Localized bases for kernel spaces on the unit sphere

Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered data, and is central to many meshless methods. For a set of $N$ scattered sites, the standard basis for such a space utilizes $N$ \emph{globally} supported kernels; computing with it is prohibitively expensive for large $N$. Easily computable, well-localized bases, with "small-footprint" basis elements - i.e., elements using only a small number of kernels -- have been unavailable. Working on \sphere, with focus on the restricted surface spline kernels (e.g. the thin-plate splines restricted to the sphere), we construct easily computable, spatially well-localized, small-footprint, robust bases for the associated kernel spaces. Our theory predicts that each element of the local basis is constructed by using a combination of only $\mathcal{O}((\log N)^2)$ kernels, which makes the construction computationally cheap. We prove that the new basis is $L_p$ stable and satisfies polynomial decay estimates that are stationary with respect to the density of the data sites, and we present a quasi-interpolation scheme that provides optimal $L_p$ approximation orders. Although our focus is on $\mathbb{S}^2$, much of the theory applies to other manifolds - $\mathbb{S}^d$, the rotation group, and so on. Finally, we construct algorithms to implement these schemes and use them to conduct numerical experiments, which validate our theory for interpolation problems on $\mathbb{S}^2$ involving over one hundred fifty thousand data sites.Comment: This article supersedes arXiv:1111.1013 "Better bases for kernel spaces," which proved existence of better bases for various kernel spaces. This article treats a smaller class of kernels, but presents an algorithm for constructing better bases and demonstrates its effectiveness with more elaborate examples. A quasi-interpolation scheme is introduced that provides optimal linear convergence rate

Wavelet and Multiscale Methods

[no abstract available

Schnelle Löser für partielle Differentialgleichungen

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New Discretization Methods for the Numerical Approximation of PDEs

The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems. The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on 2-7 Apr. 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection shows its rapid trend to further diversity and depth