Grid Points In Space Research Articles

Evolution of a Long-Track Violent Tornado within a Simulated Supercell

Abstract Tornadoes are among nature’s most destructive forces. The most violent, long-lived tornadoes form within supercell thunderstorms. Tornadoes ranked EF4 and EF5 on the Enhanced Fujita scale that exhibit long paths are the least common but most damaging and deadly type of tornado. In this article we describe an ultra-high-resolution (30-m gridpoint spacing) simulation of a supercell that produces a long-track tornado that exhibits instantaneous near-surface storm-relative winds reaching as high as 143 m s−1. The computational framework that enables this work is described, including the Blue Waters supercomputer, the CM1 cloud model, a data management framework built around the HDF5 scientific data format, and the VisIt and Vapor visualization tools. We find that tornadogenesis occurs in concert with processes not clearly seen in previous supercell simulations, including the consolidation of numerous vortices and vorticity patches along the storm’s forward-flank downdraft boundary and the intensification of a feature we call a streamwise vorticity current (SVC), a current of horizontal vorticity that is tilted upward into the storm’s low-level mesocyclone. The SVC is found throughout the genesis and much of the maintenance phase of the tornado, where it appears to help drive the storm’s vigorous low-level updraft. We compare stages of the storm’s maintenance phase to observations. We find that tornado decay occurs rapidly throughout the depth of the tornado and is associated with a weakening of the SVC and the development of a strong rainy downdraft that encircles the tornado, which has moved rearward into the storm’s cold pool.

ACCELERATED FITTING OF STELLAR SPECTRA

ABSTRACT Stellar spectra are often modeled and fitted by interpolating within a rectilinear grid of synthetic spectra to derive the stars’ labels: stellar parameters and elemental abundances. However, the number of synthetic spectra needed for a rectilinear grid grows exponentially with the label space dimensions, precluding the simultaneous and self-consistent fitting of more than a few elemental abundances. Shortcuts such as fitting subsets of labels separately can introduce unknown systematics and do not produce correct error covariances in the derived labels. In this paper we present a new approach—Convex Hull Adaptive Tessellation (chat)—which includes several new ideas for inexpensively generating a sufficient stellar synthetic library, using linear algebra and the concept of an adaptive, data-driven grid. A convex hull approximates the region where the data lie in the label space. A variety of tests with mock data sets demonstrate that chat can reduce the number of required synthetic model calculations by three orders of magnitude in an eight-dimensional label space. The reduction will be even larger for higher dimensional label spaces. In chat the computational effort increases only linearly with the number of labels that are fit simultaneously. Around each of these grid points in the label space an approximate synthetic spectrum can be generated through linear expansion using a set of “gradient spectra” that represent flux derivatives at every wavelength point with respect to all labels. These techniques provide new opportunities to fit the full stellar spectra from large surveys with 15–30 labels simultaneously.

Mapped Grid Methods Applied to the Slow Variable Discretization--Enhanced Renormalized Numerov Approach.

We introduce a hyperspherical coordinate mapping procedure to the slow variable discretization-enhanced renormalized Numerov method that allows for more accurate and cost-effective calculations of cold and ultracold atom-dimer scattering. The mapping procedure allows optimization of the numerical grid point spacing by adjusting to the shape of the interaction potential. We show results for elastic scattering in HeH2 and compare the results to previous MOLSCAT calculations by Forrey et al. [ Phys. Rev. A 1999, 59, 2146 ].

The numerical simulation of the tempered fractional Black–Scholes equation for European double barrier option

In recent years, the Finite Moment Log Stable(FMLS), KoBoL and CGMY models, which follow a jump process or a Lévy process, have become the most popular modeling frameworks in the financial field because they can capture some of the important characteristics in the dynamic process of stock price changes, such as large movements or jumps over small time steps. In this paper, we consider the numerical simulation of these three models. We construct a discrete implicit numerical scheme with second order accuracy, and provide a stability and convergence analysis of the numerical scheme. Furthermore, a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) is used to reduce the storage space from O(M2) to O(M) and the computational cost from O(M3) to O(Mlog M) per iteration, where M is the number of space grid points. Some numerical examples are chosen in order to demonstrate the accuracy and efficiency of the proposed method and technique. Finally, as an application, we use the above numerical technique to price a European double-knock-out barrier option, and then the characteristics of the three fractional Black–Scholes (B–S) models are analyzed through comparison with the classical B–S model.

Geographical model-derived grid-based directional routing for massively dense WSNs

This paper presents the grid-based directional routing algorithms for massively dense wireless sensor networks. These algorithms have their theoretical foundation in numerically solving the minimum routing cost problems, which are formulated as continuous geodesic problems via the geographical model. The numerical solutions provide the routing directions at equally spaced grid points in the region of interest, and then, the directions can be used as guidance to route information. In this paper, we investigate two types of routing costs, position-only-dependent costs (e.g., hops, throughput, or energy) and traffic-proportional costs (which correspond to energy-load-balancing). While position-only-dependent costs can be approached directly from geodesic problems, traffic-proportional costs are more easily tackled by transforming the geodesic problem into a set of equations with regard to the routing vector field. We also investigate two numerical approaches for finding the routing direction, the fast marching method for position-only-dependent costs and the finite element method (and its derived distributed algorithm, Gauss-Seidel iteration with finite element method (DGSI-FEM)) for traffic-proportional costs. Finally, we present the numerical results to demonstrate the quality of the derived routing directions.

Black Holes and the Nature of the Expansion of the Universe

Black Holes absorb matter and because matter has structure in space and time, Black Holes absorb the grid points of space and time. It is this property that creates a void for the remaining visible space-time to expand into. This expansion is roughly proportional to the density of the black holes in space and this, because also time is expanding, leads to an apparent exponential expansion for the observer, who is unaware of this. Further the Quantum Mechanical effects: the Heisenberg uncertainty principle and the Tunnelling of particles lead to a flow of matter from the present time into the past. The red-shift is thus not just a sign of an expanding universe but also (and perhaps all of it) that of matter hurling down into the black holes.

Block-structured grids for Eulerian gyrokinetic simulations

In order to predict the turbulent transport in magnetic fusion experiments, global (i.e., full-torus) gyrokinetic simulations are often carried out. In this context, one frequently encounters situations in which the plasma temperature varies by large factors across the radial simulation domain. In grid-based Eulerian codes, this enforces the use of a very large number of grid points in the two-dimensional velocity space, and, thus, an enormous computational effort. To minimize the computational requirements, one may employ block-structured grids, adapted to the radial changes of the temperature. As the block-structured grids rely on a general approach, they can be applied to different Eulerian gyrokinetic implementations. In this paper, we explain the construction and implementation of such grids in the gyrokinetic code GENE, F. Jenko et al. (2000), and present corresponding simulation results.

A New Integral Representation of the Coverage Probability of a Random Convex Hull

In this paper, the probability that a given point is covered by a random convex hull generated by independent and identically-distributed random points in a plane is studied. It is shown that such probability can be expressed in terms of an integral that can be approximated numerically by function-evaluations over the grid-points in a 2-dimensional space. The new integral representation allows such probability be computed efficiently. The computational burdens under the proposed integral representation and those in the existing literature are compared. The proposed method is illustrated through numerical examples where the random points are drawn from (i) uniform distribution over a square and (ii) bivariate normal distribution over the two-dimensional Euclidean space. The applications of the proposed method in statistics are are discussed.

Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science. The computational complexities of time fractional, space fractional, and space-time fractional equations areO(N2M),O(NM2), andO(NM(M+N)) compared withO(MN) for the classical partial differential equations with finite difference methods, whereM,Nare the number of space grid points and time steps. The potential solutions for this challenge include, but are not limited to, parallel computing, memory access optimization (fractional precomputing operator), short memory principle, fast Fourier transform (FFT) based solutions, alternating direction implicit method, multigrid method, and preconditioner technology. The relationships of these solutions for both space fractional derivative and time fractional derivative are discussed. The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcome this challenge, and some attention should be paid to the fractional killer applications, high performance iteration methods, high order schemes, and Monte Carlo methods. Since the computation of fractional equations with high dimension and variable order is even heavier, the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in the area of fractional calculus.

Increasing horizontal resolution in numerical weather prediction and climate simulations: illusion or panacea?

The steady path of doubling the global horizontal resolution approximately every 8 years in numerical weather prediction (NWP) at the European Centre for Medium Range Weather Forecasts may be substantially altered with emerging novel computing architectures. It coincides with the need to appropriately address and determine forecast uncertainty with increasing resolution, in particular, when convective-scale motions start to be resolved. Blunt increases in the model resolution will quickly become unaffordable and may not lead to improved NWP forecasts. Consequently, there is a need to accordingly adjust proven numerical techniques. An informed decision on the modelling strategy for harnessing exascale, massively parallel computing power thus also requires a deeper understanding of the sensitivity to uncertainty--for each part of the model--and ultimately a deeper understanding of multi-scale interactions in the atmosphere and their numerical realization in ultra-high-resolution NWP and climate simulations. This paper explores opportunities for substantial increases in the forecast efficiency by judicious adjustment of the formal accuracy or relative resolution in the spectral and physical space. One path is to reduce the formal accuracy by which the spectral transforms are computed. The other pathway explores the importance of the ratio used for the horizontal resolution in gridpoint space versus wavenumbers in spectral space. This is relevant for both high-resolution simulations as well as ensemble-based uncertainty estimation.

Efficient implementation of the three-dimensional reference interaction site model method in the fragment molecular orbital method.

The three-dimensional reference interaction site model (3D-RISM) method was efficiently implemented in the fragment molecular orbital (FMO) method. The method is referred to as the FMO/3D-RISM method, and allows us to treat electronic structure of the whole of a macromolecule, such as a protein, as well as the solvent distribution around a solute macromolecule. The formalism of the FMO/3D-RISM method, for the computationally available form and variational expressions, are proposed in detail. A major concern leading to the implementation of the method was decreasing the computational costs involved in calculating the electrostatic potential, because the electrostatic potential is calculated on numerous grid points in three-dimensional real space in the 3D-RISM method. In this article, we propose a procedure for decreasing the computational costs involved in calculating the electrostatic potential in the FMO method framework. The strategy involved in this procedure is to evaluate the electrostatic potential and the solvated Fock matrix in different manners, depending on the distance between the solute and the solvent. The electrostatic potential is evaluated directly in the vicinity of the solute molecule by integrating the molecular orbitals of monomer fragments of the solute molecule, whereas the electrostatic potential is described as the sum of multipole interactions when an analog of the fast multipole method is used. The efficiency of our method was demonstrated by applying it to a water trimer system and three biomolecular systems. The FMO/3D-RISM calculation can be performed within a reasonable computational time, retaining the accuracy of some physical properties.

A Gaussian graphical model approach to climate networks.

Distinguishing between direct and indirect connections is essential when interpreting network structures in terms of dynamical interactions and stability. When constructing networks from climate data the nodes are usually defined on a spatial grid. The edges are usually derived from a bivariate dependency measure, such as Pearson correlation coefficients or mutual information. Thus, the edges indistinguishably represent direct and indirect dependencies. Interpreting climate data fields as realizations of Gaussian Random Fields (GRFs), we have constructed networks according to the Gaussian Graphical Model (GGM) approach. In contrast to the widely used method, the edges of GGM networks are based on partial correlations denoting direct dependencies. Furthermore, GRFs can be represented not only on points in space, but also by expansion coefficients of orthogonal basis functions, such as spherical harmonics. This leads to a modified definition of network nodes and edges in spectral space, which is motivated from an atmospheric dynamics perspective. We construct and analyze networks from climate data in grid point space as well as in spectral space, and derive the edges from both Pearson and partial correlations. Network characteristics, such as mean degree, average shortest path length, and clustering coefficient, reveal that the networks posses an ordered and strongly locally interconnected structure rather than small-world properties. Despite this, the network structures differ strongly depending on the construction method. Straightforward approaches to infer networks from climate data while not regarding any physical processes may contain too strong simplifications to describe the dynamics of the climate system appropriately.

A fast algorithm for the energy space boson Boltzmann collision operator

This paper introduces a fast algorithm for the energy space boson Boltzmann collision operator. Compared to the direct O ( N 3 ) O(N^3) calculation and the previous O ( N 2 log ⁡ N ) O(N^2\log N) method [Markowich and Pareschi, 2005], the new algorithm runs in complexity O ( N log 2 ⁡ N ) O(N\log ^2N) , which is optimal up to a logarithmic factor ( N N is the number of grid points in energy space). The basic idea is to partition the 3-D summation domain recursively into elementary shapes so that the summation within each shape becomes a special double convolution that can be computed efficiently by the fast Fourier transform. Numerical examples are presented to illustrate the efficiency and accuracy of the proposed algorithm.

A method to determine optimal grid parameters based on characteristic analysis of discrete data distribution

We evaluated a method that determines the optimal grid parameters of potential-field data such as the intervals of grid nodes and the grid azimuth. Optimal grid parameters produced the best gridding results with the least amount of data distortions. We used spectral analysis of the data statistics on azimuth projections. The direction with the largest degree of periodicity was defined as the optimal azimuth of the grid. The grid line interval was determined according to the frequency point with the largest amplitude value. By spectral analysis of the data sequence projected in the direction normal to the survey lines, grid point spacing was also determined. Correlative analysis of the template matching method was used to determine the optimal grid range. Tests on simulated and field data proved that this method is rather robust and can be incorporated into gridding software packages.

Emulation to simulate low-resolution atmospheric data

Climate model development, testing, and analysis involve running the model extensively to tune the subgrid- scale parameters that provide closure to the system. This process demands substantial time and computational resources even for typical spatial resolutions and becomes in feasibly expensive for high-resolution studies. This paper presents alternative, computationally feasible methods to emulate the simulations within acceptable error bounds. This strategy can be easily implemented to obtain an ensemble of model runs. The paper outlines three approximation strategies: (1) interpolation with Lagrange basis functions, (2) least-squares (LS) approximation, and (3) interpolation with radial basis functions. As a proof of concept, a suite of relevant physical quantities are evaluated at unknown grid points of parameters, space, and time. The values obtained by emulation are compared against the simulated values to check the feasibility of the method. The advantages and shortcomings of the above-mentioned approximation schemes are discussed, including the savings of time and computational resources.

Vlasov on GPU (VOG project)

This work concerns the numerical simulation of the Vlasov-Poisson equation using semi-Lagrangian methods on Graphics Processing Units (GPU). To accomplish this goal, modifications to traditional methods had to be implemented. First and foremost, a reformulation of semi-Lagrangian methods is performed, which enables us to rewrite the governing equations as a circulant matrix operating on the vector of unknowns. This product calculation can be performed efficiently using FFT routines. Nowadays GPU is no more limited to single precision; however, single precision may still be preferred with respect to performance and available memory. So, in order to be able to deal with single precision, a δf type method is adopted which only needs refinement in specialized areas of phase space but not throughout. Thus, a GPU Vlasov-Poisson solver can indeed perform high precision simulations (since it uses very high order of reconstruction and a large number of grid points in phase space). We show results for more academic test cases and also for physically relevant phenomena such as the bump on tail instability and the simulation of Kinetic Electrostatic Electron Nonlinear (KEEN) waves.

A pseudo‐spectral method that uses an overlapping multidomain technique for the numerical solution of sine‐Gordon equation in one and two spatial dimensions

In this article, we study an explicit scheme for the solution of sine‐Gordon equation when the space discretization is carried out by an overlapping multidomain pseudo‐spectral technique. By using differentiation matrices, the equation is reduced to a nonlinear system of ordinary differential equations in time that can be discretized with the explicit fourth‐order Runge–Kutta method. To achieve approximation with high accuracy in large domains, the number of space grid points must be large enough. This yields very large and full matrices in the pseudo‐spectral method that causes large memory requirements. The domain decomposition approach provides sparsity in the matrices obtained after the discretization, and this property reduces storage for large matrices and provides economical ways of performing matrix–vector multiplications. Therefore, we propose a multidomain pseudo‐spectral method for the numerical simulation of the sine‐Gordon equation in large domains. Test examples are given to demonstrate the accuracy and capability of the proposed method. Numerical experiments show that the multidomain scheme has an excellent long‐time numerical behavior for the sine‐Gordon equation in one and two dimensions. Copyright © 2013 John Wiley & Sons, Ltd.

TU‐C‐141‐02: Development of a Hyrbid Biomechanical Model Based Deformable Image Registration; Application in Lung

Purpose: To investigate the reduction in maximum errors in 4D CT lung Deformable Image Registration (DIR) using a hybrid biomechanical model based algorithm with an intensity based registration refinement. Methods: Morfeus, a biomechanical modeling algorithm for deformable image registration was used to register 3D inhale to exhale images of the lungs acquired via 4DCT. The predicted exhale image was obtained by interpolation. An intensity based registration algorithm based on Sum of Absolute Differences (SAD) and BSpline free form deformations (drop3D, Munich, Germany) was then employed to refine the results by a second registration from the Morfeus predicted exhale to the actual exhale image. The BSpline grid point spacing for the refinement was set to 9mm. To measure the accuracy, Target Registration Error (TRE) was calculated based on the Euclidean distance of corresponding bifurcation points on both images. Improvements in the overall TRE as well as reduction in maximum errors were evaluated. Results: The hybrid method has been evaluated on 5 patients to date with 45 to 76 identified bifurcation points per patient. The Morfeus TRE ranged from 2.2±1.0 mm (mean±SD) to 3.6±1.9 mm. This was reduced to 1.4±0.8 mm up to 1.6±2.0 mm with the hybrid method. More importantly, the number of bifurcation points (% of total points) with an error greater than 5 mm was reduced from 54 (22%) points (across all patients, range 3 to 24 points per patient) with Morfeus to 10 (4%) points (range 0 to 4 points per patient) with the hybrid method. Conclusion: The proposed hybrid method combines the flexibility of the biomechanical model based algorithm, including the ability to model the sliding interface between the lungs and the chest wall, with the advantages of intensity based registration for the contrast rich lung images to provide a highly accurate DIR algorithm. This research is supported by the NIH 5RO1CA124714‐02. KK Brock has financial interest in the deformable registration technology through a licensing agreement with RaySearch Laboratories.

A kinematic classification of the cosmic web

A new approach for the classification of the cosmic web is presented. In extension of the previous work of Hahn et al. (2007) and Forero-Romero et al. (2009) the new algorithm is based on the analysis of the velocity shear tensor rather than the gravitational tidal tensor. The procedure consists of the construction of the the shear tensor at each (grid) point in space and the evaluation of its three eigenvectors. A given point is classified to be either a void, sheet, filament or a knot according to the number of eigenvalues above a certain threshold, 0, 1, 2, or 3 respectively. The threshold is treated as a free parameter that defines the web. The algorithm has been applied to a dark matter only, high resolution simulation of a box of side-length 64$h^{-1}$Mpc and N = $1024^3$ particles with the framework of the WMAP5/LCDM model. The resulting velocity based cosmic web resolves structures down to <0.1$h^{-1}$Mpc scales, as opposed to the ~1$h^{-1}$Mpc scale of the tidal based web. The under-dense regions are made of extended voids bisected by planar sheets, whose density is also below the mean. The over-dense regions are vastly dominated by the linear filaments and knots. The resolution achieved by the velocity based cosmic web provides a platform for studying the formation of halos and galaxies within the framework of the cosmic web.

Comparison and Druggability Prediction of Protein–Ligand Binding Sites from Pharmacophore-Annotated Cavity Shapes

Estimating the pairwise similarity of protein-ligand binding sites is a fast and efficient way of predicting cross-reactivity and putative side effects of drug candidates. Among the many tools available, three-dimensional (3D) alignment-dependent methods are usually slow and based on simplified representations of binding site atoms or surfaces. On the other hand, fast and efficient alignment-free methods have recently been described but suffer from a lack of interpretability. We herewith present a novel binding site description (VolSite), coupled to an alignment and comparison tool (Shaper) combining the speed of alignment-free methods with the interpretability of alignment-dependent approaches. It is based on the comparison of negative images of binding cavities encoding both shape and pharmacophoric properties at regularly spaced grid points. Shaper approximates the resulting molecular shape with a smooth Gaussian function and aligns protein binding sites by optimizing their volume overlap. Volsite and Shaper were successfully applied to compare protein-ligand binding sites and to predict their structural druggability.