Annals of mathematics studies

With respect to space-time tensor-product wavelet bases, parabolic initial boundary value problems are equivalently formulated as bi-infinite matrix problems. Adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate. In case the spatial domain is of product type, the use of spatial tensor product wavelet bases is proved to overcome the so-called curse of dimensionality, i.e., the reduction of the convergence rate with increasing spatial dimension.

https://www.hcm.uni-bonn.de/fileadmin/user_upload/sparcity_and_computation/Stevenson.pdf

Space-time adaptive wavelet methods for parabolic evolution problems

Flow visualization research has made rapid advances in recent years, especially in the area of topology-based flow visualization. The ever increasing size of scientific data sets favors algorithms that are capable of extracting important subsets of the data, leaving the scientist with a more manageable representation that may be visualized interactively. Extracting the topology of a flow achieves the goal of obtaining a compact representation of a vector or tensor field while simultaneously retaining its most important features. We present the state of the art in topology-based flow visualization techniques. We outline numerous topology-based algorithms categorized according to the type and dimensionality of data on which they operate and according to the goal-oriented nature of each method. Topology tracking algorithms are also discussed. The result serves as a useful introduction and overview to research literature concerned with the study of topology-based flow visualization.

/pdf/topology-based-flow-visualization-the-state-of-the-art-5dm3tq5bsk.pdf

Topology-Based Flow Visualization, The State of the Art

This paper presents a method for filtered ridge extraction based on adaptive mesh refinement. It is applicable in situations where the underlying scalar field can be refined during ridge extraction. This requirement is met by the concept of Lagrangian coherent structures which is based on trajectories started at arbitrary sampling grids that are independent of the underlying vector field. The Lagrangian coherent structures are extracted as ridges in finite Lyapunov exponent fields computed from these grids of trajectories. The method is applied to several variants of finite Lyapunov exponents, one of which is newly introduced. High computation time due to the high number of required trajectories is a main drawback when computing Lyapunov exponents of 3-dimensional vector fields. The presented method allows a substantial speed-up by avoiding the seeding of trajectories in regions where no ridges are present or do not satisfy the prescribed filter criteria such as a minimum finite Lyapunov exponent.

/pdf/efficient-visualization-of-lagrangian-coherent-structures-by-kfpd2u5j5i.pdf

Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction

Feature tracking algorithms for instationary vector fields are usually based on a correspondence analysis of the features at different time steps. This paper introduces a method for feature tracking which is based on the integration of stream lines of a certain vector field called feature flow field. We analyze for which features the method of feature flow fields can be applied, we show how events in the flow can be detected using feature flow fields, and we show how to construct the feature flow fields for particular classes of features. Finally, we apply the technique to track critical points in a 2D instationary vector field.

Feature Flow Fields

While vortex region quantities are Galilean invariant, most methods for extracting vortex cores depend on the frame of reference. We present an approach to extracting vortex core lines independently of the frame of reference by extracting ridge and valley lines of Galilean invariant vortex region quantities. We discuss a generalization of this concept leading to higher dimensional features. For the visualization of extracted line features we use an iconic representation indicating their scale and extent. We apply our approach to datasets from numerical simulations and experimental measurements.

/pdf/galilean-invariant-extraction-and-iconic-representation-of-4nfr3cbi8y.pdf

Galilean invariant extraction and iconic representation of vortex core lines

In nature and in flow experiments particles form patterns of swirling motion in certain locations. Existing approaches identify these structures by considering the behavior of stream lines. However, in unsteady flows particle motion is described by path lines which generally gives different swirling patterns than stream lines. We introduce a novel mathematical characterization of swirling motion cores in unsteady flows by generalizing the approach of Sujudi/Haimes to path lines. The cores of swirling particle motion are lines sweeping over time, i.e., surfaces in the space-time domain. They occur at locations where three derived 4D vectors become coplanar. To extract them, we show how to re-formulate the problem using the parallel vectors operator. We apply our method to a number of unsteady flow fields.

/pdf/cores-of-swirling-particle-motion-in-unsteady-flows-35nx4g6kt8.pdf

Cores of Swirling Particle Motion in Unsteady Flows

We present an approach to analyze mixing in flow fields by extracting vortex and strain features as extremal structures of derived scalar quantities that satisfy a duality property: They indicate vortical as well as high-strain (saddle-type) regions. Specifically, we consider the Okubo-Weiss criterion and the recently introduced MZ criterion. Although the first is derived from a purely Eulerian framework, the latter is based on Lagrangian considerations. In both cases, high values indicate vortex activity, whereas low values indicate regions of high strain. By considering the extremal features of those quantities, we define the notions of a vortex and a strain skeleton in a hierarchical manner: The collection of maximal zero-dimensional, one-dimensional, and 2D structures assemble the vortex skeleton; the minimal structures identify the strain skeleton. We extract those features using scalar field topology and apply our method to a number of steady and unsteady 3D flow fields.

Vortex and Strain Skeletons in Eulerian and Lagrangian Frames

We introduce an approach to tracking vortex core lines in time-dependent 3D flow fields which are defined by the parallel vectors approach. They build surface structures in the 4D space-time domain. To extract them, we introduce two 4D vector fields which act as feature flow fields, i.e., their integration gives the vortex core structures. As part of this approach, we extract and classify local bifurcations of vortex core lines in space-time. Based on a 4D stream surface integration, we provide an algorithm to extract the complete vortex core structure. We apply our technique to a number of test data sets.

Extraction of parallel vector surfaces in 3D time-dependent fields and applications to vortex core line tracking

http://domino.mpi-inf.mpg.de/intranet/ag4/ag4publ.nsf/a58289c9ff876be5c125675300686234/a60b02a24f4d8a3fc1257050003f02f0/$file/ag4_003.pdf

Jan Sahner

Papers

Galilean invariant extraction and iconic representation of vortex core lines

Cores of Swirling Particle Motion in Unsteady Flows

Vortex and Strain Skeletons in Eulerian and Lagrangian Frames

Extraction of parallel vector surfaces in 3D time-dependent fields and applications to vortex core line tracking

Extraction of parallel vector surfaces in 3D time-dependent fields and application to vortex core line tracking