Scattered Data Approximation

Abstract

Many practical applications require the reconstruction of a multivariate function from discrete, unstructured data. This book gives a self-contained, complete introduction into this subject. It concentrates on truly meshless methods such as radial basis functions, moving least squares, and partitions of unity. The book starts with an overview on typical applications of scattered data approximation, coming from surface reconstruction, fluid-structure interaction, and the numerical solution of partial differential equations. It then leads the reader from basic properties to the current state of research, addressing all important issues, such as existence, uniqueness, approximation properties, numerical stability, and efficient implementation. Each chapter ends with a section giving information on the historical background and hints for further reading. Complete proofs are included, making this perfectly suited for graduate courses on multivariate approximation and it can be used to support courses in computer-aided geometric design, and meshless methods for partial differential equations.

Multidimensional Lobachevsky Spline Integration on Scattered Data

“…Finally, we observe that, since Lobachevsky splines are (univariate) strictly positive definite functions for even n ≥ 2, we can construct multivariate strictly positive definite functions from univariate ones (see, e.g., [25]), expressing them as products of Lobachevsky splines [4].…”

“…Taking into account the convergence results of Gaussian interpolation (see, e.g., [25]), we can deduce that, for sufficiently regular g, the error |I(g) − I(F n )| may decrease exponentially as n → ∞ and h → 0, where h denotes the so-called fill distance, i.e.…”

“…Now, in the literature lower bounds for the smallest eigenvalue of the interpolation matrix A of the Gaussian have been extensively studied (see, e.g., [25]), providing upper bounds for A −1 2 in terms of the so-called separation distance of the nodes…”

On the Construction of Kernel-Based Adaptive Particle Methods in Numerical Flow Simulation

“…Rather than dwelling much on explaining conditionally positive definite kernel functions and the structure of their native reproducing kernel Hilbert spaces, we refer to the text books [5,7,12,24]. For the following of our discussion, it is sufficient to say that scattered data interpolation by positive definite kernels (where m = 0) leads to a unique reconstruction of the form (6).…”

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

“…is positive definite; see [29,Theorem 9.13]. Since x − y = √ 2 − 2x · y for any x, y ∈ S n , the kernel Φ defined by (12) with (17) φ…”