Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree
01 Dec 1995-Advances in Computational Mathematics (Baltzer Science Publishers, Baarn/Kluwer Academic Publishers)-Vol. 4, Iss: 1, pp 389-396
TL;DR: A new class of positive definite and compactly supported radial functions which consist of a univariate polynomial within their support is constructed, it is proved that they are of minimal degree and unique up to a constant factor.
Abstract: We construct a new class of positive definite and compactly supported radial functions which consist of a univariate polynomial within their support. For given smoothness and space dimension it is proved that they are of minimal degree and unique up to a constant factor. Finally, we establish connections between already known functions of this kind.
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TL;DR: The Virgo Consortium's EAGLE project as discussed by the authors is a suite of hydrodynamical simulations that follow the formation of galaxies and black holes in representative volumes, where thermal energy is injected into the gas, allowing winds to develop without predetermined speed or mass loading factors.
Abstract: We introduce the Virgo Consortium's EAGLE project, a suite of hydrodynamical simulations that follow the formation of galaxies and black holes in representative volumes. We discuss the limitations of such simulations in light of their finite resolution and poorly constrained subgrid physics, and how these affect their predictive power. One major improvement is our treatment of feedback from massive stars and AGN in which thermal energy is injected into the gas without the need to turn off cooling or hydrodynamical forces, allowing winds to develop without predetermined speed or mass loading factors. Because the feedback efficiencies cannot be predicted from first principles, we calibrate them to the z~0 galaxy stellar mass function and the amplitude of the galaxy-central black hole mass relation, also taking galaxy sizes into account. The observed galaxy mass function is reproduced to ≲0.2 dex over the full mass range, 108
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...…of the conservative pressure-entropy formulation of SPH derived by Hopkins (2013), the artificial viscosity switch from Cullen & Dehnen (2010), an artificial conduction switch similar to that of Price (2008), the C2 Wendland (1995) kernel and the time step limiters of Durier & Dalla Vecchia (2012)....
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TL;DR: This paper attempts to give an overview of deformable registration methods, putting emphasis on the most recent advances in the domain, and provides an extensive account of registration techniques in a systematic manner.
Abstract: Deformable image registration is a fundamental task in medical image processing. Among its most important applications, one may cite: 1) multi-modality fusion, where information acquired by different imaging devices or protocols is fused to facilitate diagnosis and treatment planning; 2) longitudinal studies, where temporal structural or anatomical changes are investigated; and 3) population modeling and statistical atlases used to study normal anatomical variability. In this paper, we attempt to give an overview of deformable registration methods, putting emphasis on the most recent advances in the domain. Additional emphasis has been given to techniques applied to medical images. In order to study image registration methods in depth, their main components are identified and studied independently. The most recent techniques are presented in a systematic fashion. The contribution of this paper is to provide an extensive account of registration techniques in a systematic manner.
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...[121] investigated the use of Wendland functions [122, 123] that exhibit the desired locality property, for elastic registration....
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TL;DR: This paper gives a selective but up-to-date survey of several recent developments that explains their usefulness from the theoretical point of view and contributes useful new classes of radial basis function.
Abstract: From the Publisher:
"In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing it is necessary to estimate parameters, usually multidimensional, by approximation and interpolation. Radial basis functions are a modern and powerful tool which work well in very general circumstances, and so are becoming of widespread use, as the limitations of other methods, such as least squares, polynomial interpolation or wavelet-based, become apparent." This is the first book devoted to the subject and the author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence, and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.
1,335 citations
Cites background or methods from "Piecewise polynomial, positive defi..."
...Starting from this, Wendland (1995, 1998), in particular, developed an entire theory of the radial basis functions of compact support which are piecewise polynomial and are positive definite....
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...Next, Wendland (1995) and Wu (1995) use the fact that the said interpolation matrix A for the truncated power φ0 remains positive definite if the basis function φ(r) = φn,k(r) = I kφ0(r), r ≥ 0, (3.1) is used when ` = k+[n/2]+1....
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...Next, Wendland (1995) and Wu (1995) use the fact that the said interpolation matrix A for the truncated power φ0 remains positive definite if the basis function...
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TL;DR: In this article, the authors present results from thirteen cosmological simulations that explore the parameter space of the "Evolution and Assembly of GaLaxies and their Environments" (EAGLE) simulation project.
Abstract: We present results from thirteen cosmological simulations that explore the parameter space of the "Evolution and Assembly of GaLaxies and their Environments" (EAGLE) simulation project. Four of the simulations follow the evolution of a periodic cube L = 50 cMpc on a side, and each employs a different subgrid model of the energetic feedback associated with star formation. The relevant parameters were adjusted so that the simulations each reproduce the observed galaxy stellar mass function at z = 0.1. Three of the simulations fail to form disc galaxies as extended as observed, and we show analytically that this is a consequence of numerical radiative losses that reduce the efficiency of stellar feedback in high-density gas. Such losses are greatly reduced in the fourth simulation - the EAGLE reference model - by injecting more energy in higher density gas. This model produces galaxies with the observed size distribution, and also reproduces many galaxy scaling relations. In the remaining nine simulations, a single parameter or process of the reference model was varied at a time. We find that the properties of galaxies with stellar mass <~ M* (the "knee" of the galaxy stellar mass function) are largely governed by feedback associated with star formation, while those of more massive galaxies are also controlled by feedback from accretion onto their central black holes. Both processes must be efficient in order to reproduce the observed galaxy population. In general, simulations that have been calibrated to reproduce the low-redshift galaxy stellar mass function will still not form realistic galaxies, but the additional requirement that galaxy sizes be acceptable leads to agreement with a large range of observables.
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Cites methods from "Piecewise polynomial, positive defi..."
...…formulation of SPH derived by Hopkins (2013), the artificial viscosity switch proposed by Cullen & Dehnen (2010), an artificial conduction switch similar to that proposed by Price (2008), the C2 smoothing kernel of Wendland (1995), and the time-step limiter of Durier & Dalla Vecchia (2012)....
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TL;DR: It is shown that tapering the correct covariance matrix with an appropriate compactly supported positive definite function reduces the computational burden significantly and still leads to an asymptotically optimal mean squared error.
Abstract: Interpolation of a spatially correlated random process is used in many scientific areas. The best unbiased linear predictor, often called a kriging predictor in geostatistical science, requires the solution of a (possibly large) linear system based on the covariance matrix of the observations. In this article, we show that tapering the correct covariance matrix with an appropriate compactly supported positive definite function reduces the computational burden significantly and still leads to an asymptotically optimal mean squared error. The effect of tapering is to create a sparse approximate linear system that can then be solved using sparse matrix algorithms. Monte Carlo simulations support the theoretical results. An application to a large climatological precipitation dataset is presented as a concrete and practical illustration.
757 citations
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...Wu (1995), Wendland (1995, 1998), Gaspari and Cohn (1999), and Gneiting (2002) gave several procedures to construct compactly supported covariance functions with arbitrary degree of differentiability at the origin and at the support length....
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...This section investigates numerically the convergence of the ratios (2.5) and (2.6) for different sample sizes, shapes of covariance functions and tapers....
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References
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Journal Article•
TL;DR: The relationship between 'learning' in adaptive layered networks and the fitting of data with high dimensional surfaces is discussed, leading naturally to a picture of 'generalization in terms of interpolation between known data points and suggests a rational approach to the theory of such networks.
Abstract: : The relationship between 'learning' in adaptive layered networks and the fitting of data with high dimensional surfaces is discussed. This leads naturally to a picture of 'generalization in terms of interpolation between known data points and suggests a rational approach to the theory of such networks. A class of adaptive networks is identified which makes the interpolation scheme explicit. This class has the property that learning is equivalent to the solution of a set of linear equations. These networks thus represent nonlinear relationships while having a guaranteed learning rule. Great Britain.
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TL;DR: In this paper, the evaluation of methods for scattered data interpolation and some of the results of the tests when applied to a number of methods are presented. But the evaluation process involves evaluation of the methods in terms of timing, storage, accuracy, visual pleasantness of the surface, and ease of implementation.
Abstract: Absract. This paper is concerned with the evaluation of methods for scattered data interpolation and some of the results of the tests when applied to a number of methods. The process involves evaluation of the methods in terms of timing, storage, accuracy, visual pleasantness of the surface, and ease of implementation. To indicate the flavor of the type of results obtained, we give a summary table and representative perspective plots of several surfaces.
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"Piecewise polynomial, positive defi..." refers background in this paper
...[5]) and was studied extensively in recent years....
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IBM1
TL;DR: In this paper, it was shown that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R Franke, which is a conjecture that was later proved in the present paper.
Abstract: Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R Franke
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TL;DR: A variation of the Narcowich-Ward theory of upper bounds on the norm of the inverse of the interpolation matrix is presented in order to handle the whole set of radial basis functions that are currently in use.
Abstract: For interpolation of scattered multivariate data by radial basis functions, an "uncertainty relation" between the attainable error and the condition of the interpolation matrices is proven. It states that the error and the condition number cannot both be kept small. Bounds on the Lebesgue constants are obtained as a byproduct. A variation of the Narcowich-Ward theory of upper bounds on the norm of the inverse of the interpolation matrix is presented in order to handle the whole set of radial basis functions that are currently in use.
628 citations
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...= r - d - ~ - e l r ( r - t)et(d/2)+kJ(d/2)+k_ l ( t ) d t , JO which is, except for ¢~,0, strictly positive (see Gasper [6]), so that the techniques of Narcowich and Ward [8] and Schaback [11] can be used to obtain bounds for the condition of the interpolation matrix....
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TL;DR: The major theme of this work is the development of an iterative scheme for the construction of a smooth surface, presented by global basis functions, which approximates only the smooth components of a set of scattered noisy data.
Abstract: In many applications one encounters the problem of approximating surfaces from data given on a set of scattered points in a two-dimensional domain. The global interpolation methods with Duchon's “thin plate splines” and Hardy's multiquadrics are considered to be of high quality; however, their application is limited, due to computational difficulties, to $ \sim 150$ data points. In this work we develop some efficient iterative schemes for computing global approximation surfaces interpolating a given smooth data. The suggested iterative procedures can, in principle, handle any number of data points, according to computer capacity. These procedures are extensions of a previous work by Dyn and Levin on iterative methods for computing thin-plate spline interpolants for data given on a square grid. Here the procedures are improved significantly and generalized to the case of data given in a general configuration.The major theme of this work is the development of an iterative scheme for the construction of a smooth surface, presented by global basis functions, which approximates only the smooth components of a set of scattered noisy data. The novelty in the suggested method is in the construction of an iterative procedure for low-pass filtering based on detailed spectral properties of a preconditioned matrix. The general concepts of this approach can also be used in designing iterative computation procedures for many other problems.The interpolation and smoothing procedures are tested, and the theoretical results are verified, by many numerical experiments.
429 citations
"Piecewise polynomial, positive defi..." refers methods in this paper
...of the usual basis functions like multiquadrics or thin plate splines still some problems exist, caused by a large number N of centers xj. Special methods of computation (cf. [ 4 ]) and evaluation (cf....
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