Surface Reconstruction from Scattered Point via RBF Interpolation on GPU

In this paper we describe a parallel implicit method based on radial basis functions (RBF) for surface reconstruction. The applicability of RBF methods is hindered by its computational demand, that requires the solution of linear systems of size equal to the number of data points. Our reconstruction implementation relies on parallel scientific libraries and is supported for massively multi-core architectures, namely Graphic Processor Units (GPUs). The performance of the proposed method in terms of accuracy of the reconstruction and computing time shows that the RBF interpolant can be very effective for such problem.Comment: arXiv admin note: text overlap with arXiv:0909.5413 by other author

Evaluating the Differences of Gridding Techniques for Digital Elevation Models Generation and Their Influence on the Modeling of Stony Debris Flows Routing: A Case Study From Rovina di Cancia Basin (North-Eastern Italian Alps)

Debris \ufb02ows are among the most hazardous phenomena in mountain areas. To cope with debris \ufb02ow hazard, it is common to delineate the risk-prone areas through routing models. The most important input to debris \ufb02ow routing models are the topographic data, usually in the form of Digital Elevation Models (DEMs). The quality of DEMs depends on the accuracy, density, and spatial distribution of the sampled points; on the characteristics of the surface; and on the applied gridding methodology. Therefore, the choice of the interpolation method affects the realistic representation of the channel and fan morphology, and thus potentially the debris \ufb02ow routing modeling outcomes. In this paper, we initially investigate the performance of common interpolation methods (i.e., linear triangulation, natural neighbor, nearest neighbor, Inverse Distance to a Power, ANUDEM, Radial Basis Functions, and ordinary kriging) in building DEMs with the complex topography of a debris \ufb02ow channel located in the Venetian Dolomites (North-eastern Italian Alps), by using small footprint full- waveform Light Detection And Ranging (LiDAR) data. The investigation is carried out through a combination of statistical analysis of vertical accuracy, algorithm robustness, and spatial clustering of vertical errors, and multi-criteria shape reliability assessment. After that, we examine the in\ufb02uence of the tested interpolation algorithms on the performance of a Geographic Information System (GIS)-based cell model for simulating stony debris \ufb02ows routing. In detail, we investigate both the correlation between the DEMs heights uncertainty resulting from the gridding procedure and that on the corresponding simulated erosion/deposition depths, both the effect of interpolation algorithms on simulated areas, erosion and deposition volumes, solid-liquid discharges, and channel morphology after the event. The comparison among the tested interpolation methods highlights that the ANUDEM and ordinary kriging algorithms are not suitable for building DEMs with complex topography. Conversely, the linear triangulation, the natural neighbor algorithm, and the thin-plate spline plus tension and completely regularized spline functions ensure the best trade-off among accuracy and shape reliability. Anyway, the evaluation of the effects of gridding techniques on debris \ufb02ow routing modeling reveals that the choice of the interpolation algorithm does not signi\ufb01cantly affect the model outcomes

Comparison of Current Gravity Estimation and Determination Models

This paper will discuss the history of gravity estimation and determination models while analyzing methods that are in development. Some fundamental methods for calculating the gravity field include spherical harmonics solutions, local weighted interpolation, and global point mascon modeling (PMC). Recently, high accuracy measurements have become more accessible, and the requirements for high order geopotential modeling have become more stringent. Interest in irregular bodies, accurate models of the hydrological system, and on-board processing has demanded a comprehensive model that can quickly and accurately compute the geopotential with low memory costs. This trade study of current geopotential modeling techniques will reveal that each modeling technique has a unique use case. It is notable that the spherical harmonics model is relatively accurate but poses a cumbersome inversion problem. PMC and interpolation models, on the other hand, are computationally efficient, but require more research to become robust models with high levels of accuracy. Considerations of the trade study will suggest further research for the point mascon model. The PMC model should be improved through mascon refinement, direct solutions that stem from geodetic measurements, and further validation of the gravity gradient. Finally, the potential for each model to be implemented with parallel computation will be shown to lead to large improvements in computing time while reducing the memory cost for each technique.Aerospace Engineering and Engineering Mechanic

Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique

Based on the radial basis function (RBF), non-singular general solution and dual reciprocity principle (DRM), this paper presents an inheretnly meshless, exponential convergence, integration-free, boundary-only collocation techniques for numerical solution of general partial differential equation systems. The basic ideas behind this methodology are very mathematically simple and generally effective. The RBFs are used in this study to approximate the inhomogeneous terms of system equations in terms of the DRM, while non-singular general solution leads to a boundary-only RBF formulation. The present method is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of non-singular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does no need to introduce the artificial boundary and results in the symmetric system equations under certain conditions. It is also found that the BKM can solve nonlinear partial differential equations one-step without iteration if only boundary knots are used. The efficiency and utility of this new technique are validated through some typical numerical examples. Some promising developments of the BKM are also discussed.Comment: 36 pages, 2 figures, Welcome to contact me on this paper: Email: [email protected] or [email protected]

Chebyshev interpolation for functions with endpoint singularities via exponential and double-exponential transforms

We present five theorems concerning the asymptotic convergence rates of Chebyshev interpolation applied to functions transplanted to either a semi-infinite or an infinite interval under exponential or double-exponential transformations. This strategy is useful for approximating and computing with functions that are analytic apart from endpoint singularities. The use of Chebyshev polynomials instead of the more commonly used cardinal sinc or Fourier interpolants is important because it enables one to apply maps to semi-infinite intervals for functions which have only a single endpoint singularity. In such cases, this leads to significantly improved convergence rates

Fast and flexible interpolation via PUM with applications in population dynamics

In this paper the Partition of Unity Method (PUM) is efficiently performed using Radial Basis Functions (RBFs) as local approximants. In particular, we present a new space-partitioning data structure extremely useful in applications because of its independence from the problem geometry. Moreover, we study, in the context of wild herbivores in forests, an application of such algorithm. This investigation shows that the ecosystem of the considered natural park is in a very delicate situation, for which the animal population could become extinguished. The determination of the so-called sensitivity surfaces, obtained with the new fast and flexible interpolation tool, indicates some possible preventive measures to the park administrators

A Semi-Lagrangian Scheme with Radial Basis Approximation for Surface Reconstruction

We propose a Semi-Lagrangian scheme coupled with Radial Basis Function interpolation for approximating a curvature-related level set model, which has been proposed by Zhao et al. in \cite{ZOMK} to reconstruct unknown surfaces from sparse, possibly noisy data sets. The main advantages of the proposed scheme are the possibility to solve the level set method on unstructured grids, as well as to concentrate the reconstruction points in the neighbourhood of the data set, with a consequent reduction of the computational effort. Moreover, the scheme is explicit. Numerical tests show the accuracy and robustness of our approach to reconstruct curves and surfaces from relatively sparse data sets.Comment: 14 pages, 26 figure