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Multi-time-scale stochastic dynamical systems are ubiquitous in science and engineering, and the reduction of such systems and their models to only their slow components is often essential for scientific computation and further analysis. Rather than being available in the form of an explicit analytical model, often such systems can only be observed as a data set which embodies dynamics on several time scales. We focus on applying and adapting data-mining and manifold learning techniques to detect the slow components in a class of such multiscale data. Traditional data-mining methods are based on metrics (and thus, geometries) which are not informed of the multiscale nature of the underlying system dynamics; such methods cannot successfully recover the slow variables. Here, we present an approach which utilizes both the local geometry and the local noise dynamics within the data set through a metric which is both insensitive to the fast variables and more general than simple statistical averaging. Our analysis of the approach provides conditions for successfully recovering the underlying slow variables, as well as an empirical protocol guiding the selection of the method parameters. Interestingly, the recovered underlying variables are gauge invariant--they are insensitive to the measuring instrument/observation function. [ABSTRACT FROM AUTHOR]

We employ the diffusion map approach as a nonlinear dimensionality reduction technique to extract a dynamically relevant, low-dimensional description of n-alkane chains in the ideal-gas phase and in aqueous solution. In the case of C8 we find the dynamics to be governed by torsional motions. For C16 and C24 we extract three global order parameters with which we characterize the fundamental dynamics, and determine that the low free-energy pathway of globular collapse proceeds by a "kink and slide" mechanism, whereby a bend near the end of the linear chain migrates toward the middle to form a hairpin and, ultimately, a coiled helix. The low-dimensional representation is subtly perturbed in the solvated phase relative to the ideal gas, and its geometric structure is conserved between C16 and C24. The methodology is directly extensible to biomolecular self-assembly processes, such as protein folding. [ABSTRACT FROM AUTHOR]

Nonlinear independent component analysis is combined with diffusion-map data analysis techniques to detect good observables in high-dimensional dynamic data. These detections are achieved by integrating local principal component analysis of simulation bursts by using eigenvectors of a Markov matrix describing anisotropic diffusion. The widely applicable procedure, a crucial step in model reduction approaches, is illustrated on stochastic chemical reaction network simulations. [ABSTRACT FROM AUTHOR]

Interacting particle systems constitute the dynamic model of choice in a variety of application areas. A prominent example is pedestrian dynamics, where good design of escape routes for large buildings and public areas can improve evacuation in emergency situations, avoiding exit blocking and the ensuing panic. Here we employ diffusion maps to study the coarse-grained dynamics of two pedestrian crowds trying to pass through a door from opposite sides. These macroscopic variables and the associated smooth embeddings lead to a better description and a clearer understanding of the nature of the transition to oscillatory dynamics. We also compare the results to those obtained through intuitively chosen macroscopic variables. [ABSTRACT FROM AUTHOR]

Because of the small thermal capacity of ultrathin (∼200 nanometers) metal single crystals, it is possible to explore the coupling of catalytic and thermal action at low pressures. We analyzed a chemothermomechanical instability in this regime, in which catalytic reaction kinetics interact with heat transfer and mechanical buckling to create oscillations. These interacting components are separated and explored through experimentation, mathematical modeling, and scientific computation, and an explanation of the phenomenon emerges from their synthesis. [ABSTRACT FROM AUTHOR]