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MikaelSlevinskyMikaelSlevinsky
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Merge pull request #4 from JulienMLN/master
A rapid and well-conditioned algorithm for the Helmholtz–Hodge decomposition of vector fields on the sphere
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‎src/gradient.jl

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# This file calculates the surface gradient of a scalar field.
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function gradient!(U::Matrix{T}, ∇θU::Matrix{T}, ∇φU::Matrix{T}) where T
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@assert size(U) == size(∇θU) == size(∇φU)
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N, M = size(U)
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# The first column is easy.
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@inbounds @simd for ℓ = 1:N-1
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∇θU[ℓ, 1] = -sqrt(T(ℓ*(ℓ+1)))*U[ℓ+1, 1]
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∇φU[ℓ, 1] = 0
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end
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∇θU[N, 1] = 0
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∇φU[N, 1] = 0
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# Next, we differentiate with respect to φ, which preserves the order. It swaps sines and cosines in longitude, though.
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@inbounds for m = 1:M÷2
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@simd for ℓ = 1:N+1-m
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∇φU[ℓ, 2m] = -m*U[ℓ, 2m+1]
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∇φU[ℓ, 2m+1] = m*U[ℓ, 2m]
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end
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end
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# Then, we differentiate with respect to θ, which preserves the order but divides by sin(θ).
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@inbounds for m = 1:M÷2
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ℓ = 1
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bℓ = -(ℓ+m+1)*sqrt(T(ℓ*(ℓ+2m))/T((2ℓ+2m-1)*(2ℓ+2m+1)))
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∇θU[ℓ, 2m] = bℓ*U[ℓ+1, 2m]
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∇θU[ℓ, 2m+1] = bℓ*U[ℓ+1, 2m+1]
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@simd for ℓ = 2:N-m
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aℓ = (ℓ+m-2)*sqrt(T((ℓ-1)*(ℓ+2m-1))/T((2ℓ+2m-3)*(2ℓ+2m-1)))
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bℓ = -(ℓ+m+1)*sqrt(T(ℓ*(ℓ+2m))/T((2ℓ+2m-1)*(2ℓ+2m+1)))
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∇θU[ℓ, 2m] = aℓ*U[ℓ-1, 2m] + bℓ*U[ℓ+1, 2m]
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∇θU[ℓ, 2m+1] = aℓ*U[ℓ-1, 2m+1] + bℓ*U[ℓ+1, 2m+1]
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end
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ℓ = N-m+1
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aℓ = (ℓ+m-2)*sqrt(T((ℓ-1)*(ℓ+2m-1))/T((2ℓ+2m-3)*(2ℓ+2m-1)))
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∇θU[ℓ, 2m] = aℓ*U[ℓ-1, 2m]
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∇θU[ℓ, 2m+1] = aℓ*U[ℓ-1, 2m+1]
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end
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# Finally, we divide by sin(θ), which can be done by decrementing the order P_ℓ^m ↘ P_ℓ^{m-1}.
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@inbounds for m = 1:M÷2
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ℓ = N+1-m
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aℓ = sqrt(T((ℓ+2m-2)*(ℓ+2m-1))/T((2ℓ+2m-3)*(2ℓ+2m-1)))
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∇θU[ℓ, 2m] = ∇θU[ℓ, 2m]/aℓ
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∇θU[ℓ, 2m+1] = ∇θU[ℓ, 2m+1]/aℓ
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∇φU[ℓ, 2m] = ∇φU[ℓ, 2m]/aℓ
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∇φU[ℓ, 2m+1] = ∇φU[ℓ, 2m+1]/aℓ
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ℓ = N+1-m-1
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if ℓ > 0
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aℓ = sqrt(T((ℓ+2m-2)*(ℓ+2m-1))/T((2ℓ+2m-3)*(2ℓ+2m-1)))
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∇θU[ℓ, 2m] = ∇θU[ℓ, 2m]/aℓ
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∇θU[ℓ, 2m+1] = ∇θU[ℓ, 2m+1]/aℓ
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∇φU[ℓ, 2m] = ∇φU[ℓ, 2m]/aℓ
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∇φU[ℓ, 2m+1] = ∇φU[ℓ, 2m+1]/aℓ
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end
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@simd for ℓ = N+1-m-2:-1:1
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aℓ = sqrt(T((ℓ+2m-2)*(ℓ+2m-1))/T((2ℓ+2m-3)*(2ℓ+2m-1)))
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bℓ = -sqrt(T(ℓ*(ℓ+1))/T((2ℓ+2m-1)*(2ℓ+2m+1)))
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∇θU[ℓ, 2m] = (∇θU[ℓ, 2m] - bℓ*∇θU[ℓ+2, 2m])/aℓ
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∇θU[ℓ, 2m+1] = (∇θU[ℓ, 2m+1] - bℓ*∇θU[ℓ+2, 2m+1])/aℓ
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∇φU[ℓ, 2m] = (∇φU[ℓ, 2m] - bℓ*∇φU[ℓ+2, 2m])/aℓ
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∇φU[ℓ, 2m+1] = (∇φU[ℓ, 2m+1] - bℓ*∇φU[ℓ+2, 2m+1])/aℓ
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end
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end
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∇θU
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end
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function curl!(U::Matrix, U1::Matrix, U2::Matrix)
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gradient!(U, U2, U1)
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N, M = size(U)
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@inbounds for j = 1:M
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for i = 1:N
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U1[i,j] = -U1[i,j]
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end
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end
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U1
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end
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function partial_gradient!(U::Matrix{T}, ∇θU::Matrix{T}, ∇φU::Matrix{T}) where T
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@assert size(U) == size(∇θU) == size(∇φU)
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N, M = size(U)
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# The first column is easy.
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@inbounds @simd for ℓ = 1:N-1
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∇θU[ℓ, 1] = -sqrt(T(ℓ*(ℓ+1)))*U[ℓ+1, 1]
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∇φU[ℓ, 1] = 0
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end
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∇θU[N, 1] = 0
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∇φU[N, 1] = 0
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# Next, we differentiate with respect to φ, which preserves the order. It swaps sines and cosines in longitude, though.
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@inbounds for m = 1:M÷2
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@simd for ℓ = 1:N+1-m
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∇φU[ℓ, 2m] = -m*U[ℓ, 2m+1]
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∇φU[ℓ, 2m+1] = m*U[ℓ, 2m]
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end
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end
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# Then, we differentiate with respect to θ, which preserves the order but divides by sin(θ).
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@inbounds for m = 1:M÷2
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ℓ = 1
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bℓ = -(ℓ+m+1)*sqrt(T(ℓ*(ℓ+2m))/T((2ℓ+2m-1)*(2ℓ+2m+1)))
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∇θU[ℓ, 2m] = bℓ*U[ℓ+1, 2m]
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∇θU[ℓ, 2m+1] = bℓ*U[ℓ+1, 2m+1]
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@simd for ℓ = 2:N-m
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aℓ = (ℓ+m-2)*sqrt(T((ℓ-1)*(ℓ+2m-1))/T((2ℓ+2m-3)*(2ℓ+2m-1)))
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bℓ = -(ℓ+m+1)*sqrt(T(ℓ*(ℓ+2m))/T((2ℓ+2m-1)*(2ℓ+2m+1)))
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∇θU[ℓ, 2m] = aℓ*U[ℓ-1, 2m] + bℓ*U[ℓ+1, 2m]
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∇θU[ℓ, 2m+1] = aℓ*U[ℓ-1, 2m+1] + bℓ*U[ℓ+1, 2m+1]
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end
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ℓ = N-m+1
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aℓ = (ℓ+m-2)*sqrt(T((ℓ-1)*(ℓ+2m-1))/T((2ℓ+2m-3)*(2ℓ+2m-1)))
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∇θU[ℓ, 2m] = aℓ*U[ℓ-1, 2m]
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∇θU[ℓ, 2m+1] = aℓ*U[ℓ-1, 2m+1]
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end
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∇θU
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end

‎src/helmholtzhodge.jl

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include("gradient.jl")
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using BandedMatrices
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import BandedMatrices: BandedQ
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# Store QR factorizations required to apply the Helmholtz-Hodge decomposition.
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struct HelmholtzHodge{T}
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Q::Vector{BandedQ{T}}
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R::Vector{BandedMatrix{T, Matrix{T}}}
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X::Vector{T}
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end
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function HelmholtzHodge(::Type{T}, N::Int) where T
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Q = Vector{BandedQ{T}}(N)
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R = Vector{BandedMatrix{T, Matrix{T}}}(N)
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for m = 1:N
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Q[m], R[m] = qr(helmholtzhodgeconversion(T, N, m))
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end
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HelmholtzHodge(Q, R, zeros(T, 2N+2))
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end
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function helmholtzhodgeconversion(::Type{T}, N::Int, m::Int) where T
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A = BandedMatrix(Zeros{T}(2N+4-2m, 2N+2-2m), (2, 2))
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for ℓ = 1:N+1-m
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A[2ℓ, 2ℓ-1] = m
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A[2ℓ-1, 2ℓ] = m
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cst = (m+ℓ-1)*sqrt(T(ℓ*(ℓ+2m))/T((2ℓ+2m-1)*(2ℓ+2m+1)))
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A[2ℓ+2, 2ℓ] = cst
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A[2ℓ+1, 2ℓ-1] = cst
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end
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for ℓ = 1:N-m
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cst = -(m+ℓ+1)*sqrt(T(ℓ*(ℓ+2m))/T((2ℓ+2m-1)*(2ℓ+2m+1)))
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A[2ℓ-1, 2ℓ+1] = cst
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A[2ℓ, 2ℓ+2] = cst
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end
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A
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end
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# This function works in-place on the data stored in the factorization.
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function solvehelmholtzhodge!(HH::HelmholtzHodge{T}, m::Int) where T
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Q = HH.Q[m]
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R = HH.R[m]
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X = HH.X
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# Step 1: apply Q'
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H=Q.H
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m=Q.m
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M=size(H,1)
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x=pointer(X)
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h=pointer(H)
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st=stride(H,2)
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sz=sizeof(T)
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for k=1:min(size(H,2),m-M+1)
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wp=h+sz*st*(k-1)
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xp=x+sz*(k-1)
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dt=BandedMatrices.dot(M,wp,1,xp,1)
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BandedMatrices.axpy!(M,-2*dt,wp,1,xp,1)
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end
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for k=m-M+2:size(H,2)
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p=k-m+M-1
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wp=h+sz*st*(k-1)
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xp=x+sz*(k-1)
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dt=BandedMatrices.dot(M-p,wp,1,xp,1)
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BandedMatrices.axpy!(M-p,-2*dt,wp,1,xp,1)
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end
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# Step 2: backsolve with (square) R
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BandedMatrices.tbsv!('U', 'N', 'N', size(R.data, 2), R.u, pointer(R.data), size(R.data, 1), pointer(X), 1)
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X
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end
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function helmholtzhodge!(HH::HelmholtzHodge{T}, U1, U2, V1, V2) where T
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N, M = size(V1)
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# U1 is for e_theta and U2 is for e_phi.
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# The first columns are easy.
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U1[1, 1] = 0
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U2[1, 1] = 0
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@inbounds @simd for ℓ = 1:N-1
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U1[ℓ+1, 1] = -V1[ℓ, 1]/sqrt(T(ℓ*(ℓ+1)))
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U2[ℓ+1, 1] = -V2[ℓ, 1]/sqrt(T(ℓ*(ℓ+1)))
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end
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# First, we multiply by sin(θ), which can be done by incrementing the order P_ℓ^{m-1} ↗ P_ℓ^m.
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@inbounds for m = 1:M÷2
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@simd for ℓ = 1:N-1-m
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aℓ = sqrt(T((ℓ+2m-2)*(ℓ+2m-1))/T((2ℓ+2m-3)*(2ℓ+2m-1)))
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bℓ = -sqrt(T(ℓ*(ℓ+1))/T((2ℓ+2m-1)*(2ℓ+2m+1)))
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V1[ℓ, 2m] = aℓ*V1[ℓ, 2m] + bℓ*V1[ℓ+2, 2m]
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V1[ℓ, 2m+1] = aℓ*V1[ℓ, 2m+1] + bℓ*V1[ℓ+2, 2m+1]
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V2[ℓ, 2m] = aℓ*V2[ℓ, 2m] + bℓ*V2[ℓ+2, 2m]
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V2[ℓ, 2m+1] = aℓ*V2[ℓ, 2m+1] + bℓ*V2[ℓ+2, 2m+1]
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end
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ℓ = N-m
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if ℓ > 0
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aℓ = sqrt(T((ℓ+2m-2)*(ℓ+2m-1))/T((2ℓ+2m-3)*(2ℓ+2m-1)))
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V1[ℓ, 2m] = aℓ*V1[ℓ, 2m]
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V1[ℓ, 2m+1] = aℓ*V1[ℓ, 2m+1]
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V2[ℓ, 2m] = aℓ*V2[ℓ, 2m]
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V2[ℓ, 2m+1] = aℓ*V2[ℓ, 2m+1]
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end
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ℓ = N+1-m
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aℓ = sqrt(T((ℓ+2m-2)*(ℓ+2m-1))/T((2ℓ+2m-3)*(2ℓ+2m-1)))
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V1[ℓ, 2m] = aℓ*V1[ℓ, 2m]
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V1[ℓ, 2m+1] = aℓ*V1[ℓ, 2m+1]
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V2[ℓ, 2m] = aℓ*V2[ℓ, 2m]
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V2[ℓ, 2m+1] = aℓ*V2[ℓ, 2m+1]
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end
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# Next, we solve the banded linear systems.
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for m = 1:M÷2
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readin1!(HH.X, V1, V2, N, m)
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solvehelmholtzhodge!(HH, m)
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writeout1!(HH.X, U1, U2, N, m)
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readin2!(HH.X, V1, V2, N, m)
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solvehelmholtzhodge!(HH, m)
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writeout2!(HH.X, U1, U2, N, m)
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end
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U1
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end
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function readin1!(X, V1, V2, N, m)
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X[1] = V1[1, 2m]
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X[2N+2-2m] = V2[N+1-m, 2m+1]
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@inbounds for ℓ = 1:N-m
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X[2ℓ] = V2[ℓ, 2m+1]
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X[2ℓ+1] = V1[ℓ+1, 2m]
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end
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X[2N+3-2m] = X[2N+4-2m] = 0
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end
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function readin2!(X, V1, V2, N, m)
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X[1] = V1[1, 2m+1]
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X[2N+2-2m] = -V2[N+1-m, 2m]
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@inbounds for ℓ = 1:N-m
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X[2ℓ] = -V2[ℓ, 2m]
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X[2ℓ+1] = V1[ℓ+1, 2m+1]
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end
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X[2N+3-2m] = X[2N+4-2m] = 0
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end
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function writeout1!(X, U1, U2, N, m)
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U1[1, 2m] = X[1]
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U2[1, 2m+1] = X[N+1-m]
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@inbounds for ℓ = 1:N-1-m
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U1[ℓ+1, 2m] = X[2ℓ+1]
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U2[ℓ, 2m+1] = X[2ℓ]
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end
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U2[N-m, 2m+1] = X[2N-2m]
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end
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function writeout2!(X, U1, U2, N, m)
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U1[1, 2m+1] = X[1]
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U2[1, 2m] = X[N+1-m]
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@inbounds for ℓ = 1:N-1-m
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U1[ℓ+1, 2m+1] = X[2ℓ+1]
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U2[ℓ, 2m] = -X[2ℓ]
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end
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U2[N-m, 2m] = -X[2N-2m]
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end

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