Multivariate Approximation: Theory and Applications

A set of irregularly distributed points is fitted to a Least-Squares approximating surface constructed as linear combination of wavelets. The construction proceeds adaptively on a coarse-to-fine refinement. We discuss by numerical experiments the efficiency of the method in relation to iterative solution procedures. Furthermore, we present extensions of the method that allow to cope with wrong measurements (robust fitting) and to control the smoothness of the reconstruction by wavelet-adapted regularization methods. Stationary and Non-stationary Vector Subdivision Costanza Conti, Universita di Firenze, Italy <[email protected]> Subdivision schemes are efficient computational means for generating recursively denser and denser sequences of points in $\RR^d$. At each step of the subdivision recursion a new sequence of points is obtained simply by averaging the previously computed points. The average coefficients form the so-called refinement mask. If the averaging rules do not depend on the recursion level the scheme is said to be stationary, otherwise nonstationary. Furthermore, if the refinement coefficients are real numbers then we speak of a scalar subdivision scheme and, if the scheme has matrix coefficients, of a vector subdivision scheme. Subdivision schemes with matrix masks play an important role in the analysis of multivariate subdivision schemes, in the analysis of Hermite-type subdivision schemes or in the context of multiwavelets. One of the difficult tasks when dealing with subdivision schemes is to prove their convergence and to investigate the smoothness of the associated limit function. In the last two decades many authors have investigated the convergence and the regularity of vector subdivision schemes. This has been mainly done by studying the properties of the transition operator ([4],[6]), of the joint spectral radius ([1],[3]) and of difference operators ([2],[5]) under particular assumptions on the refinement mask. We start the talk by recalling some basic facts about stationary and non-stationary vector subdivision schemes. Then, we investigate the convergence of multivariate vector subdivision schemes with matrix masks as general as possible using the difference operator approach. In particular, we show how the derived difference and divided difference subdivision schemes can be used to study the convergence of the original subdivision scheme and the differentiability of the associated limit function. [1] D.R. Cheng, R.Q. Jia, S.D. Riemenschneider, Convergence of Vector Subdivision Schemes in Sobolev Spaces, {\sl Appl. and Comp. Harm. Anal.} 12 (2002), 128-149. [2] N. Dyn and D. Levin, Matrix subdivision-analysis by factorization, in: Approximation Theory, B.~D.~Bojanov, ed., DARBA, Sofia, (2002), 187-211. [3] R.Q. Jia, S.D. Riemenschneider, and D.X. Zhou, Smoothness of Multiple Refinable Functions and Multiple Wavelets, SIAM J. Matrix.Anal.Appl. 21,(1999), 1-28. [4] Q.T. Jiang, Multivariate matrix refinable functions with arbitrary matrix, Trans. Amer. Math. Soc. 351, (1999), 24072438. [5] C.A. Micchelli and T. Sauer, Regularity of Multiwavelets, Adv. Comp. Math. 7, (1997), 455--545. [6] Z. Shen, Refinable Functions Vectors, SIAM J. Math. Anal. 29,1, (1998) 235-250. Detecting and Approximating Fault Lines A. Crampton and J. C. Mason School of Computing and Engineering, University of Huddersfield, UK. {a.crampton, j.c.mason}@hud.ac.uk