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Adaptive thinning for bivariate scattered data
 J. Comput. Appl. Math
, 2002
"... Abstract: This paper studies adaptive thinning strategies for approximating a large set of scattered data by piecewise linear functions over triangulated subsets. Our strategies depend on both the locations of the data points in the plane, and the values of the sampled function at these points — ada ..."
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Cited by 23 (6 self)
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Abstract: This paper studies adaptive thinning strategies for approximating a large set of scattered data by piecewise linear functions over triangulated subsets. Our strategies depend on both the locations of the data points in the plane, and the values of the sampled function at these points — adaptive thinning. All our thinning strategies remove data points one by one, so as to minimize an estimate of the error that results by the removal of a point from the current set of points (this estimate is termed ”anticipated error”). The thinning process generates subsets of ”most significant ” points, such that the piecewise linear interpolants over the Delaunay triangulations of these subsets approximate progressively the function values sampled at the original scattered points, and such that the approximation errors are small relative to the number of points in the subsets. We design various methods for computing the anticipated error at reasonable cost, and compare and test the performance of the methods. It is proved that for data sampled from a convex function, with the strategy of convex triangulation, the actual error is minimized by minimizing the best performing measure of anticipated error. It is also shown that for data sampled from certain quadratic polynomials, adaptive thinning is equivalent to thinning which depends only on the locations of the data points — nonadaptive thinning. Based on our numerical tests and comparisons, two practical adaptive thinning algorithms are proposed for thinning large data sets, one which is more accurate and another which is faster. Key words: thinning, adaptive data reduction, simplification, bivariate scattered data, triangulation, piecewise linear approximation
Nested Latin Hypercube Design
 Biometrika
, 2009
"... We propose an approach to constructing nested Latin hypercube designs. Such designs are useful for conducting multiple computer experiments with different levels of accuracy. A nested Latin hypercube design with two layers is defined to be a special Latin hypercube design that contains a smaller Lat ..."
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Cited by 17 (8 self)
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We propose an approach to constructing nested Latin hypercube designs. Such designs are useful for conducting multiple computer experiments with different levels of accuracy. A nested Latin hypercube design with two layers is defined to be a special Latin hypercube design that contains a smaller Latin hypercube design as a subset. Our method is easy to implement and can accommodate any number of factors. We also extend this method to construct nested Latin hypercube designs with more than two layers. Illustrative examples are given. Some statistical properties of the constructed designs are derived.
Thinning algorithms for scattered data interpolation
 BIT
, 1998
"... Abstract: Multistep interpolation of scattered data by compactly supported radial basis functions requires hierarchical subsets of the data. This paper analyzes thinning algorithms for generating evenly distributed subsets of scattered data in a given domain in IR d. ..."
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Cited by 16 (8 self)
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Abstract: Multistep interpolation of scattered data by compactly supported radial basis functions requires hierarchical subsets of the data. This paper analyzes thinning algorithms for generating evenly distributed subsets of scattered data in a given domain in IR d.
Numerical techniques based on radial basis functions
 Curve and Surface Fitting: SaintMalo 1999
, 2000
"... Radial basis functions are tools for reconstruction of multivariate functions from scattered data. This includes, for instance, reconstruction of surfaces from large sets of measurements, and solving partial differential equations by collocation. The resulting very large linear N x N systems require ..."
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Cited by 16 (4 self)
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Radial basis functions are tools for reconstruction of multivariate functions from scattered data. This includes, for instance, reconstruction of surfaces from large sets of measurements, and solving partial differential equations by collocation. The resulting very large linear N x N systems require efficient techniques for their solution, preferably of O(N) or O(N log N) computational complexity. This contribution describes some special lines of research towards this future goal. Theoretical results are accompanied by numerical examples, and various open problems are pointed out.
On Smoothing for Multilevel Approximation with Radial Basis Functions
 Vanderbilt University Press, Nashville TN
, 1999
"... . In a recent paper with Jerome we have suggested the use of a smoothing operation at each step of the basic multilevel approximation algorithm to improve the convergence rate of the algorithm. In our original paper the smoothing was defined via convolution, and its actual implementation was done vi ..."
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Cited by 15 (7 self)
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. In a recent paper with Jerome we have suggested the use of a smoothing operation at each step of the basic multilevel approximation algorithm to improve the convergence rate of the algorithm. In our original paper the smoothing was defined via convolution, and its actual implementation was done via numerical quadrature. In this paper we suggest a different approach to smoothing, namely the use of a precomputed hierarchy of smooth functions. This essentially reduces the cost of the smoothing to zero. x1. Background and Motivation Multilevel approximation with radial basis functions (RBFs) was first suggested in [4], and since then also investigated theoretically in [2,6]. The basic idea is to work with locally supported basis functions at different levels of resolution. This ensures stability and accuracy, something which is difficult to achieve with globally supported RBFs (cf. the wellknown tradeoff principle [7]). One starts with a coarselevel approximation, and then approximat...
Trefftz methods for time dependent partial differential equations
 Comput. Mat. Cont
, 2004
"... Abstract: In this paper we present a meshfree approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolichyperbolic types. For numerical purposes, a variety of transformations is used to conv ..."
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Cited by 13 (0 self)
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Abstract: In this paper we present a meshfree approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolichyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to standard reactiondiffusion and wave equation forms. To solve initial boundary value problems for these equations, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a sequence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then solved by a combination of the method of particular solutions and Trefftz methods. To do this, a variety of techniques is proposed for numerically computing a particular solution for the inhomogeneous modified Helmholtz equation. Here, we focus on the Dual Reciprocity Method where the source term is approximated by radial basis functions, polynomial or trigonometric functions. Analytic particular solutions are presented for each of these approximations. The Trefftz method is then used to solve the resulting homogenous equation obtained after the approximate particular solution is subtracted off. Two types of Trefftz bases are considered, FTrefftz bases based on the fundamental solution of the modified Helmholtz equation, and TTrefftz bases based on separation of variables solutions. Various techniques for satisfying the boundary conditions are considered, and a discussion is given of techniques for mitigating the illconditioning of the resulting linear systems. Finally, some numerical results are presented illustrating the accuracy and efficacy of this methodology.
Multiscale data sampling and function extension. SampTA 2011 proceedins
, 2011
"... We introduce a multiscale scheme for sampling scattered data and extending functions de ned on the sampled data points, which overcomes some limitations of the Nyström interpolation method. The multiscale extension (MSE) method is based on mutual distances between data points. It uses a coarseto n ..."
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Cited by 12 (7 self)
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We introduce a multiscale scheme for sampling scattered data and extending functions de ned on the sampled data points, which overcomes some limitations of the Nyström interpolation method. The multiscale extension (MSE) method is based on mutual distances between data points. It uses a coarseto ne hierarchy of the multiscale decomposition of a Gaussian kernel. It generates a sequence of subsamples, which we refer to as adaptive grids, and a sequence of approximations to a given empirical function on the data, as well as their extensions to any newlyarrived data point. The subsampling is done by a special decomposition of the associated Gaussian kernel matrix in each scale in the hierarchical procedure.
ACCURATE EMULATORS FOR LARGESCALE COMPUTER EXPERIMENTS
, 1203
"... Largescale computer experiments are becoming increasingly important in science. A multistep procedure is introduced to statisticians for modeling such experiments, which builds an accurate interpolator inmultiple steps. Inpractice, the procedureshows substantial improvements in overall accuracy, b ..."
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Cited by 11 (2 self)
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Largescale computer experiments are becoming increasingly important in science. A multistep procedure is introduced to statisticians for modeling such experiments, which builds an accurate interpolator inmultiple steps. Inpractice, the procedureshows substantial improvements in overall accuracy, but its theoretical properties are not well established. We introduce the terms nominal and numeric error and decompose the overall error of an interpolator into nominal and numeric portions. Bounds on the numeric and nominal error are developed to show theoretically that substantial gains in overall accuracy can be attained with the multistep approach.
Remarks on Meshless Local Construction of Surfaces
 In Proceedings of IMA Mathematics of Surfaces IX Conference
, 2000
"... This contribution deals with techniques for the construction of surfaces from N given data at irregularly distributed locations. Such methods should ideally have the properties computational eciency, smoothness of the resulting surface, if required, and quality of reproduction, but these go ..."
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Cited by 9 (0 self)
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This contribution deals with techniques for the construction of surfaces from N given data at irregularly distributed locations. Such methods should ideally have the properties computational eciency, smoothness of the resulting surface, if required, and quality of reproduction, but these goals turn out to be hard to meet by a single algorithm. Methods are split into a single construction or precalculation part and subsequent pointwise evaluations. Both parts are analyzed with respect to their complexity. It turns out that one has to expect the main workload on the side of geometric subproblems rather than within numerical techniques. Furthermore, if exact reconstruction at the data locations is required, and if the user wants to avoid solving non{local linear systems, there is no way around localized Langrange{type interpolation formulae. Thus two instances of such techniques are studied in some detail: interpolation by weighted local Lagrangians based on radial ...