# FaVeST: Fast Vector Spherical Harmonic Transforms

@article{Gia2019FaVeSTFV, title={FaVeST: Fast Vector Spherical Harmonic Transforms}, author={Quoc Thong Le Gia and Ming Li and Yu Guang Wang}, journal={ArXiv}, year={2019}, volume={abs/1908.00041} }

Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ 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 paper, 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… Expand

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#### References

SHOWING 1-10 OF 76 REFERENCES

Fast Algorithms for Spherical Harmonic Expansions

- Mathematics, Computer Science
- SIAM J. Sci. Comput.
- 2006

An algorithm is introduced for the rapid evaluation at appropriately chosen nodes on the two-dimensional sphere of functions specified by their spherical harmonic expansions (known as the inverse spherical harmonic transform); the performance of the algorithm is illustrated via several numerical examples. Expand

Fast Tensor Needlet Transforms for Tangent Vector Fields on the Sphere

- Computer Science, Mathematics
- ArXiv
- 2019

The proposed tight tensor needlets provide a multiscale representation of any square integrable tangent vector field on $\mathbb{S}^2$, which leads to a multiresolution analysis (MRA) for the field. Expand

A fast transform for spherical harmonics

- Mathematics
- 1997

AbstractSpherical harmonics arise on the sphere S2 in the same way that the (Fourier) exponential functions {eikθ}k∈ℤ arise on the circle. Spherical harmonic series have many of the same wonderful… Expand

Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids

- Mathematics, Computer Science
- Adv. Comput. Math.
- 2018

A spectrally accurate numerical method for computing the spherical/vector spherical harmonic expansion of a function/vector field with given (elemental) nodal values on a spherical surface that is robust for high mode expansions. Expand

Algorithm 888: Spherical Harmonic Transform Algorithms

- Mathematics, Computer Science
- TOMS
- 2008

A collection of MATLAB classes for computing and using spherical harmonic transforms is presented and the use of the spectral synthesis and analysis algorithms using fast Fourier transforms and Legendre transforms with the associated Legendre functions is demonstrated. Expand

Fast evaluation of quadrature formulae on the sphere

- Computer Science, Mathematics
- Math. Comput.
- 2008

A new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules and results of numerical tests are provided showing the stability of the obtained algorithm. Expand

Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2009

A new numerical technique based on radial basis functions (RBFs) is presented for fitting a vector field tangent to the sphere from samples of the field at “scattered” locations on S2, providing a way to decompose the reconstructed field into its individual Helmholtz-Hodge components. Expand

On the computation of spherical designs by a new optimization approach based on fast spherical Fourier transforms

- Mathematics, Computer Science
- Numerische Mathematik
- 2011

This paper considers the problem of finding numerical spherical t-designs on the sphere for high polynomial degree t, and shows that by means of the nonequispaced fast spherical Fourier transforms, gradient and Hessian evaluations in arithmetic operations are performed. Expand

Fully discrete needlet approximation on the sphere

- Mathematics
- 2015

Spherical needlets are highly localized radial polynomials on the sphere $\mathbb{S}^{d}\subset \mathbb{R}^{d+1}$, $d\ge 2$, with centers at the nodes of a suitable cubature rule. The original… Expand

Generalized Discrete Spherical Harmonic Transforms

- Mathematics
- 2000

Two generalizations of the spherical harmonic transforms are provided. First, they are generalized to an arbitrary distribution of latitudinal points ?i. This unifies transforms for Gaussian and… Expand