Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems

In this paper we develop new techniques for revealing geometrical structures in phase space that are valid for aperiodically time dependent dynamical systems, which we refer to as Lagrangian descriptors. These quantities are based on the integration, for a finite time, along trajectories of an intrinsic bounded, positive geometrical and/or physical property of the trajectory itself. We discuss a general methodology for constructing Lagrangian descriptors, and we discuss a "heuristic argument" that explains why this method is successful for revealing geometrical structures in the phase space of a dynamical system. We support this argument by explicit calculations on a benchmark problem having a hyperbolic fixed point with stable and unstable manifolds that are known analytically. Several other benchmark examples are considered that allow us the assess the performance of Lagrangian descriptors in revealing invariant tori and regions of shear. Throughout the paper "side-by-side" comparisons of the performance of Lagrangian descriptors with both finite time Lyapunov exponents (FTLEs) and finite time averages of certain components of the vector field ("time averages") are carried out and discussed. In all cases Lagrangian descriptors are shown to be both more accurate and computationally efficient than these methods. We also perform computations for an explicitly three dimensional, aperiodically time-dependent vector field and an aperiodically time dependent vector field defined as a data set. Comparisons with FTLEs and time averages for these examples are also carried out, with similar conclusions as for the benchmark examples.Comment: 52 pages, 25 figure

Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of "transport barriers" in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterizing transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate, in a concrete manner, the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of "flow transition" which occurs when finite-time hyperbolicity is lost, or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing, and important, new area of dynamical systems theory

UVaFTLE: Lagrangian finite time Lyapunov exponent extraction for fluid dynamic applications

Producción CientíficaThe determination of Lagrangian Coherent Structures (LCS) is becoming very important in several disciplines, including cardiovascular engineering, aerodynam- ics, and geophysical fluid dynamics. From the computational point of view, the extraction of LCS consists of two main steps: The flowmap computation and the resolution of Finite Time Lyapunov Exponents (FTLE). In this work, we focus on the design, implementation, and parallelization of the FTLE resolution. We offer an in-depth analysis of this procedure, as well as an open source C implementation (UVaFTLE) parallelized using OpenMP directives to attain a fair parallel efficiency in shared-memory environments. We have also implemented CUDA kernels that allow UVaFTLE to leverage as many NVIDIA GPU devices as desired in order to reach the best parallel efficiency. For the sake of reproducibility and in order to con- tribute to open science, our code is publicly available through GitHub. Moreover, we also provide Docker containers to ease its usage.Ministerio de Economía, Industria y Competitividad, Consejo Asesor de Educación de Castilla y León y Programas del Fondo de Desarrollo (FEDER): Proyecto PCAS (TIN2017-88614-R) y Proyecto PROPHET-2 (VA226P20).Ministerio de Ciencia e Innovación, Agencia Estatal de Investigación y “European Union NextGenerationEU/PRTR” : (MCIN/ AEI/10.13039/501100011033) - (grant TED2021-130367B-I00)Junta de Castilla y León (project VA182P20)Red Española de Supercomputación (RES) (projects IM-2022-2-0015 and IM-2022-3-0021)Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

Transport in fluid flows unveiled by Lagrangian Structures

The interpretation of flows in steady systems is straightforward: streamlines and trajectories coincide. Unsteady flows are much more challenging. A natural framework is the Lagrangian one since it allows the study of the flow in terms of particle trajectories. Mixing, i.e. dispersion, plays a fundamental role from this point of view. A usual approach consists in the computation of Lagrangian statistics as absolute and relative dispersion

Advancing the theory and applications of Lagrangian Coherent Structures methods for oceanic surface flows

Submitted in partial fulfillment of the requirements for the degree of Doctor of Science at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution June 2019.Ocean surface transport is at the core of many environmental disasters, including the spread of marine plastic pollution, the Deepwater Horizon oil spill and the Fukushima nuclear contamination. Understanding and predicting flow transport, however, remains a scientific challenge, because it operates on multiple length- and time-scales that are set by the underlying dynamics. Building on the recent emergence of Lagrangian methods, this thesis investigates the present-day abilities to describe and understand the organization of flow transport at the ocean surface, including the abilities to detect the underlying key structures, the regions of stirring and regions of coherence within the flow. Over the past four years, the field of dynamical system theory has adapted several algorithms from unsupervised machine learning for the detection of Lagrangian Coherent Structures (LCS). The robustness and applicability of these tools is yet to be proven, especially for geophysical flows. An updated, parameter-free spectral clustering approach is developed and a noise-based cluster coherence metric is proposed to evaluate the resulting clusters. The method is tested against benchmarks flows of dynamical system theory: the quasi-periodic Bickley jet, the Duffng oscillator and a modified, asymmetric Duffing oscillator. The applicability of this newly developed spectral clustering method, along with several common LCS approaches, such as the Finite-Time Lyapunov Exponent, is tested in several field studies. The focus is on the ability to predict these LCS in submesoscale ocean surface flows, given all the uncertainties of the modeled and observed velocity fields, as well as the sparsity of Lagrangian data. This includes the design and execution of field experiments targeting LCS from predictive models and their subsequent Lagrangian analysis. These experiments took place in Scott Reef, an atoll system in Western Australia, and off the coast of Martha's Vineyard, Massachusetts, two case studies with tidally-driven channel flows. The FTLE and spectral clustering analyses were particularly helpful in describing key transient flow features and how they were impacted by tidal forcing and vertical velocities. This could not have been identified from the Eulerian perspective, showing the utility of the Lagrangian approach in understanding the organization of transport

A quasi-objective single-buoy approach for understanding Lagrangian coherent structures and sea ice dynamics

Sea ice drift and deformation, namely sea ice dynamics, play a significant role in atmosphere–ice–ocean coupling. Deformation patterns in sea ice can be observed over a wide range of spatial and temporal scales, though high-resolution objective quantification of these features remains difficult. In an effort to better understand local deformation of sea ice, we adapt the trajectory-stretching exponents (TSEs), quasi-objective measures of Lagrangian stretching in continuous media, to sea ice buoy data and develop a temporal analysis of TSE time series. Our work expands on previous ocean current studies that have shown TSEs provide an approximation of Lagrangian coherent structure diagnostics when only sparse trajectory data are available. As TSEs do not require multiple buoys, we find they have an expanded range of use when compared with traditional Eulerian buoy-array deformation metrics and provide local-stretching information below the length scales possible when averaging over buoy arrays. We verify the ability of TSEs to temporally and spatially identify dynamic features for three different sea ice datasets. The ability of TSEs to quantify trajectory stretching is verified by concurrent ice fracture in buoy neighborhoods ranging from tens to hundreds of kilometers in diameter, as well as the temporal concurrence of significant storm events.</p

Visualization challenges in distributed heterogeneous computing environments

Large-scale computing environments are important for many aspects of modern life. They drive scientific research in biology and physics, facilitate industrial rapid prototyping, and provide information relevant to everyday life such as weather forecasts. Their computational power grows steadily to provide faster response times and to satisfy the demand for higher complexity in simulation models as well as more details and higher resolutions in visualizations. For some years now, the prevailing trend for these large systems is the utilization of additional processors, like graphics processing units. These heterogeneous systems, that employ more than one kind of processor, are becoming increasingly widespread since they provide many benefits, like higher performance or increased energy efficiency. At the same time, they are more challenging and complex to use because the various processing units differ in their architecture and programming model. This heterogeneity is often addressed by abstraction but existing approaches often entail restrictions or are not universally applicable. As these systems also grow in size and complexity, they become more prone to errors and failures. Therefore, developers and users become more interested in resilience besides traditional aspects, like performance and usability. While fault tolerance is well researched in general, it is mostly dismissed in distributed visualization or not adapted to its special requirements. Finally, analysis and tuning of these systems and their software is required to assess their status and to improve their performance. The available tools and methods to capture and evaluate the necessary information are often isolated from the context or not designed for interactive use cases. These problems are amplified in heterogeneous computing environments, since more data is available and required for the analysis. Additionally, real-time feedback is required in distributed visualization to correlate user interactions to performance characteristics and to decide on the validity and correctness of the data and its visualization. This thesis presents contributions to all of these aspects. Two approaches to abstraction are explored for general purpose computing on graphics processing units and visualization in heterogeneous computing environments. The first approach hides details of different processing units and allows using them in a unified manner. The second approach employs per-pixel linked lists as a generic framework for compositing and simplifying order-independent transparency for distributed visualization. Traditional methods for fault tolerance in high performance computing systems are discussed in the context of distributed visualization. On this basis, strategies for fault-tolerant distributed visualization are derived and organized in a taxonomy. Example implementations of these strategies, their trade-offs, and resulting implications are discussed. For analysis, local graph exploration and tuning of volume visualization are evaluated. Challenges in dense graphs like visual clutter, ambiguity, and inclusion of additional attributes are tackled in node-link diagrams using a lens metaphor as well as supplementary views. An exploratory approach for performance analysis and tuning of parallel volume visualization on a large, high-resolution display is evaluated. This thesis takes a broader look at the issues of distributed visualization on large displays and heterogeneous computing environments for the first time. While the presented approaches all solve individual challenges and are successfully employed in this context, their joint utility form a solid basis for future research in this young field. In its entirety, this thesis presents building blocks for robust distributed visualization on current and future heterogeneous visualization environments.Große Rechenumgebungen sind für viele Aspekte des modernen Lebens wichtig. Sie treiben wissenschaftliche Forschung in Biologie und Physik, ermöglichen die rasche Entwicklung von Prototypen in der Industrie und stellen wichtige Informationen für das tägliche Leben, beispielsweise Wettervorhersagen, bereit. Ihre Rechenleistung steigt stetig, um Resultate schneller zu berechnen und dem Wunsch nach komplexeren Simulationsmodellen sowie höheren Auflösungen in der Visualisierung nachzukommen. Seit einigen Jahren ist die Nutzung von zusätzlichen Prozessoren, z.B. Grafikprozessoren, der vorherrschende Trend für diese Systeme. Diese heterogenen Systeme, welche mehr als eine Art von Prozessor verwenden, finden zunehmend mehr Verbreitung, da sie viele Vorzüge, wie höhere Leistung oder erhöhte Energieeffizienz, bieten. Gleichzeitig sind diese jedoch aufwendiger und komplexer in der Nutzung, da die verschiedenen Prozessoren sich in Architektur und Programmiermodel unterscheiden. Diese Heterogenität wird oft durch Abstraktion angegangen, aber bisherige Ansätze sind häufig nicht universal anwendbar oder bringen Einschränkungen mit sich. Diese Systeme werden zusätzlich anfälliger für Fehler und Ausfälle, da ihre Größe und Komplexität zunimmt. Entwickler sind daher neben traditionellen Aspekten, wie Leistung und Bedienbarkeit, zunehmend an Widerstandfähigkeit gegenüber Fehlern und Ausfällen interessiert. Obwohl Fehlertoleranz im Allgemeinen gut untersucht ist, wird diese in der verteilten Visualisierung oft ignoriert oder nicht auf die speziellen Umstände dieses Feldes angepasst. Analyse und Optimierung dieser Systeme und ihrer Software ist notwendig, um deren Zustand einzuschätzen und ihre Leistung zu verbessern. Die verfügbaren Werkzeuge und Methoden, um die erforderlichen Informationen zu sammeln und auszuwerten, sind oft vom Kontext entkoppelt oder nicht für interaktive Szenarien ausgelegt. Diese Probleme sind in heterogenen Rechenumgebungen verstärkt, da dort mehr Daten für die Analyse verfügbar und notwendig sind. Für verteilte Visualisierung ist zusätzlich Rückmeldung in Echtzeit notwendig, um Interaktionen der Benutzer mit Leistungscharakteristika zu korrelieren und um die Gültigkeit und Korrektheit der Daten und ihrer Visualisierung zu entscheiden. Diese Dissertation präsentiert Beiträge für all diese Aspekte. Zunächst werden zwei Ansätze zur Abstraktion im Kontext von generischen Berechnungen auf Grafikprozessoren und Visualisierung in heterogenen Umgebungen untersucht. Der erste Ansatz verbirgt Details verschiedener Prozessoren und ermöglicht deren Nutzung über einheitliche Schnittstellen. Der zweite Ansatz verwendet pro-Pixel verkettete Listen (per-pixel linked lists) zur Kombination von Pixelfarben und zur Vereinfachung von ordnungsunabhängiger Transparenz in verteilter Visualisierung. Übliche Fehlertoleranz-Methoden im Hochleistungsrechnen werden im Kontext der verteilten Visualisierung diskutiert. Auf dieser Grundlage werden Strategien für fehlertolerante verteilte Visualisierung abgeleitet und in einer Taxonomie organisiert. Beispielhafte Umsetzungen dieser Strategien, ihre Kompromisse und Zugeständnisse, und die daraus resultierenden Implikationen werden diskutiert. Zur Analyse werden lokale Exploration von Graphen und die Optimierung von Volumenvisualisierung untersucht. Herausforderungen in dichten Graphen wie visuelle Überladung, Ambiguität und Einbindung zusätzlicher Attribute werden in Knoten-Kanten Diagrammen mit einer Linsenmetapher sowie ergänzenden Ansichten der Daten angegangen. Ein explorativer Ansatz zur Leistungsanalyse und Optimierung paralleler Volumenvisualisierung auf einer großen, hochaufgelösten Anzeige wird untersucht. Diese Dissertation betrachtet zum ersten Mal Fragen der verteilten Visualisierung auf großen Anzeigen und heterogenen Rechenumgebungen in einem größeren Kontext. Während jeder vorgestellte Ansatz individuelle Herausforderungen löst und erfolgreich in diesem Zusammenhang eingesetzt wurde, bilden alle gemeinsam eine solide Basis für künftige Forschung in diesem jungen Feld. In ihrer Gesamtheit präsentiert diese Dissertation Bausteine für robuste verteilte Visualisierung auf aktuellen und künftigen heterogenen Visualisierungsumgebungen