[Paper Review] FaVeST: Fast Vector Spherical Harmonic Transforms

Quôc Thông Lê Gia, Ming Li+1 more|arXiv (Cornell University)|Jul 31, 2019

Geophysics and Gravity Measurements68 references4 citations

TL;DR

This paper introduces FaVeST, a fast algorithm for computing forward and adjoint vector spherical harmonic transforms (FwdVSHT and AdjVSHT) on the unit sphere, leveraging scalar spherical harmonic transforms and Clebsch-Gordan coefficients to achieve near-linear computational complexity. The method reduces cost to 𝒪(N log √N) for FwdVSHT and 𝒪(M log √M) for AdjVSHT, with validation showing high accuracy, efficiency, and numerical stability on simulated tangent fields.

ABSTRACT

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 $N\log \sqrt{N}$ for $N$ number of evaluation points. The adjoint FaVeST which evaluates a linear combination of vector spherical harmonics with a degree up to $\sqrt{M}$ for $M$ evaluation points has cost proportional to $M\log\sqrt{M}$. Numerical examples of simulated tangent fields illustrate the accuracy, efficiency and stability of FaVeST.

Motivation & Objective

• To address the lack of efficient algorithms for computing vector spherical harmonic transforms in applications involving tangent vector fields on the sphere.
• To reduce the computational cost of FwdVSHT and AdjVSHT from 𝒪(N²) and 𝒪(M²) to near-linear complexity.
• To develop a practical, software-implemented solution for fast transforms applicable in geophysics, astrophysics, and fluid dynamics.
• To validate the accuracy, stability, and efficiency of the proposed algorithm through numerical experiments on simulated tangent fields.

Proposed method

• Representing divergence-free and curl-free vector spherical harmonics explicitly in terms of scalar spherical harmonics using Clebsch-Gordan coefficients.
• Formulating the FwdVSHT and AdjVSHT as compositions of scalar spherical harmonic transforms via these representations.
• Applying fast Fourier transform (FFT) techniques on the sphere to accelerate the evaluation of scalar harmonic coefficients.
• Using polynomial-exact quadrature rules to ensure high-accuracy discretization of integrals in the transform computations.
• Implementing the algorithm in a MATLAB software package for practical use and integration into existing solvers.
• Validating the method through numerical experiments with simulated tangent fields across varying node counts.

Experimental results

Research questions

• RQ1 Can vector spherical harmonic transforms for tangent fields be computed in near-linear time using existing fast scalar harmonic transforms?
• RQ2 How can Clebsch-Gordan coefficients be leveraged to express vector spherical harmonics in terms of scalar harmonics for computational acceleration?
• RQ3 What is the achievable accuracy and stability of the proposed fast algorithm across different numbers of evaluation points?
• RQ4 Can the algorithm be efficiently implemented in software and integrated into existing computational frameworks for PDEs on the sphere?

Key findings

• The forward FaVeST achieves a computational cost of 𝒪(N log √N) for N evaluation points, significantly reducing the standard 𝒪(N²) complexity.
• The adjoint FaVeST operates in 𝒪(M log √M) time for M evaluation points, enabling fast reconstruction of vector fields from harmonic coefficients.
• Numerical experiments show that the relative errors in the forward transform grow slowly with N, with the normalized error 𝒪(1/N) indicating good convergence and stability.
• The algorithm demonstrates high numerical stability, as evidenced by decreasing normalized error envelopes with increasing N in both forward and adjoint transforms.
• The software package is functional and has already been applied to accelerate tensor needlet transforms for tangent fields on the sphere.
• The method is suitable for integration into solvers for PDEs such as Stokes and Navier-Stokes equations on the sphere, with future work targeting HEALPix grid support in Python.