ACM Transactions on Mathematical Software, Sep 28, 2021
Vector spherical harmonics on the unit sphere of ℝ3have broad applications in geophysics, quantum... more Vector spherical harmonics on the unit sphere of ℝ3have broad applications in geophysics, quantum mechanics, and astrophysics. In the representation of a tangent vector field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this article, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional toNlog √NforNnumber of evaluation points. The adjoint FaVeST, which evaluates a linear combination of vector spherical harmonics with a degree up to ⊡MforMevaluation points, has cost proportional toMlog √M. Numerical examples of simulated tangent fields illustrate the accuracy, efficiency, and stability of FaVeST.
Book synopsis: This book is the first to be devoted to the theory and applications of spherical (... more Book synopsis: 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.
In this chapter we consider the following parabolic partial differential equation defined on the ... more In this chapter we consider the following parabolic partial differential equation defined on the unit sphere \(S^{d-1} \subset {\mathbb R}^{d}\).
We open this chapter by establishing the basic notation that will be used regularly throughout th... more We open this chapter by establishing the basic notation that will be used regularly throughout the book. We then provide some motivation behind what we aim to achieve in this book, namely to develop the theory and explore the applications of spherical basis functions (SBFs).
ACM Transactions on Mathematical Software, Sep 28, 2021
Vector spherical harmonics on the unit sphere of ℝ3have broad applications in geophysics, quantum... more Vector spherical harmonics on the unit sphere of ℝ3have broad applications in geophysics, quantum mechanics, and astrophysics. In the representation of a tangent vector field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this article, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional toNlog √NforNnumber of evaluation points. The adjoint FaVeST, which evaluates a linear combination of vector spherical harmonics with a degree up to ⊡MforMevaluation points, has cost proportional toMlog √M. Numerical examples of simulated tangent fields illustrate the accuracy, efficiency, and stability of FaVeST.
Book synopsis: This book is the first to be devoted to the theory and applications of spherical (... more Book synopsis: 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.
In this chapter we consider the following parabolic partial differential equation defined on the ... more In this chapter we consider the following parabolic partial differential equation defined on the unit sphere \(S^{d-1} \subset {\mathbb R}^{d}\).
We open this chapter by establishing the basic notation that will be used regularly throughout th... more We open this chapter by establishing the basic notation that will be used regularly throughout the book. We then provide some motivation behind what we aim to achieve in this book, namely to develop the theory and explore the applications of spherical basis functions (SBFs).
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