Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph

A standard way to approximate the distance between any two vertices $p$ and $q$ on a mesh is to compute, in the associated graph, a shortest path from $p$ to $q$ that goes through one of $k$ sources, which are well-chosen vertices. Precomputing the distance between each of the $k$ sources to all vertices of the graph yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources, which has been used extensively and successfully for isometry-invariant surface processing, is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources. In this paper, we analyze the stretch factor $\mathcal{F}_{FPS}$ of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that $\mathcal{F}_{FPS}$ can be bounded in terms of the minimal value $\mathcal{F}^*$ of the stretch factor obtained using an optimal placement of $k$ sources as $\mathcal{F}_{FPS}\leq 2 r_e^2 \mathcal{F}^*+ 2 r_e^2 + 8 r_e + 1$, where $r_e$ is the ratio of the lengths of the longest and the shortest edges of the graph. This provides some evidence explaining why farthest point sampling has been used successfully for isometry-invariant shape processing. Furthermore, we show that it is NP-complete to find $k$ sources that minimize the stretch factor.Comment: 13 pages, 4 figure

Interference and k-point sampling in the supercell approach to phase-coherent transport

We present a systematic study of interference and k-point sampling effects in the supercell approach to phase-coherent electron transport. We use a representative tight-binding model to show that interference between the repeated images is a small effect compared to the error introduced by using only the Gamma-point for a supercell containing (3,3) sites in the transverse plane. An insufficient k-point sampling can introduce strong but unphysical features in the transmission function which can be traced to the presence of van Hove singularities in the lead. We present a first-principles calculation of the transmission through a Pt contact which shows that the k-point sampling is also important for realistic systems.Comment: 4 pages, 5 figures. Accepted for Phys. Rev. B (Brief Report

Simple and efficient way of speeding up transmission calculations with kk-point sampling

The transmissions as functions of energy are central for electron or phonon transport in the Landauer transport picture. We suggest a simple and computationally "cheap" post-processing scheme to interpolate transmission functions over $k$-points to get smooth well-converged average transmission functions. This is relevant for data obtained using typical "expensive" first principles calculations where the leads/electrodes are described by periodic boundary conditions. We show examples of transport in graphene structures where a speed-up of an order of magnitude is easily obtained.Comment: 6 pages, 4 figure

Operator Bias with Optical Wedges in Point Sampling

Point sampling, or sampling with probability proportional to size (PPS), is an efficient technique often used in timber cruising, but care must be taken to avoid operator bias

Forestry Bulletin No. 6: Point-Sampling from Two Angles

Point-sampling is a valuable tool in the kit of the practicing forester. It is employed for permanent and temporary sampling and for growth studies. Since the concept of point-sampling is new in American forestry, different approaches have been employed to explain its application.https://scholarworks.sfasu.edu/forestrybulletins/1023/thumbnail.jp

Carbon nanotube, graphene, nanowire, and molecule-based electron and spin transport phenomena using the non-equilibrium Green function method at the level of first principles theory

Based on density functional theory (DFT), we have developed algorithms and a program code to investigate the electron transport characteristics for a variety of nanometer scaled devices in the presence of an external bias voltage. We employed basis sets comprised of linear combinations of numerical type atomic orbitals and k-point sampling for the realistic modeling of the bulk electrode. The scheme coupled with the matrix version of the non-equilibrium Green function method enables determination of the transmission coefficients at a given energy and voltage in a self-consistent manner, as well as the corresponding current-voltage (I-V) characteristics. This scheme has advantages because it is applicable to large systems, easily transportable to different types of quantum chemistry packages, and extendable to describe time-dependent phenomena or inelastic scatterings. It has been applied to diverse types of practical electronic devices such as carbon nanotubes, graphene nano-ribbons, metallic nanowires, and molecular electronic devices. The quantum conductance phenomena for systems involving quantum point contacts and I-V curves are described for the dithiol-benzene molecule in contact with two Au electrodes using the k-point sampling method.Comment: 20 pages, 14 figures. submitte

Hydroacoustic Survey and Point Sampling of Macrophytes In Diamond Lake 2009

MaxDepth Aquatics, Inc. was contracted to conduct a hydroacoustic survey of macrophyte distribution in Diamond Lake in 2009. The survey essentially repeated surveys conducted in 2002 and 2007, allowing for a detailed assessment of conditions in 2009 and comparisons among previous years. In addition, Portland State University was contracted to conduct a depth stratified random point sample survey of macrophyte species presence and absence. The point sample survey was similar to surveys conducted in 2005 and 2007. The results of the 2009 hydroacoustic survey showed that macrophytes were widely distributed throughout the lake at depths less than 8 meters, although some shorter aggregations of macrophytes were found at depths down to 14 meters. Average canopy height corresponded closely to macrophyte density in 2009. The recent survey showed that macrophytes had extended deeper throughout the lake compared to 2002 and 2007 and that canopy height had increased substantially in some locations. The 2009 distribution showed that recolonization of the near shore areas was proceeding, albeit at a relatively slow pace since the lake drawdown completed in 2006. The maximum density of macrophytes in 2009 was found between 4 to 6 meters. Five macrophyte species, one macroalgal species, and filamentous algae were present in the 2009 random point survey. The occurrence rates of the macrophytes Elodea canadensis, Ceratophyllum demersum, and Potamogeton praelongus in 2009 were similar to 2005 and 2007 while Potamogeton pusillus occurrence increased and Myriophyllum verticillatum decreased. Macrophytes were present in a few samples greater than 9 meters; however, biomass, as measured by the fullness of a sampling rake, was highest between 2 and 6 meters. The comparison of the grab sampling conducted in August with the hydroacoustic survey in early September showed poor correspondence in macrophyte density obtained by the two methods. This is likely due to differences in spatial scales of collected samples (10 m2 grid for hydroacoustics compared to \u3c 1 m2 grab samples), comparison of a continuous analytic tool (hydroacoustic) versus an ordinal ranking of density (rake), and possibly some changes in the macrophytes community between the two sampling dates