An early warning system for multivariate time series with sparse and non-uniform sampling

In this paper we propose a new early warning test statistic, the ratio of deviations (RoD), which is defined to be the root mean squared of successive differences divided by the standard deviation. We show that RoD and autocorrelation are asymptotically related, and this relationship motivates the use of RoD to predict Hopf bifurcations in multivariate systems before they occur. We validate the use of RoD on synthetic data in the novel situation where the data is sparse and non-uniformly sampled. Additionally, we adapt the method to be used on high-frequency time series by sampling, and demonstrate the proficiency of RoD as a classifier.Comment: 14 pages, 8 figure

Coarse -graining dynamics of interacting particle systems

This thesis is concerned with coarse-graining dynamics of interacting particle systems. We study two different coarse-graining approximations of the microscopic process. In the first part of the thesis we study coarse-graining schemes for stochastic many body microscopic models focusing on the dynamics for the spin adsorption/desorption mechanism. We show that these coarse-graining schemes which are derived using cluster expansion are able to describe complex phenomena. We also study the role of multi-body interactions in coarse-graining schemes for achieving higher order accuracy. The multi-body interactions are often not included in coarse-graining schemes as they can be computational expensive. On the other hand the numerical experiments show that the inclusion of multi-body interactions is critical in accurately reproducing dynamical properties such as rare events and switching times. Here we propose strategies to compress multi-body interactions within a specified error tolerance, making such corrections computationally feasible. In the second part of the thesis we present an alternate way of modeling the noise in the CGMC/MC simulations for the diffusion mechanisms. Computational efficiency has been a motivation for Langevin approximations of the microscopic process in many fields. The Langevin approximation we derive is based upon the coarse-grained Markov process, so it automatically inherits the coarse-graining. We show that the long time behavior of the Langevin approximation is asymptotically equivalent to the long time behavior of the microscopic process. We establish this connection using a calculus of variations perspective and large deviations principles. We derive a time dependent action functional for the Langevin approximation using a Taylor series expansion of the drift and diffusion coefficients. We show that, using Γ-convergence arguments, asymptotically this action functional is equivalent to the action functional of the mesoscopic limit of the microscopic diffusion process. In the end, we show that the CGSDE model is capable of further enhancing the CPU savings achieved by CGMC and also that it is independent of the temperature and the interaction potential radius and other parameters of the problem

Application of the Turbulent Potential Model to Unsteady Flows and Three-Dimensional Boundary Layers

The turbulent potential model is a Reynolds-averaged (RANS) turbulence model that is theoretically capable of capturing nonequilibrium turbulent flows at a computational cost and complexity comparable to two-equation models. The ability of the turbulent potential model to predict nonequilibrium turbulent flows accurately is evaluated in this work. The flow in a spanwise-driven channel flow and over a swept bump are used to evaluate the turbulent potential model's ability to predict complex three-dimensional boundary layers. Results of turbulent vortex shedding behind a triangular and a square cylinder are also presented in order to evaluate the model's ability to predict unsteady flows. Early indications suggest that models of this type may be capable of significantly enhancing current numerical predictions of turbomachinery components at little extra computational cost or additional code complexity