Spectral, Combinatorial, and Probabilistic Methods in Analyzing and Visualizing Vector Fields and Their Associated Flows

In this thesis, we introduce several tools, each coming from a different branch of mathematics, for analyzing real vector fields and their associated flows. Beginning with a discussion about generalized vector field decompositions, that mainly have been derived from the classical Helmholtz-Hodge-decomposition, we decompose a field into a kernel and a rest respectively to an arbitrary vector-valued linear differential operator that allows us to construct decompositions of either toroidal flows or flows obeying differential equations of second (or even fractional) order and a rest. The algorithm is based on the fast Fourier transform and guarantees a rapid processing and an implementation that can be directly derived from the spectral simplifications concerning differentiation used in mathematics. Moreover, we present two combinatorial methods to process 3D steady vector fields, which both use graph algorithms to extract features from the underlying vector field. Combinatorial approaches are known to be less sensitive to noise than extracting individual trajectories. Both of the methods are extensions of an existing 2D technique to 3D fields. We observed that the first technique can generate overly coarse results and therefore we present a second method that works using the same concepts but produces more detailed results. Finally, we discuss several possibilities for categorizing the invariant sets with respect to the flow. Existing methods for analyzing separation of streamlines are often restricted to a finite time or a local area. In the frame of this work, we introduce a new method that complements them by allowing an infinite-time-evaluation of steady planar vector fields. Our algorithm unifies combinatorial and probabilistic methods and introduces the concept of separation in time-discrete Markov chains. We compute particle distributions instead of the streamlines of single particles. We encode the flow into a map and then into a transition matrix for each time direction. Finally, we compare the results of our grid-independent algorithm to the popular Finite-Time-Lyapunov-Exponents and discuss the discrepancies. Gauss\'' theorem, which relates the flow through a surface to the vector field inside the surface, is an important tool in flow visualization. We are exploiting the fact that the theorem can be further refined on polygonal cells and construct a process that encodes the particle movement through the boundary facets of these cells using transition matrices. By pure power iteration of transition matrices, various topological features, such as separation and invariant sets, can be extracted without having to rely on the classical techniques, e.g., interpolation, differentiation and numerical streamline integration

Computational Electromagnetism and Acoustics

The challenge inherent in the accurate and efficient numerical modeling of wave propagation phenomena is the common grand theme in both computational electromagnetics and acoustics. Many excellent contributions at this Oberwolfach workshop were devoted to this theme and a wide range of numerical techniques and algorithms were mustered to tackle this challenge

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Feature-based Vector Field Representation and Comparison

In recent years, simulations have steadily replaced real world experiments in science and industry. Instead of performing numerous arduous experiments in order to develop new products or test a hypothesis, the system to be examinded is described by a set of equations which are subsequently solved within the simulation. The produced vector fields describe the system's behavior under the conditions of the experiment. While simulations steadily increase in terms of complexity and precision, processing and analysis are still approached by the same long-standing visual techniques. However, these are limited by the capability of the human visual system and its abilities to depict large, multi-dimensional data sets. In this thesis, we replace the visual processing of data in the traditional workflow with an automated, statistical method. Cluster algorithms are able to process large, multi-dimensional data sets efficiently and therefore resolve the limitations we faced so far. For their application to vector fields we define a special feature vector that describes the data comprehensively. After choosing an appropriate clustering method, the vector field is split into its features. Based on these features the novel flow graph is constructed. It serves as an abstract representation of the vector field and gives a detailed description of its parts as well as their relations. This new representation enables a quantitative analysis and describes the input data. Additionally, the flow graphs are comparable to each other through a uniform description, since techniques of graph theory may be applied. In the traditional workflow, visualization is the bottleneck, because it is built manually by the user for a specific data set. In consequence the output is diminished and the results are likely to be biased by the user. Both issues are solved by our approach, because both the feature extraction and the construction of the flow graph are executed in an un-supervised manner. We will compare our newly developed workflow with visualization techniques based on different data sets and discuss the results. The concluding chapter on the similarity and comparison of graphs applies techniques of graph theory and demonstrates the advantages of the developed representation and its use for the analysis of vector fields using flow graphs

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

13th Annual Review of Progress in Applied Computational Electromagnetics at the Naval Postgraduate School, Monterey, CA, March 17-21, 1997, Conference Proceedings Volumes I & II

Includes Volumes 1 &