Polynomial splines become less attractive as the problem dimension increases, as finding a basis of the interpolating space becomes very difficult (if not impossible), thus affecting the uniqueness of the interpolation. In addition, it is well known that polynomial interpolation in high dimensions can easily lead to singular problems.
This is the first book on this subject that presents a thorough treatment from both the theoretical and implementation viewpoints. It has been shown that interpolations based on radial basis functions are effective when the functions to be approximated are of multiple variables, or are given only by a great amount of data or by data that is scattered. More specifically, a radial basis function, &PHgr;: Rn → R being continuous, defines a space S ⊂ C(Rn), which depends on &Xgr; = (hZ)n ⊂ Rn and is spanned by &PHgr;(||.-&xgr;||), &xgr; ∈ &Xgr;. Thus, an approximation has the general form
Some commonly used radial basis functions are r2log r, √r2 + c2 and . Practically, this primarily concerns two kinds of interpolations: quasi-interpolation, given by where and the Lagrange type, for which &psgr; satisfies &psgr;(j) = &dgr;0j, j ∈ Zn.
Chapter 2 briefly introduces, using a 1-dimensional model, the methods used, the type of mathematical analysis needed, and a typical approximation process for partial differential equation (PDE) boundary value problems using collocation. Chapter 3 lays out an environment for multidimensional approximations. A detailed account for the existence, uniqueness, interpolation-matrix conditioning, polynomial reproducing, and convergence of the approximations, with respect to the equally spaced grid is given in chapter 4. Chapter 5 generalizes the results of chapter 4 to scattered data, confirming some parallel results. Convergence in the presence of boundaries is also addressed. Chapter 6 is appropriate for anyone who needs an approximation to deal with dynamic data, and who is interested in meshless or compact supported approximations. Chapter 7 describes several modern techniques to evaluate and compute radial basis interpolants efficiently to a degree that supports real-time surface rendering. Least-square approximation and its wavelet expansions are discussed in chapters 8 and 9. Finally, Chapter 10 reviews the most recent results in the area, and provides a view of future research developments.
The applications of radial basis function approximations are wide ranging, from pattern reconstruction (image, weather, or environment), artificial intelligence (fire detection by a fire detector or object construction so that a robot can recognize objects), to simply solving mathematical PDEs based on irregular data distributions.
