Naratip Santitissadeekorn

Transport in time-dependent dynamical systems: Finite-time coherent sets

We study the transport properties of nonautonomous chaotic dynamical systems over a finite-time duration. We are particularly interested in those regions that remain coherent and relatively nondispersive over finite periods of time, despite the chaotic nature of the system. We develop a novel probabilistic methodology based upon transfer operators that automatically detect maximally coherent sets. The approach is very simple to implement, requiring only singular vector computations of a matrix of transitions induced by the dynamics. We illustrate our new methodology on an idealized stratospheric flow and in two and three-dimensional analyses of European Centre for Medium Range Weather Forecasting (ECMWF) reanalysis data.

Applied and Computational Measurable Dynamics

Until recently, measurable dynamics has been held as a highly theoretcal mathematical topic with few generally known obvious links for practitioners in areas of applied mathematics. However, the advent of high-speed computers, rapidly developing algorithms, and new numerical methods has allowed for a tremendous amount of progress and sophistication in efforts to represent the notion of a transfer operator discretely but to high resolution. This book connects many concepts in dynamical systems with mathematical tools from areas such as graph theory and ergodic theory. The authors introduce practical tools for applications related to measurable dynamical systems, coherent structures, and transport problems. The new and fast-developing computational tools discussed throughout the book allow for detailed analysis of real-world problems that are simply beyond the reach of traditional methods. Audience: Applied and Computational Measurable Dynamics is intended for advanced undergraduate and graduate students and researchers in applied dynamical systems, computational ergodic theory, geosciences, and fluid dynamics. Contents Chapter 1: Dynamical Systems, Ensembles, and Transfer Operators; Chapter 2: Dynamical Systems Terminology and Definitions; Chapter 3: Frobenius-Perron Operator and Infintesimal Generator; Chapter 4: Graph Theoretic Methods and Markov Models of Dynamical Transport; Chapter 5: Graph Partition Methods and Their Relationship to Transport in Dynamical Systems; Chapter 6: The Topological Dynamics Perspective of Symbol Dynamics; Chapter 7: Transport Mechanism, Lobe Dynamics, Flux Rates, and Escape; Chapter 8: Finite Time Lyapunov Exponents; Chapter 9: Information Theory in Dynamical Systems; Appendix A: Computation, Codes, and Computational Complexity; Bibliography; Index

Two-stage filtering for joint state-parameter estimation

AbstractThis paper presents an approach for the simultaneous estimation of the state and unknown parameters in a sequential data assimilation framework. The state augmentation technique, in which the state vector is augmented by the model parameters, has been investigated in many previous studies and some success with this technique has been reported in the case where model parameters are additive. However, many geophysical or climate models contain nonadditive parameters such as those arising from physical parameterization of subgrid-scale processes, in which case the state augmentation technique may become ineffective. This is due to the fact that the inference of parameters from partially observed states based on the cross covariance between states and parameters is inadequate if states and parameters are not linearly correlated. In this paper, the authors propose a two-stage filtering technique that runs particle filtering (PF) to estimate parameters while updating the state estimate using an ensemble...

Optimal Mixing Enhancement

We introduce a general-purpose method for optimizing the mixing rate of advective fluid flows. An existing velocity field is perturbed in a

ANALYSIS AND MODELING OF AN EXPERIMENTAL DEVICE BY FINITE-TIME LYAPUNOV EXPONENT METHOD

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The infinitesimal operator for the semigroup of the Frobenius-Perron operator from image sequence data: Vector fields and transport barriers from movies

neighborhood to maximize the mixing rate for flows generated by velocity fields in this neighborhood. Our numerical approach is based on the infinitesimal generator of the flow and is solved by standard linear programming methods. The perturbed flow may be easily constrained to preserve the same steady state distribution as the original flow, and various natural geometric constraints can also be simply applied. The same technique can also be used to optimize the mixing rate of advection-diffusion flow models by manipulating the drift term in a small neighborhood.

Characterization of mixing in a simple paddle mixer using experimentally derived velocity fields

In this paper, we investigate the transport and mixing process of the batch mixers with two different configurations, the centered-blade and offset-blade mixers, by using a dynamical system approach. The 2-D velocity fields of the mixers measured using Particle Image Velocimetry (PIV) are used to identify the Lagrangian coherent structures (LCSs). The results show that the LCSs separate the physical space into two portions. In the case of the center-blade mixer the portion bounded inside the LCS experiences a relatively slow mixing relative to the portion outside of the LCS boundary. However, when the blade position is located near the wall, the LCS becomes more complicated but it still separates regions of fast mixing from a slower one. We develop a heuristic dynamical system model of our mixers to understand how the vorticity strength at the blade tips influences the variation of the LCSs. Finally, we define an appropriate notion of mixing to study the mixing rate of our mixing devices.

IMAGE EDGE RESPECTING DENOISING WITH EDGE DENOISING BY A DESIGNER NONISOTROPIC STRUCTURE TENSOR METHOD

In this paper, we present an approach to approximate the Frobenius-Perron transfer operator from a sequence of time-ordered images, that is, a movie dataset. Unlike time-series data, successive images do not provide a direct access to a trajectory of a point in a phase space; more precisely, a pixel in an image plane. Therefore, we reconstruct the velocity field from image sequences based on the infinitesimal generator of the Frobenius-Perron operator. Moreover, we relate this problem to the well-known optical flow problem from the computer vision community and we validate the continuity equation derived from the infinitesimal operator as a constraint equation for the optical flow problem. Once the vector field and then a discrete transfer operator are found, then, in addition, we present a graph modularity method as a tool to discover basin structure in the phase space. Together with a tool to reconstruct a velocity field, this graph-based partition method provides us with a way to study transport behavior and other ergodic properties of measurable dynamical systems captured only through image sequences.

Coherent sets for nonautonomous dynamical systems

The flow field in a cylindrical container driven by a flat bladed impeller was investigated using particle image velocimetry (PIV). Three Reynolds numbers (0.02, 8, 108) were investigated for different impeller locations within the cylinder. The results showed that vortices were formed at the tips of the blades and rotated with the blades. As the blades were placed closer to the wall the vortices interacted with the induced boundary layer on the wall to enhance both regions of vorticity. Finite time lyapunov exponents (FTLE) were used to determine the lagrangian coherent structure (LCS) fields for the flow. These structures highlighted the regions where mixing occurred as well as barriers to fluid transport. Mixing was estimated using zero mass particles convected by numeric integration of the experimentally derived velocity fields. The mixing data confirmed the location of high mixing regions and barriers shown by the LCS analysis. The results indicated that mixing was enhanced within the region described by the blade motion as the blade was positioned closed to the cylinder wall. The mixing average within the entire tank was found to be largely independent of the blade location and flow Reynolds number.

Three-dimensional characterization and tracking of an Agulhas Ring

Abstract We consider image denoising as the problem of removing spurious oscillations due to noise while preserving edges in the images. We will suggest here how to directly make infinitesimal adjustment to standard variational methods of image denoising, to enhance desirable target assumption of the noiseless image. The standard regularization method is used to define a suitable energy functional to penalize the data fidelity and the smoothness of the solution. This energy functional is tailored so that the region with small gradient is isotropically smoothed whereas in a neighborhood of an edge presented by a large gradient smoothing is allowed only along the edge contour. The regularized solution that arises in this fashion is then the solution of a variational principle. To this end the associated Euler — Lagrange equation needs to be solved numerically and the half-quadratic minimization is generally used to linearize the equation and to derive an iterative scheme. We describe here a method to modify Euler — Largrange equation from commonly used energy functionals, in a way to enhance certain desirable preconceived assumptions of the image, such as edge preservation. From an algorithmic point of view, we may deem this algorithm as a smoothing by a local average with an adaptive gradient-based weight. However, this algorithm may result in noisy edges although the edge is preserved and noise is suppressed in the low-gradient regions of the image. The main focus here is to present an edge-preserving regularization in the aforementioned view point, and to provide an alternative and simple way to modify the existing algorithm to mitigate the phenomena of noisy edges without explicitly defining step where we specify an energy functional to be minimized.