ahead Potential

We study the statistical properties of a cellular automata model of traffic flow with the look-ahead potential. The model defines stochastic rules for the movement of cars on a lattice. We analyze the underlying statistical assumptions needed for the derivation of the coarse-grained model and demonstrate that it is possible to relax some of them to obtain an improved coarse-grained ODE model. We also demonstrate that spatial correlations play a crucial role in the presence of the look-ahead potential and propose a simple empirical correction to account for the spatial dependence between neighboring cells.

Markov Chain Stochastic Parametrizations of Essential Variables

We analyze the performance of the novel Markov chain stochastic modeling technique for derivation of effective equations for a set of essential variables. This technique is an empirical approach where the right-hand side of the essential variables is modeled by a Markov chain. We demonstrate that the Markov chain modeling approach performs well in a prototype model without scale separation between the essential and the nonessential variables. Moreover, we utilize the truncated Burgers–Hopf model to show that the Markov chain should be properly conditioned on the essential variables to reproduce the structure of two-point statistical quantities. On the other hand, the conditioning can be rather straightforward and unsophisticated.

Homoclinic orbits and chaos in a second-harmonic generating optical cavity

Abstract We present two large families of Silnikov-type homoclinic orbits in a two-mode model that describes second-harmonic generation in a passive optical cavity. These families of homoclinic orbits give rise to chaotic dynamics in the model.