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Error Estimates and Condition Numbers for Radial Basis Function Interpolation
 Adv. Comput. Math
, 1994
"... : 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 constant ..."
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Cited by 81 (20 self)
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: 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 NarcowichWard 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. 1 Introduction Interpolation by "radial" basis functions requires a function \Phi : IR d ! IR, a space IP d m of dvariate polynomials of degree less than m, and interpolates data values y 1 ; . . . ; yN 2 IR at data locations ("centers") x 1 ; . . . ; xN 2 IR d by solving the system N X j=1 ff j \Phi(x j \Gamma x k ) + Q X `=1 fi ` p ` (x k ) = y k ; 1 k N N X j=1 ff j p i (x j ) + 0 = 0; 1 i Q (1:1) for a basis p 1 ; . . . ; pQ...
Multivariate Interpolation and Approximation by Translates of a Basis Function
, 1995
"... . This contribution will touch the following topics: ffl Short introduction into the theory of multivariate interpolation and approximation by finitely many (irregular) translates of a (not necessarily radial) basis function, motivated by optimal recovery of functions from discrete samples. ffl Na ..."
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Cited by 35 (7 self)
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. This contribution will touch the following topics: ffl Short introduction into the theory of multivariate interpolation and approximation by finitely many (irregular) translates of a (not necessarily radial) basis function, motivated by optimal recovery of functions from discrete samples. ffl Native spaces of functions associated to conditionally positive definite functions, and relations between such spaces. ffl Error bounds and condition numbers for interpolation of functions from native spaces. ffl Uncertainty Relation: Why are good error bounds always tied to bad condition numbers? ffl Shift and Scale: How to cope with the Uncertainty Relation? x1. Introduction and Overview This contribution contains the author's view of a certain area of multivariate interpolation and approximation. It is not intended to be a complete survey of a larger area of research, and it will not account for the history of the theory it deals with. Related surveys are [15, 21, 22, 27, 30, 47, 48, 58...
A Numerical Study of some Radial Basis Function based Solution Methods for Elliptic PDEs
 Comput. Math. Appl
, 2003
"... During the last decade, three main variations have been proposed for solving elliptic PDEs by means of collocation with radial basis functions (RBFs). In this study, we have implemented them for infinitely smooth RBFs, and then compared them across the full range of values for the shape parameter ..."
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Cited by 31 (4 self)
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During the last decade, three main variations have been proposed for solving elliptic PDEs by means of collocation with radial basis functions (RBFs). In this study, we have implemented them for infinitely smooth RBFs, and then compared them across the full range of values for the shape parameter of the RBFs. This was made possible by a recently discovered numerical procedure that bypasses the illconditioning, which has previously limited the range that could be used for this parameter. We find that the best values for it often fall outside the range that was previously available. We have also looked at piecewise smooth versus infinitely smooth RBFs, and found that for PDE applications with smooth solutions, the infinitely smooth RBFs are preferable, mainly because they lead to higher accuracy. In a comparison of RBFbased methods against two standard techniques (a secondorder finite di#erence method and a pseudospectral method), the former gave a much superior accuracy.
Kernel Techniques: From Machine Learning to Meshless Methods
, 2006
"... Kernels are valuable tools in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers ..."
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Cited by 29 (8 self)
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Kernels are valuable tools in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers the links between them, as far as they are related to kernel techniques. It addresses nonexpert readers and focuses on practical guidelines for using kernels in applications.
Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions
, 2003
"... ..."
Improved error bounds for scattered data interpolation by radial basis functions
 Math. Comp
, 1999
"... Abstract. If additional smoothness requirements and boundary conditions are met, the well–known approximation orders of scattered data interpolants by radial functions can roughly be doubled. 1. ..."
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Cited by 25 (6 self)
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Abstract. If additional smoothness requirements and boundary conditions are met, the well–known approximation orders of scattered data interpolants by radial functions can roughly be doubled. 1.
The Runge phenomenon and spatially variable shape parameters
 Comput. Math. Appl
, 2006
"... Abstract. Many studies, mostly empirical, have been devoted to finding an optimal shape parameter for radial basis functions (RBF). When exploring the underlying factors that determine what is a good such choice, we are led to consider the Runge phenomenon (RP; best known in case of high order polyn ..."
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Cited by 23 (6 self)
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Abstract. Many studies, mostly empirical, have been devoted to finding an optimal shape parameter for radial basis functions (RBF). When exploring the underlying factors that determine what is a good such choice, we are led to consider the Runge phenomenon (RP; best known in case of high order polynomial interpolation) as a key error mechanism. This observation suggests that it can be advantageous to let the shape parameter vary spatially, rather than assigning a single value to it. Benefits typically include improvements in both accuracy and numerical conditioning. Still another benefit arises if one wishes to improve local accuracy by clustering nodes in select areas. This idea is routinely used when working with splines or finite element methods. However, local refinement with RBFs may cause RPtype errors unless we use a spatially variable shape paremeter. With this enhancement, RBF approximations combine freedom from meshes with spectral accuracy on irregular domains, and furthermore permit local node clustering to improve the resolution wherever this might be needed.
Approximation by Radial Basis Functions with Finitely Many Centers
, 1996
"... : Interpolation by translates of "radial" basis functions \Phi is optimal in the sense that it minimizes the pointwise error functional among all comparable quasiinterpolants on a certain "native" space of functions F \Phi . Since these spaces are rather small for cases where \Phi is smooth, we st ..."
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Cited by 23 (8 self)
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: Interpolation by translates of "radial" basis functions \Phi is optimal in the sense that it minimizes the pointwise error functional among all comparable quasiinterpolants on a certain "native" space of functions F \Phi . Since these spaces are rather small for cases where \Phi is smooth, we study the behavior of interpolants on larger spaces of the form F \Phi 0 for less smooth functions \Phi 0 . It turns out that interpolation by translates of \Phi to mollifications of functions f from F \Phi 0 yields approximations to f that attain the same asymptotic error bounds as (optimal) interpolation of f by translates of \Phi 0 on F \Phi 0 . AMS Classification: 41A15, 41A25, 41A30, 41A63, 65D10 Keywords: Radial Basis Functions, Multivariate Approximation, Approximation Order 1 Introduction Given a continuous realvalued function \Phi on IR d and a nonnegative integer m, we consider approximations by finitely many translates \Phi(\Delta \Gamma x j ); 1 j N , of \Phi together with...
A stable algorithm for flat radial basis functions
 SIAM J. Sci. Comp
"... Abstract. When radial basis functions (RBFs) are made increasingly flat, the interpolation error typically decreases steadily until some point when Rungetype oscillations either halt or reverse this trend. Because the most obvious method to calculate an RBF interpolant becomes a numerically unstabl ..."
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Cited by 19 (6 self)
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Abstract. When radial basis functions (RBFs) are made increasingly flat, the interpolation error typically decreases steadily until some point when Rungetype oscillations either halt or reverse this trend. Because the most obvious method to calculate an RBF interpolant becomes a numerically unstable algorithm for a stable problem in case of nearflat basis functions, there will typically also be a separate point at which disasterous illconditioning enters. We introduce here a new method, RBFQR, which entirely eliminates such illconditioning, and we apply it in the special case when the data points are distributed over the surface of a sphere. This algorithm works even for thousands of node points, and it allows the RBF shape parameter to be optimized without the limitations imposed by stability concerns. Since interpolation in the flat RBF limit on a sphere is found to coincide with spherical harmonics interpolation, new insights are gained as to why the RBF approach (with nonflat basis functions) often is the more accurate of the two methods. Key words. Radial basis functions, RBF, shape parameter, sphere, spherical harmonics. 1. Introduction. Numerical
The dangers of extreme counterfactuals
 Political Analysis
, 2006
"... We address the problem that occurs when inferences about counterfactuals—predictions, ‘‘whatif’ ’ questions, and causal effects—are attempted far from the available data. The danger of these extreme counterfactuals is that substantive conclusions drawn from statistical models that fit the data well ..."
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Cited by 13 (7 self)
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We address the problem that occurs when inferences about counterfactuals—predictions, ‘‘whatif’ ’ questions, and causal effects—are attempted far from the available data. The danger of these extreme counterfactuals is that substantive conclusions drawn from statistical models that fit the data well turn out to be based largely on speculation hidden in convenient modeling assumptions that few would be willing to defend. Yet existing statistical strategies provide few reliable means of identifying extreme counterfactuals. We offer a proof that inferences farther from the data allow more model dependence and then develop easytoapply methods to evaluate how model dependent our answers would be to specified counterfactuals. These methods require neither sensitivity testing over specified classes of models nor evaluating any specific modeling assumptions. If an analysis fails the simple tests we offer, then we know that substantive results are sensitive to at least some modeling choices that are not based on empirical evidence. Free software that accompanies this article implements all the methods developed. 1