390 research outputs found

    A meshless interpolation algorithm using a cell-based searching procedure

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    In this paper we propose a fast algorithm for bivariate interpolation of large scattered data sets. It is based on the partition of unity method for constructing a global interpolant by blending radial basis functions as local approximants and using locally supported weight functions. The partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cell-based searching procedure. More precisely, we construct a cell structure, which partitions the domain and strictly depends on the dimension of the subdomains, thus providing a meaningful improvement in the searching process compared to the nearest neighbour searching techniques presented in Allasia et al. (2011) and Cavoretto and De Rossi (2010, 2012). In fact, this efficient algorithm and, in particular, the new searching procedure enable us a fast computation also in several applications, where the amount of data to be interpolated is often very large, up to many thousands or even millions of points. Analysis of computational complexity shows the high efficiency of the proposed interpolation algorithm. This is also supported by numerical experiments

    A trivariate interpolation algorithm using a cube-partition searching procedure

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    In this paper we propose a fast algorithm for trivariate interpolation, which is based on the partition of unity method for constructing a global interpolant by blending local radial basis function interpolants and using locally supported weight functions. The partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cube-partition searching procedure. More precisely, we construct a cube structure, which partitions the domain and strictly depends on the size of its subdomains, so that the new searching procedure and, accordingly, the resulting algorithm enable us to efficiently deal with a large number of nodes. Complexity analysis and numerical experiments show high efficiency and accuracy of the proposed interpolation algorithm

    Partition of Unity Interpolation on Multivariate Convex Domains

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    In this paper we present a new algorithm for multivariate interpolation of scattered data sets lying in convex domains \Omega \subseteq \RR^N, for any N≥2N \geq 2. To organize the points in a multidimensional space, we build a kdkd-tree space-partitioning data structure, which is used to efficiently apply a partition of unity interpolant. This global scheme is combined with local radial basis function approximants and compactly supported weight functions. A detailed description of the algorithm for convex domains and a complexity analysis of the computational procedures are also considered. Several numerical experiments show the performances of the interpolation algorithm on various sets of Halton data points contained in Ω\Omega, where Ω\Omega can be any convex domain like a 2D polygon or a 3D polyhedron

    Two-dimensional interpolation using a cell-based searching procedure

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    In this paper we present an efficient algorithm for bivariate interpolation, which is based on the use of the partition of unity method for constructing a global interpolant. It is obtained by combining local radial basis function interpolants with locally supported weight functions. In particular, this interpolation scheme is characterized by the construction of a suitable partition of the domain in cells so that the cell structure strictly depends on the dimension of its subdomains. This fact allows us to construct an efficient cell-based searching procedure, which provides a significant reduction of CPU times. Complexity analysis and numerical results show such improvements on the algorithm performances

    Fast and flexible interpolation via PUM with applications in population dynamics

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    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

    Efficient computation of partition of unity interpolants through a block-based searching technique

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    In this paper we propose a new efficient interpolation tool, extremely suitable for large scattered data sets. The partition of unity method is used and performed by blending Radial Basis Functions (RBFs) as local approximants and using locally supported weight functions. In particular we present a new space-partitioning data structure based on a partition of the underlying generic domain in blocks. This approach allows us to examine only a reduced number of blocks in the search process of the nearest neighbour points, leading to an optimized searching routine. Complexity analysis and numerical experiments in two- and three-dimensional interpolation support our findings. Some applications to geometric modelling are also considered. Moreover, the associated software package written in \textsc{Matlab} is here discussed and made available to the scientific community

    Probabilistic and parallel algorithms for centroidal Voronoi tessellations with application to meshless computing and numerical analysis on surfaces

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    Centroidal Voronoi tessellations (CVT) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. Such tessellations are of use in very diverse applications, including data compression, clustering analysis, cell biology, territorial behavior of animals, optimal allocation of resources, and grid generation. A detailed review is given in chapter 1. In chapter 2, some probabilistic methods for determining centroidal Voronoi tessellations and their parallel implementation on distributed memory systems are presented. The results of computational experiments performed on a CRAY T3E-600 system are given for each algorithm. These demonstrate the superior sequential and parallel performance of a new algorithm we introduce. Then, new algorithms are presented in chapter 3 for the determination of point sets and associated support regions that can then be used in meshless computing methods. The algorithms are probabilistic in nature so that they are totally meshfree, i.e., they do not require, at any stage, the use of any coarse or fine boundary conforming or superimposed meshes. Computational examples are provided that show, for both uniform and non-uniform point distributions that the algorithms result in high-quality point sets and high-quality support regions. The extensions of centroidal Voronoi tessellations to general spaces and sets are also available. For example, tessellations of surfaces in a Euclidean space may be considered. In chapter 4, a precise definition of such constrained centroidal Voronoi tessellations (CCVT\u27s) is given and a number of their properties are derived, including their characterization as minimizers of a kind of energy. Deterministic and probabilistic algorithms for the construction of CCVT\u27s are presented and some analytical results for one of the algorithms are given. Some computational examples are provided which serve to illustrate the high quality of CCVT point sets. CCVT point sets are also applied to polynomial interpolation and numerical integration on the sphere. Finally, some conclusions are given in chapter 5
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