Algebraic Multilevel Methods for Markov Chains

A new algebraic multilevel algorithm for computing the second eigenvector of a column-stochastic matrix is presented. The method is based on a deflation approach in a multilevel aggregation framework. In particular a square and stretch approach, first introduced by Treister and Yavneh, is applied. The method is shown to yield good convergence properties for typical example problems

Regular map smoothing

A regular map is a family of equivalent polygons, glued together to form a closed surface without boundaries which is vertex, edge and face transitive. The commonly known regular maps are derived from the Platonic solids and some tessellations of the torus. There are also regular maps of genus greater than 1 which are traditionally viewed as finitely generated groups. RMS (Regular Map Smoothing) is a tool for visualizing a geometrical realization of such a group either as a cut-out in the hyperbolic space or as a compact surface in 3−space. It provides also a tool to make the resulting regular map more appealing than before. RMS achieves that by the use of a coloring scheme based on coset enumeration, a Catmull-Clark smoothing scheme and a force-directed algorithm with topology preservation

Surface Denoising based on Normal Filtering in a Robust Statistics Framework

During a surface acquisition process using 3D scanners, noise is inevitable and an important step in geometry processing is to remove these noise components from these surfaces (given as points-set or triangulated mesh). The noise-removal process (denoising) can be performed by filtering the surface normals first and by adjusting the vertex positions according to filtered normals afterwards. Therefore, in many available denoising algorithms, the computation of noise-free normals is a key factor. A variety of filters have been introduced for noise-removal from normals, with different focus points like robustness against outliers or large amplitude of noise. Although these filters are performing well in different aspects, a unified framework is missing to establish the relation between them and to provide a theoretical analysis beyond the performance of each method. In this paper, we introduce such a framework to establish relations between a number of widely-used nonlinear filters for face normals in mesh denoising and vertex normals in point set denoising. We cover robust statistical estimation with M-smoothers and their application to linear and non-linear normal filtering. Although these methods originate in different mathematical theories - which include diffusion-, bilateral-, and directional curvature-based algorithms - we demonstrate that all of them can be cast into a unified framework of robust statistics using robust error norms and their corresponding influence functions. This unification contributes to a better understanding of the individual methods and their relations with each other. Furthermore, the presented framework provides a platform for new techniques to combine the advantages of known filters and to compare them with available methods

Weaving patterns inspired by the pentagon snub subdivision scheme

Various computer simulations regarding, e.g., the weather or structural mechanics, solve complex problems on a two-dimensional domain. They mostly do so by splitting the input domain into a finite set of smaller and simpler elements on which the simulation can be run fast and efficiently. This process of splitting can be automatized by using subdivision schemes. Given the wide range of simulation problems to be tackled, an equally wide range of subdivision schemes is available. They create subdivisions that are (mainly) comprised of triangles, quadrilaterals, or hexagons. Furthermore, they ensure that (almost) all vertices have the same number of neighboring vertices. This paper illustrates a subdivision scheme that splits the input domain into pentagons. Repeated application of the scheme gives rise to fractal-like structures. Furthermore, the resulting subdivided domain admits to certain weaving patterns. These patterns are subsequently generalized to several other subdivision schemes. As a final contribution, we provide paper models illustrating the weaving patterns induced by the pentagonal subdivision scheme. Furthermore, we present a jigsaw puzzle illustrating both the subdivision process and the induced weaving pattern. These transform the visual and abstract mathematical algorithms into tactile objects that offer exploration possibilities aside from the visual.Comment: Submitted for publication to the Journal of Mathematics and the Arts (2022

Extraction and Visualization of Swirl and Tumble Motion from Engine Simulation Data

Figure 1: Unsteady visualization of vortices from in-cylinder tumble motion in a gas engine and its relationship to the boundary. During the valve cycle (left to right), the piston head that shapes the bottom of the geometry moves down (not shown). The volume rendering shows vortices using a two-dimensional transfer function of λ2 and normalized helicity (legend). The main tumble vortex is extracted and visible as off-center and with an undesired diagonal orientation. The flow structure on the boundary is visualized using boundary topology. A direct correspondence between the volume and boundary visualizations can be observed. In the third image, the intersection of the main vortex with the boundary results in critical points on the front and back walls. Optimizing the combustion process within an engine block is central to the performance of many motorized vehicles. Associated with this process are two important patterns of flow: swirl and tumble motion, which optimize the mixing of fluid within each of an engine’s cylinders. Good visualizations are necessary to analyze the simulation data of these in-cylinder flows. We present a range of methods including integral, feature-based, and imagebased schemes with the goal of extracting and visualizing these tw