Multiscale, Nonlinear and Adaptive Approximation

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Includes supplementary material: sn.pub/extras

Autorentext

Ronald DeVore's speciality is Nonlinear Approximation Theory. He is The Walter E. Koss Professor of Mathematics at Texas A&M University.He was elected a member of the American Academy of Arts and Sciences in 2001 and received an Honorary Doctorate from RWTH Aachen in 2004. In 2006, he was a Plenary Lecturer at the International Congress of Mathematicians in Madrid.

Angela Kunoth is working on wavelet and multiscale methods for solving partial differential equations and for data analysis purposes. She holds the Chair of Complex Systems at Universitaet Paderborn since 2007 and is an editor of five journals in applied mathematics and numerics.

Inhalt
Introduction: Wolfgang Dahmen's mathematical work.- The way things were in multivariate splines: A personal view.- On the efficient computation of high-dimensional integrals and the approximation by exponential sums.- Adaptive and anisotropic piecewise polynomial approximation.- Anisotropic function spaces with applications.- Nonlinear approximation and its applications.- Univariate subdivision and multi-scale transforms: The nonlinear case.- Rapid solution of boundary integral equations by wavelet Galerkin schemes.- Learning out of leaders.- Optimized wavelet preconditioning.- Multiresolution schemes for conservation laws.- Theory of adaptive finite element methods: An introduction.- Adaptive wavelet methods for solving operator equations: An overview.- Optimal multilevel methods for (grad), (curl), and (div) systems on graded and unstructured grids.