Algorithm 1018: FaVeST—Fast Vector Spherical Harmonic Transforms

2021 ◽  
Vol 47 (4) ◽  
pp. 1-24
Author(s):  
Quoc T. Le Gia ◽  
Ming Li ◽  
Yu Guang Wang
Keyword(s):  
Spherical Harmonics ◽  
Spherical Harmonic ◽  
Fourier Coefficients ◽  
Computational Cost ◽  
Tangent Vector ◽  
Tangent Vector Field ◽  
Vector Spherical Harmonics ◽  
Numerical Examples ◽  
Vector Spherical Harmonic ◽  
Tangent Fields

Vector spherical harmonics on the unit sphere of ℝ 3 have 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 to N log √ N for N number of evaluation points. The adjoint FaVeST, which evaluates a linear combination of vector spherical harmonics with a degree up to ⊡ M for M evaluation points, has cost proportional to M log √ M . Numerical examples of simulated tangent fields illustrate the accuracy, efficiency, and stability of FaVeST.

Analysis of radiative scattering for multiple sphere configurations

1991 ◽  
Vol 433 (1889) ◽  
pp. 599-614 ◽  
Keyword(s):  
Spherical Harmonics ◽  
Scattered Field ◽  
Spherical Harmonic ◽  
Recurrence Relations ◽  
Solution Technique ◽  
Addition Theorems ◽  
Vector Spherical Harmonics ◽  
Vector Spherical Harmonic ◽  
Arbitrary Configuration ◽  
Radiative Scattering

An analysis of radiative scattering for an arbitrary configuration of neighbouring spheres is presented. The analysis builds upon the previously developed superposition solution, in which the scattered field is expressed as a superposition of vector spherical harmonic expansions written about each sphere in the ensemble. The addition theorems for vector spherical harmonics, which transform harmonics from one coordinate system into another, are rederived, and simple recurrence relations for the addition coefficients are developed. The relations allow for a very efficient implementation of the ‘order of scattering’ solution technique for determining the scattered field coefficients for each sphere.

New tensor spherical harmonics, for application to the partial differential equations of mathematical physics

1976 ◽  
Vol 281 (1302) ◽  
pp. 195-221 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Spherical Harmonics ◽  
Magnetic Induction ◽  
Spherical Harmonic ◽  
Mathematical Physics ◽  
Partial Differential ◽  
Vector Spherical Harmonics ◽  
Tensor Spherical Harmonics ◽  
Rank 2

Rank-A cartesian-tensor spherical harmonics are defined recursively by the ClebschGordan coupling of rank-( k — 1) tensor spherical harmonics with certain complex basis vectors. By taking the rank-0 tensor harmonics to be the usual scalar spherical harmonics, the new definition generates rank-1 harmonics equivalent to the vector spherical harmonics commonly employed in the quantum theory of angular momentum. A second application of the definition generates new rank-2 harmonics which are orthogonal transformations of the symmetric and antisymmetric rank-2 harmonics defined by Zerilli (1970). Continued application of the definition generates new rank-A harmonics which are orthogonally related to tensors used by Burridge (1966). The main advantage of the new tensor harmonics is that the numerous standard properties (for example, completeness; orthogonality; gradient, divergence and curl formulae; addition formulae) of scalar and vector spherical harmonics, generalize, essentially unchanged in form, to the rank- k case. Furthermore, the recursive definition allows systematic evaluation of integrals of products of three tensor harmonics in terms of Wigner coefficients, the latter immediately implying selection rules and symmetries for the integrals. Together, these generalized properties and coupling integrals permit straightforward spherical harmonic analysis of many partial differential equations in mathematical physics. Application of the new harmonics is demonstrated by analysis of the tensor equations of Laplace and Helmhotz, stress-strain equations for free vibrations of an elastic sphere, the Euler and Navier-Stokes equations for a rotating fluid, and the magnetic induction and mean-field magnetic-induction equations for a conducting fluid. Finally, the method of Orszag (1970) for the fast computation of spherical harmonic coefficients of nonlinear interactions is generalized for the tensor-harmonic case.

scholarly journals On some expressions for the radial and axial components of the magnetic force in the interior of solenoids of circular cross-section

1899 ◽  
Vol 64 (402-411) ◽  
pp. 192-202 ◽  
Keyword(s):  
Cross Section ◽  
Spherical Harmonics ◽  
Spherical Harmonic ◽  
Magnetic Force ◽  
Circular Cross Section ◽  
Total Force ◽  
Harmonic Method ◽  
Zonal Harmonics ◽  
Derivatives Of ◽  
Spherical Harmonic Method

In the present paper, certain expressions are arrived at, in terms of zonal spherical harmonics and their first derivatives, by which the values of the two components of the magnetic force may be calculated for any point in the interior of a coil, and hence the total force may be found both in magnitude and direction. The resulting series suffer from the well-known defect in the spherical harmonic method, in that they are not very rapidly converging for points near the boundary of the space for which they apply. A table of the values of the first derivatives of the first seven zonal harmonics is added.

scholarly journals Data-driven internal multiple elimination and its consequences for imaging: A comparison of strategies

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. S365-S372 ◽  
Author(s):  
Lele Zhang ◽  
Jan Thorbecke ◽  
Kees Wapenaar ◽  
Evert Slob
Keyword(s):  
Inverse Scattering ◽  
Computational Cost ◽  
Multiple Reflection ◽  
Data Driven ◽  
Transmission Losses ◽  
Multiple Reflections ◽  
Data Set ◽  
Numerical Examples ◽  
Inverse Scattering Series ◽  
Multiple Elimination

We have compared three data-driven internal multiple reflection elimination schemes derived from the Marchenko equations and inverse scattering series (ISS). The two schemes derived from Marchenko equations are similar but use different truncation operators. The first scheme creates a new data set without internal multiple reflections. The second scheme does the same and compensates for transmission losses in the primary reflections. The scheme derived from ISS is equal to the result after the first iteration of the first Marchenko-based scheme. It can attenuate internal multiple reflections with residuals. We evaluate the success of these schemes with 2D numerical examples. It is shown that Marchenko-based data-driven schemes are relatively more robust for internal multiple reflection elimination at a higher computational cost.

scholarly journals An Operator Approach to Tangent Vector Field Processing

2013 ◽  
Vol 32 (5) ◽  
pp. 73-82 ◽  
Author(s):  
Omri Azencot ◽  
Mirela Ben-Chen ◽  
Frédéric Chazal ◽  
Maks Ovsjanikov
Keyword(s):  
Vector Field ◽  
Tangent Vector ◽  
Tangent Vector Field ◽  
Operator Approach
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