Angela Kunoth - Böcker

This book presents a collection of high-quality papers in applied and numerical mathematics, as well as approximation theory, all closely related to Wolfgang Dahmen’s scientific contributions.

This book takes readers on a multi-perspective tour through state-of-the-art mathematical developments related to the numerical treatment of PDEs based on splines, and in particular isogeometric methods.

While wavelets have since their discovery mainly been applied to problems in signal analysis and image compression, their analytic power has more and more also been recognized for problems in Numerical Analysis. Together with the functional analytic framework for different differential and integral quations, one has been able to conceptu­ ally discuss questions which are relevant for the fast numerical solution of such problems: preconditioning issues, derivation of stable discretizations, compression of fully popu­ lated matrices, evaluation of non-integer or negative norms, and adaptive refinements based on A-posteriori error estimators. This research monograph focusses on applying wavelet methods to elliptic differential equations. Particular emphasis is placed on the treatment of the boundary and the boundary conditions. Moreover, a control problem with an elliptic boundary problem as contraint serves as an example to show the conceptual strengths of wavelet techniques for some of the above mentioned issues. At this point, I would like to express my gratitude to several people before and during the process of writing this monograph. Most of all, I wish to thank Prof. Dr. Wolfgang Dahmen to whom I personally owe very much and with whom I have co-authored a large part of my work. He is responsible for the very stimulating and challenging scientific atmosphere at the Institut fiir Geometrie und Praktische Mathematik, RWTH Aachen. We also had an enjoyable collaboration with Prof. Dr. Reinhold Schneider from the Technical University of Chemnitz.

On the occasion of his 60th birthday in October 2009, friends, collaborators, and admirers of Wolfgang Dahmen have organized this volume which touches on va- ous of his research interests. This volume will provide an easy to read excursion into many important topics in applied and computational mathematics. These include nonlinear and adaptive approximation, multivariate splines, subdivision schemes, multiscale and wavelet methods, numerical schemes for partial differential and boundary integral equations, learning theory, and high-dimensional integrals. College Station, Texas, USA Ronald A. DeVore Paderborn, Germany Angela Kunoth June 2009 vii Acknowledgements We are deeply grateful to Dr. Martin Peters and Thanh-Ha Le Thi from Springer for realizing this book project and to Frank Holzwarth for technical support. ix Contents Introduction: Wolfgang Dahmen's mathematical work...1 Ronald A. DeVore and Angela Kunoth 1 Introduction ...1 2 The early years: Classical approximation theory...2 3 Bonn, Bielefeld, Berlin, and multivariate splines ...2 3. 1 Computer aided geometric design ...3 3. 2 Subdivision and wavelets ...4 4 Wavelet and multiscale methods for operator equations...5 4.1 Multilevel preconditioning ...5 4. 2 Compression of operators...5 5 Adaptive solvers ...6 6 Constructionandimplementation...7 7 Hyperbolic partial differential equations and conservation laws ...8 8 Engineering collaborations ...9 9 Thepresent ...9 10 Finalremarks...10 Publications by Wolfgang Dahmen (as of summer 2009)...10 The way things were in multivariate splines: A personal view...19 Carl de Boor 1 Tensor product spline interpolation...19 2 Quasiinterpolation ...20 3 MultivariateB-splines ...21 4 Kergininterpolation ...