: 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. 1 Introduction Interpolation by "radial" basis functions requires a function \Phi : IR d ! IR, a space IP d m of d--variate 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...

. 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...

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.

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 ill-conditioning, 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 RBF-based methods against two standard techniques (a second-order finite di#erence method and a pseudospectral method), the former gave a much superior accuracy.

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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 quasi--interpolants 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 real--valued 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...

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 non-expert readers and focuses on practical guidelines for using kernels in applications.

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 RP-type 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.

Abstract. When radial basis functions (RBFs) are made increasingly flat, the interpolation error typically decreases steadily until some point when Runge-type 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 near-flat basis functions, there will typically also be a separate point at which disasterous ill-conditioning enters. We introduce here a new method, RBF-QR, which entirely eliminates such ill-conditioning, 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 non-flat 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

We address the problem that occurs when inferences about counterfactuals—predictions, ‘‘what-if’ ’ 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 easyto-apply 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