Numerical Techniques Based on Radial Basis Functions

Radial basis functions are tools for reconstruction of mul-tivariate functions from scattered data. This includes, for instance, reconstruction of surfaces from large sets of measurements, and solving partial diierential equations by collocation. The resulting very large linear N N systems require eecient techniques for their solution, preferably of O(N) or O(N log N) computational complexity. This contribution describes some special lines of research towards this future goal. Theoretical results are accompanied by numerical examples, and various open problems are pointed out. x1. Introduction Many problems of numerical analysis take the form of a generalized interpolation in spaces of multivariate functions 21]. Due to the Mairhuber-Curtis theorem 12], such spaces cannot be xed beforehand but must necessarily depend on the given data. For a plain multivariate interpolation problem on a nite set X = fx 1 ; : : : ; x N g of pairwise diierent points in a domain IR d , there is an easy possibility to generate a data-dependent space via linear combinations of something that depends on a free variable x 2 IR d and the data locations x j , namely (1) with a xed function : ! IR. The numerical generation of the space can be simpliied considerably in the special situations 1) (x; y) = (x y) with : IR d ! IR (translation invariance) 2) (x; y) = (kx yk 2) with : 0; 1) ! IR (radiality), and this is how the notion of a radial basis function came up. To assure that the interpolation in the points of X = fx 1 ; : : : ; x N g is uniquely deened, the matrix A ;X := (((x j ; x k)) 1j;kN (2) must be nonsingular. By deenition, positive deenite functions even make this matrix symmetric and positive deenite, and the positive deenite radial functions (r) = exp(r 2) on IR d for all d (Gaussian) (r) = (1 r) 4 + (1 + 4r) on IR d ; d 3; see 25] are typical examples. See the review articles 4,15,18,16] for details. Though the above functions are scalar, the positive deeniteness property of the second depends on the dimension d of the space containing x and y when forming the scalar argument r = kx yk 2. These two examples already show that the matrix A ;X in (2) can be sparse or have a strong oo-diagonal …