This book is the first to be devoted to the theory and applications of spherical (radial) basis functions (SBFs), which is rapidly emerging as one of the most promising techniques for solving problems where approximations are needed on the surface of a sphere. The aim of the book is to provide enough theoretical and practical details for the reader to be able to implement the SBF methods to solve real world problems. The authors stress the close connection between the theory of SBFs and that of the more well-known family of radial basis functions (RBFs), which are well-established tools for solving approximation theory problems on more general domains. The unique solvability of the SBF interpolation method for data fitting problems is established and an in-depth investigation of its accuracy is provided. Two chapters are devoted to partial differential equations (PDEs). One deals with the practical implementation of an SBF-based solution to an elliptic PDE and another which describes an SBF approach for solving a parabolic time-dependent PDE, complete with error analysis. The theory developed is illuminated with numerical experiments throughout. Spherical Radial Basis Functions, Theory and Applications will be of interest to graduate students and researchers in mathematics and related fields such as the geophysical sciences and statistics.

The contributions by leading experts in this book focus on a variety of topics of current interest related to information-based complexity, ranging from function approximation, numerical integration, numerical methods for the sphere, and algorithms with random information, to Bayesian probabilistic numerical methods and numerical methods for stochastic differential equations.

Delve into an in-depth description and analysis of quasi-interpolation, starting from various areas of approximation theory.

?Adapted from a series of lectures given by the authors, this monograph focuses on radial basis functions (RBFs), a powerful numerical methodology for solving PDEs to high accuracy in any number of dimensions. This method applies to problems across a wide range of PDEs arising in fluid mechanics, wave motions, astro- and geosciences, mathematical biology, and other areas and has lately been shown to compete successfully against the very best previous approaches on some large benchmark problems. Using examples and heuristic explanations to create a practical and intuitive perspective, the authors address how, when, and why RBF-based methods work.? The authors trace the algorithmic evolution of RBFs, starting with brief introductions to finite difference (FD) and pseudospectral (PS) methods and following a logical progression to global RBFs and then to RBF-generated FD (RBF-FD) methods. The RBF-FD method, conceived in 2000, has proven to be a leading candidate for numerical simulations in an increasingly wide range of applications, including seismic exploration for oil and gas, weather and climate modeling, and electromagnetics, among others.? This is the first survey in book format of the RBF-FD methodology and is suitable as the text for a one-semester first-year graduate class.

Due to their high stiffness and strength and their good processing properties short fibre reinforced thermoplastics are well-established construction materials. Up to now, simulation of engineering parts consisting of short fibre reinforced thermoplastics has often been based on macroscopic phenomenological models, but deformations, damage and failure of composite materials strongly depend on their microstructure. The typical modes of failure of short fibre thermoplastics enriched with glass fibres are matrix failure, rupture of fibres and delamination, and pure macroscopic consideration is not sufficient to predict those effects. The typical predictive phenomenological models are complex and only available for very special failures. A quantitative prediction on how failure will change depending on the content and orientation of the fibres is generally not possible, and the direct involvement of the above effects in a numerical simulation requires multi-scale modelling. One the one hand, this makes it possible to take into account the properties of the matrix material and the fibre material, the microstructure of the composite in terms of fibre content, fibre orientation and shape as well as the properties of the interface between fibres and matrix. On the other hand, the multi-scale approach links these local properties to the global behaviour and forms the basis for the dimensioning and design of engineering components. Furthermore, multi-scale numerical simulations are required to allow efficient solution of the models when investigating three-dimensional problems of dimensioning engineering parts. Bringing together mathematical modelling, materials mechanics, numerical methods and experimental engineering, this book provides a unique overview of multi-scale modelling approaches, multi-scale simulations and experimental investigations of short fibre reinforced thermoplastics. The first chapters focus on two principal subjects: the mathematical and mechanical models governing composite properties and damage description. The subsequent chapters present numerical algorithms based on the Finite Element Method and the Boundary Element Method, both of which make explicit use of the composite’s microstructure. Further, the results of the numerical simulations are shown and compared to experimental results. Lastly, the book investigates deformation and failure of composite materials experimentally, explaining the applied methods and presenting the results for different volume fractions of fibres. This book is a valuable resource for applied mathematics, theoretical and experimental mechanical engineers as well as engineers in industry dealing with modelling and simulation of short fibre reinforced composites.

This self-contained text/reference provides a basic foundation for practitioners, researchers, and students interested in any of the diverse areas of multiscale (geo)potential theory. New mathematical methods are developed enabling the gravitational potential of a planetary body to be modeled using a continuous flow of observations from land or satellite devices. Harmonic wavelets methods are introduced, as well as fast computational schemes and various numerical test examples. Presented are multiscale approaches for numerous geoscientific problems, including geoidal determination, magnetic field reconstruction, deformation analysis, and density variation modelling With exercises at the end of each chapter, the book may be used as a textbook for graduate-level courses in geomathematics, applied mathematics, and geophysics. The work is also an up-to-date reference text for geoscientists, applied mathematicians, and engineers.

This book constitutes the proceedings of the 9th International Conference on Discrete Optimization and Operations Research, DOOR 2016, held in Vladivostok, Russia, in September 2016. The 39 full papers presented in this volume were carefully reviewed and selected from 181 submissions. They were organized in topical sections named: discrete optimization; scheduling problems; facility location; mathematical programming; mathematical economics and games; applications of operational research; and short communications.