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Radial Basis Functions Acta Numerica (2000), pp. 1–38 c Cambridge University Press, 2000 Radial basis functions M. D. Buhmann Mathematical Institute, Justus Liebig University, 35392 Giessen, Germany E-mail: [email protected] Radial basis function methods are modern ways to approximate multivariate functions, especially in the absence of grid data. They have been known, tested and analysed for several years now and many positive properties have been identified. 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. We consider particularly the new results on convergence rates of interpolation with radial basis functions, as well as some of the various achievements on approximation on spheres, and the efficient numerical computation of interpolants for very large sets of data. Several examples of useful applications are stated at the end of the paper. CONTENTS 1 Introduction 1 2 Convergence rates 5 3 Compact support 11 4 Iterative methods for implementation 16 5 Interpolation on spheres 25 6 Applications 28 References 34 1. Introduction There is a multitude of ways to approximate a function of many variables: multivariate polynomials, splines, tensor product methods, local methods and global methods. All of these approaches have many advantages and some disadvantages, but if the dimensionality of the problem (the number of variables) is large, which is often the case in many applications from stat- istics to neural networks, our choice of methods is greatly reduced, unless 2 M. D. Buhmann we resort solely to tensor product methods. In fact, sometimes we are given scattered data, immediately excluding the use of tensor product methods. Also, tensor product methods in high dimensions always require many, of- ten too many data. In this situation, the method of choice is often a radial basis function approach which is, incidentally, also highly useful in lower- dimensional problems and as an alternative to (piecewise) polynomials, be- cause of its excellent approximation properties; at any rate it is universally applicable independent of dimension. In order to formulate the problem as we will see it in this review, let Ξ be a finite set of distinct points in Rn, which are traditionally called centres in radial basis function jargon, because our basis functions will be radially symmetric about these points. The goal of our work is to approximate an unknown function that is only given at those centres via a set of real numbers fξ, ξ Ξ. These are almost always interpreted as function evaluations of ∈ n some smooth function f : R Ω R, so that fξ = f(ξ). Here, Ω is a domain in Rn. This point of view⊃ will→ allow us to measure conveniently the uniform approximation error between f and its approximant s. This error depends on the choice of the approximant, on Ξ, on f, and in particular on its smoothness. In order to approximate with s, which is usually by interpolation, we take a univariate continuous function φ that is radialized by composition with the Euclidean norm on Rn, or a suitable replacement thereof when we are working on a sphere in n-dimensional Euclidean space, for instance. This φ : R+ R is the radial basis function. Additionally, we take the given centres →ξ from the given finite set Ξ of distinct points and use them simultaneously for shifting the radial basis function and as interpolation (collocation) points. Therefore, our standard radial function approximants now have the form s(x)= λ φ( x ξ ),xRn, (1.1) ξ k − k ∈ ξ Ξ X∈ suitable adjustments being made when x is not from the whole space, and Ξ the coefficient vector λ =(λξ)ξ Ξ is an element of R . In many instances, particularly those that will interest∈ us in Section 3, the interpolation re- quirements s = f (1.2) |Ξ |Ξ for given data f lead to a positive definite interpolation matrix A = |Ξ φ( ξ ζ ) ξ,ζ Ξ. In that case, we call the radial basis function ‘positive definite’{ k − ask well.} ∈ If it is, the linear system of equations that comes from (1.1) and (1.2) and uses precisely that matrix A yields a unique coefficient vector λ RΞ for the interpolant (1.1). ∈All radial basis functions of Section 3 have this property of positive defin- Radial basis functions 3 c2r2 iteness, as does for instance the Gaussian radial basis function φ(r)=e− for all positive parameters c and the inverse multiquadric function φ(r)= 1/√r2 + c2. However, in some instances such as the so-called thin-plate spline radial basis function, the radial function φ is only conditionally positive definite of some order k on Rn, say, a notion that we shall explain and use in the Pk 1 subsequent section. Now, in this event, polynomials p(x) n− (x) of degree k 1inn unknowns are augmented to the right-hand side∈ of (1.1) so as to render− the interpolation problem again uniquely solvable. Consequently we have as approximant s(x)= λ φ( x ξ )+p(x),xRn. (1.3) ξ k − k ∈ ξ Ξ X∈ Then the extra degrees of freedom are taken up by requiring that the coeffi- RΞ Pk 1 cient vector λ is orthogonal to the polynomial space n− (Ξ), that is, all polynomials∈ of total degree less than k in n variables restricted to Ξ: Ξ k 1 k 1 R λ P − (Ξ) λ q(ξ)=0, q P − . (1.4) 3 ⊥ n ⇐⇒ ξ ∀ ∈ n ξ Ξ X∈ Pk 1 In order to retain uniqueness, Ξ has to contain a n− -unisolvent subset in this case. When k = 2, for instance, this means that the centre-set must not be a subset of a straight line. This is a requirement that can easily be met in most cases. The two probably best-known and most often applied radial basis func- tions are called multiquadrics and thin-plate splines, respectively. The former is, for a positive parameter c, φ(r)=√r2 + c2 and the latter is φ(r)=r2 log r, where in the second case (1.3) and (1.4) with k = 2 are applied. The multiquadric is, subject to a sign change, conditionally posit- ive definite of order k = 1, but it turns out that the original interpolation problem without augmentation by constants is also nonsingular because of special properties of the multiquadric function (Micchelli 1986). One more well-known example comes up when we set c = 0 in the mul- tiquadric example: then we have the so-called linear radial basis function φ(r)=r which also gives a nonsingular interpolation problem without aug- mentation by constants. All these examples are useful for various forms of interpolation and approximation and they all allow this interpolation pro- cedure in all dimensions n and all sets of distinct centres, independently of the geometry of the points. In contrast to spline approximation in more than one dimension, for instance, there is no triangulation or tessellation of the data points required, nor is there any restriction on the dimensionality of the problem. We note, however, that these nonsingularity properties are strictly linked to the fact that we use Euclidean norms; for p-norms with 4 M. D. Buhmann p>2, including p = ,orp = 1, singularity can occur in any dimension for unfortunate choices∞ of data points ξ (Baxter 1991). While it has been known for a long time (see Hardy (1990) for references and the history of the approach) that approximants of the above form exist and approximate well if the centres are sufficiently close together and the function f is smooth enough, it took quite a while to underpin these empir- ical results theoretically, that is, get existence, uniqueness and convergence results, and extend the known classes of useful radial basis functions to fur- ther examples. Now, however, research into radial basis functions is a very active and fruitful area and it is timely to stand back and summarize its new developments in this article. Among the plethora of new papers that are published every year on radial basis functions we have made out and selected five major directions which have had important new developments and which we will review in this work. One major feature, for example, that has been looked at recently (again) and that we address, is the accuracy of approximation with radial basis functions when the centres are scattered points in a domain, and, ultimately, form a dense subset of the domain – a subject, incidentally, well fitted to begin this review paper in the next section, because the whole theoretical de- velopment started some 20 years ago in France with Duchon’s contributions (1976, 1978, 1979) to exactly this question in a special context. Apart from the fundamental question of unique solvability of the ensuing linear system – which has by now been completely answered for large classes of radial basis functions that are conditionally positive definite, mostly fol- lowing Micchelli (1986) – an important question is that of convergence and convergence rates of those interpolants to the function f that is being ap- proximated by collocation to f Ξ if f is in a suitable smoothness space. Duchon (1976, 1978, 1979) gave| answers to this question that address the important special case of thin-plate splines φ(r)=r2 log r and its siblings, for instance the odd powers of r (‘pseudo-cubics’ φ(r)=r3, etc.), and recent research has improved some of his 20 year-old results in various directions, including inverse theorems, and theorems about optimality of convergence orders, the sets of centres still always being finite.
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