Bulletin of the American Physical Society, Nov 19, 2007
Lagrangian coherent structures in geophysical flows PHILIP DU TOIT, California Institute of Techn... more Lagrangian coherent structures in geophysical flows PHILIP DU TOIT, California Institute of Technology -In aperiodic flows, particle trajectories may appear to be chaotic and unstructured. However, using finite time Lyapunov exponents we are able to detect sharp separatrices that determine the underlying structure of the flow. These separatrices are barriers to transport and form the boundaries of almost invariant regions. Computing these Lagrangian Coherent Structures (LCS) for oceanic and atmospheric flows in two and three dimensions allows us to visualize the mixing processes and elucidates the underlying mechanisms by which particle transport and mixing occur. We will show that the LCS reveal many of the familiar elements from classical geometric dynamics including hyperbolic points, intersections of stable and unstable manifolds, homoclinic tangles, and transport via lobe dynamics. These methods have broad application to many geophysical flows. In particular, we observe that mixing in hurricanes and tropical storms is dominated by transport via lobe dynamics. Identifying the LCS allows us to quantify material entrainment and detrainment from the storm center. Precisely the same transport mechanism is also observed in flows surrounding ocean eddies.
We address the inverse problem of designing isotropic pairwise particle interaction potentials th... more We address the inverse problem of designing isotropic pairwise particle interaction potentials that lead to the formation of a desired lattice when a system of particles is cooled. The design problem is motivated by the desire to produce materials with pre-specified structure and properties. We present a heuristic computation-free geometric method, as well as a fast and robust trend optimization method that lead to the formation of high quality honeycomb lattices. The trend optimization method is particularly successful
This study applies variational integrators to the three-body problem, a classic problem from cele... more This study applies variational integrators to the three-body problem, a classic problem from celestial mechanics that asks for the motion of three masses in space under mutual gravitational interaction. The use of variational integrators, a class of numerical methods used to simulate mechanical systems, is of value because they are known to exhibit accurate behavior with respect to the preservation of physical constants of motion when applied to systems with conserved quantities like energy and momentum. This contrasts with the performance of other numerical differential equation-solving algorithms like the RungeKutta method, which typically experience artificial dissipation of conserved quantities over successive iterations. A comparison of the performance of variational integrators versus the fourth-order RungeKutta method applied to the planar circular restricted three-body problem composes the core of this study. In particular, the algorithms are evaluated based on their ability...
In this paper, we present an approach to autonomous navigation of a balloon by optimally exploiti... more In this paper, we present an approach to autonomous navigation of a balloon by optimally exploiting wind elds to minimize control (i.e. power requirements), time of travel, or other cost functionals. We use the principles of Discrete Mechanics and Optimal Control (DMOC) to compute optimal trajectories for a simpli ed model of balloon dynamics in a two-dimensional, time-dependent wind velocity eld. The wind eld was produced using the Weather Research and Forecasting (WRF) model for a region of the Mojave Desert on July 5, 2005. Due to inherent inaccuracies of the global wind model and the need for e cient optimization, we approach the problem of optimizing medium-scale (i.e., distances of the order approximately 100km) balloon trajectories using a simpli ed model of balloon dynamics. The results presented in this paper provide a framework to extend this approach to nding optimal trajectories in real-time for a three-dimensional wind eld. We hope to test our approach during a balloon ...
Visualizing and Quantifying Transport in Hurricanes
Flows surrounding hurricane storms are manifestly complex. The chaotic nature of individual parti... more Flows surrounding hurricane storms are manifestly complex. The chaotic nature of individual particle trajectories forestalls attempts to understand material transport by appealing to the Eulerian velocity field or particle trajectories alone. However, even in these complex flows, we demonstrate the existence of time-dependent coherent structures that govern the pathways for transport. These structures, called Lagrangian Coherent Structures (LCS), are moving barriers to transport in the flow and define the boundaries of almost invariant regions. They also act as separatrices in that they separate flow regions with different dynamical behavior. Given the velocity field, we extract the LCS numerically through computation of the finite time Lyapunov exponent which is a measure of local stretching in the flow. Once computed, the LCS allow us to visualize and understand the essential mechanisms underlying transport in the flow. We illustrate these results through several examples in both ...
ABSTRACT In this paper we describe an approach for the detection and classication of weak, distri... more ABSTRACT In this paper we describe an approach for the detection and classication of weak, distributed patterns in sensor networks. Of course, before one can begin development of a pattern detection algorithm, one must rst dene the term "pattern", which by nature is a broad and inclusive term. One of the key aspects of our work is a denition of pattern that has already proven eective in detecting anomalies in real world data. While designing detection algorithms for all classes of patterns in all types of networks sounds appealing, this approach would almost certainly require heuristic methods and only cursory statements of performance. Rather, we have specically studied the problem of intrusion detection in computer networks in which a pattern is an abnormal or unexpected spatio-temporal dependence in the data collected across the nodes. We do not attempt to match an a priori template, but instead have developed algorithms that allow the pattern to reveal itself in the data by way of dependence or independence of observed time series. Although the problem is complex and challenging, recent advances in l1 techniques for robust matrix completion, compressed sensing, and correlation detection provide promising opportunities for progress. Our key contribution to this body of work is the development of methods that make an accounting of uncertainty in the measurements on which the inferences are based. The performance of our methods will be demonstrated on real world data, including measured data from the Abilene Internet2 network.
Journal of Fixed Point Theory and Applications, 2010
The method of using Finite Time Liapunov Exponents (FTLE) to extract Lagrangian Coherent Structur... more The method of using Finite Time Liapunov Exponents (FTLE) to extract Lagrangian Coherent Structures (LCS) in aperiodic flows, as originally developed by Haller, is applied to geophysical flows. In this approach, the LCS are identified as surfaces of greatest separation that parse the flow into regions with different dynamical behavior. In this way, the LCS reveal the underlying skeleton of turbulence. The time-dependence of the LCS provides insight into the mechanisms by which fluid is transported from one region to another. Of especial interest in this study is the utility with which the FTLE-LCS method can be used to reveal homoclinic and horseshoe dynamics in aperiodic flows. The FTLE-LCS method is applied to turbulent flow in hurricanes and reveals LCS that delineate sharp boundaries to a storm. Moreover, intersections of the LCS define lobes that mediate transport into and out of a storm through the action of homoclinic lobe dynamics. Using the FTLE-LCS method, the same homoclinic structures are seen to be a dominant transport mechanism in the Global Ocean, and provide insights into the role of mesoscale eddies in enhancing lateral mixing.
ESAIM: Mathematical Modelling and Numerical Analysis, 2012
The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff eq... more The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and for simulating collisions between wavefronts. Discretization by means of the particle method is shown to preserve the basic Hamiltonian, the weak and variational structure of the original problem, and to respect the conservation laws associated with symmetry under the Euclidean group. Numerical results illustrate that the particle method has superior features in both one and two dimensions, and can also be effectively implemented when the initial data of interest lies on a submanifold.
This study investigates Lagrangian coherent structures (LCS) in the planar elliptic restricted th... more This study investigates Lagrangian coherent structures (LCS) in the planar elliptic restricted three-body problem (ER3BP), a generalization of the circular restricted three-body problem (CR3BP) that asks for the motion of a test particle in the presence of two elliptically orbiting point masses. Previous studies demonstrate that an understanding of transport phenomena in the CR3BP, an autonomous dynamical system (when viewed in a rotating frame), can be obtained through analysis of the stable and unstable manifolds of certain periodic solutions to the CR3BP equations of motion. These invariant manifolds form cylindrical tubes within surfaces of constant energy that act as separatrices between orbits with qualitatively different behaviors. The computation of LCS, a technique typically applied to fluid flows to identify transport barriers in the domains of time-dependent velocity fields, provides a convenient means of determining the time-dependent analogues of these invariant manifolds for the ER3BP, whose equations of motion contain an explicit dependency on the independent variable. As a direct application, this study uncovers the contribution of the planet Mercury to the Interplanetary Transport Network, a network of tubes through the solar system that can be exploited for the construction of low-fuel spacecraft mission trajectories.
Abstract—A theory and algorithm for detecting and classifying weak, distributed patterns in netwo... more Abstract—A theory and algorithm for detecting and classifying weak, distributed patterns in network data is presented. The patterns we consider are anomalous temporal correlations between signals recorded at sensor nodes in a network. We use robust matrix completion and second order analysis to detect distributed patterns that are not discernible at the level of individual sensors. When viewed independently, the data at each node cannot provide a definitive determination of the underlying pattern, but when fused with data from across the network the relevant patterns emerge. We are specifically interested in detecting weak patterns in computer networks where the nodes (terminals, routers, servers, etc.) are sensors that provide measurements (of packet rates, user activity, central processing unit usage, etc.). The approach is applicable to many other types of sensor networks including wireless networks, mobile sensor networks, and social networks where correlated phenomena are of in...
arXiv: Adaptation and Self-Organizing Systems, 2009
We address the inverse problem of designing isotropic pairwise particle interaction potentials th... more We address the inverse problem of designing isotropic pairwise particle interaction potentials that lead to the formation of a desired lattice when a system of particles is cooled. The design problem is motivated by the desire to produce materials with pre-specified structure and properties. We present a heuristic computation-free geometric method, as well as a fast and robust trend optimization method that lead to the formation of high quality honeycomb lattices. The trend optimization method is particularly successful since it is well-suited to efficient optimization of the noisy and expensive objective functions encountered in the self-assembly design problem. We also present anisotropic potentials that robustly lead to the formation of the kagome lattice, a lattice that has not previously been obtained with isotropic potentials.
Lagrangian coherent structures in geophysical flows
ABSTRACT In aperiodic flows, particle trajectories may appear to be chaotic and unstructured. How... more ABSTRACT In aperiodic flows, particle trajectories may appear to be chaotic and unstructured. However, using finite time Lyapunov exponents we are able to detect sharp separatrices that determine the underlying structure of the flow. These separatrices are barriers to transport and form the boundaries of almost invariant regions. Computing these Lagrangian Coherent Structures (LCS) for oceanic and atmospheric flows in two and three dimensions allows us to visualize the mixing processes and elucidates the underlying mechanisms by which particle transport and mixing occur. We will show that the LCS reveal many of the familiar elements from classical geometric dynamics including hyperbolic points, intersections of stable and unstable manifolds, homoclinic tangles, and transport via lobe dynamics. These methods have broad application to many geophysical flows. In particular, we observe that mixing in hurricanes and tropical storms is dominated by transport via lobe dynamics. Identifying the LCS allows us to quantify material entrainment and detrainment from the storm center. Precisely the same transport mechanism is also observed in flows surrounding ocean eddies.
We study the Astrojax Pendulum and the N-body problem on the sphere in the light of Lagrangian re... more We study the Astrojax Pendulum and the N-body problem on the sphere in the light of Lagrangian reduction theory, variational integrators, and pattern evocation.
On the derivation of the quantum mechanical probability current for wave equations describing particles with spin
ABSTRACT We show how the standard procedure for deriving the probability current for the Schrodin... more ABSTRACT We show how the standard procedure for deriving the probability current for the Schrodinger equation becomes invalid when applied to the Pauli equation. Instead the correct expression for the current for an electron can be obtained from a nonrelativistic limit of the current for the Dirac equation. This procedure introduces an extra term, known as the spin current, and which is independent of the external potential. That the spin current is not, however, a relativistic effect can be seen from our derivation of this term in the expression for the current using the nonrelativistic Levy-Leblond equation.
The principle of Robust Principal Component Analysis (RPCA) is to additively resolve a matrix int... more The principle of Robust Principal Component Analysis (RPCA) is to additively resolve a matrix into a low-rank and a sparse component. The question that arises in the application of this principle to experimental data is, "when is this resolution an identification of the actual low-rank and sparse components of the data?" We report several experimental findings: (1) while successful recoveries can only be expected when the low-rank component is of low fractional rank and the sparse component is of low fractional sparsity, the subset of matrices that successfully recover is significantly larger than the subset of matrices that satisfy the currently established sufficient conditions; (2) where recovery is unsuccessful, the returned matrices tend to be near half-rank and half-sparsity, suggesting a cross validation principle; (3) the demarcation between the region of consistent recovery and consistent failure is narrow, indicating a phase change in recoverability; and (4) recovery is relatively invariant to matrix distributions, thus synthetic matrices can closely predict recoverability of real matrices. We demonstrate these findings with a variety of synthetic matrices that are faithful to matrices appearing in practice. Furthermore, we apply and verify these results on real-world matrices. HighlightsReport empirical recovery regions of Robust PCA in rank-sparsity plane.Present a cross validation principle for identifying a successful decomposition.Demonstrate applicability of the found bounds to real world problems.
The structure of transport and mixing in vortical ows cannot be readily discerned by considering ... more The structure of transport and mixing in vortical ows cannot be readily discerned by considering velocity or vorticity alone. However, the underlying Lagrangian coherent structures that govern transport in the ow can be extracted via computation of nite time Liapunov exponents. Visualizations of the Lagrangian coherent structures extracted in this way are provided for three geophysical vortices. We observe that for the apparently disparate cases of hurricanes and ocean eddies, the underlying transport mechanism in each case is identical and occurs via lobe dynamics. Introduction. Over the last several years, various methods have been developed to study coherent struc- tures in time-dependent ows. The primary goal of these methods is to identify the key structures and manifolds within the
Path Prediction for an Earth-Based Demonstration Balloon Flight
In this paper, we present an approach to autonomous navigation of a balloon by optimally exploiti... more In this paper, we present an approach to autonomous navigation of a balloon by optimally exploiting wind elds to minimize control (i.e. power requirements), time of travel, or other cost functionals. We use the princi-ples of Discrete Mechanics and Optimal Control (DMOC) to compute optimal trajectories for a simplied model of balloon dynamics in a two-dimensional, time-dependent wind velocity eld. The wind eld was produced using the Weather Research and Forecasting (WRF) model for a region of the Mojave Desert on July 5, 2005. Due to inherent inaccuracies of the global wind model and the need for ecient optimization, we approach the problem of optimizing medium-scale (i.e., distances of the order approximately 100km) balloon trajectories using a simplied model of balloon dynamics. The results presented in this paper provide a framework to extend this approach to nding optimal trajectories in real-time for a three-dimensional wind eld. We hope to test our approach during a balloon de...
On the derivation of the quantum mechanical probability current for wave equations describing particles with spin
We show how the standard procedure for deriving the probability current for the Schrodinger equat... more We show how the standard procedure for deriving the probability current for the Schrodinger equation becomes invalid when applied to the Pauli equation. Instead the correct expression for the current for an electron can be obtained from a nonrelativistic limit of the current for the Dirac equation. This procedure introduces an extra term, known as the spin current, and which is independent of the external potential. That the spin current is not, however, a relativistic effect can be seen from our derivation of this term in the expression for the current using the nonrelativistic Levy-Leblond equation.
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