--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Gradient.jl Mon Nov 18 11:00:54 2019 -0500
@@ -0,0 +1,185 @@
+########################
+# Discretised gradients
+########################
+
+module Gradient
+
+##############
+# Our exports
+##############
+
+export ∇₂!, ∇₂ᵀ!, ∇₂fold!,
+ ∇₂_norm₂₂_est, ∇₂_norm₂₂_est²,
+ ∇₂_norm₂∞_est, ∇₂_norm₂∞_est²,
+ ∇₂c!,
+ ∇₃!, ∇₃ᵀ!,
+ vec∇₃!, vec∇₃ᵀ!
+
+##################
+# Helper routines
+##################
+
+@inline function imfold₂′!(f_aa!, f_a0!, f_ab!,
+ f_0a!, f_00!, f_0b!,
+ f_ba!, f_b0!, f_bb!,
+ n, m, state)
+ # First row
+ state = f_aa!(state, (1, 1))
+ for j = 2:m-1
+ state = f_a0!(state, (1, j))
+ end
+ state = f_ab!(state, (1, m))
+
+ # Middle rows
+ for i=2:n-1
+ state = f_0a!(state, (i, 1))
+ for j = 2:m-1
+ state = f_00!(state, (i, j))
+ end
+ state = f_0b!(state, (i, m))
+ end
+
+ # Last row
+ state = f_ba!(state, (n, 1))
+ for j =2:m-1
+ state = f_b0!(state, (n, j))
+ end
+ return f_bb!(state, (n, m))
+end
+
+#########################
+# 2D forward differences
+#########################
+
+∇₂_norm₂₂_est² = 8
+∇₂_norm₂₂_est = √∇₂_norm₂₂_est²
+∇₂_norm₂∞_est² = 2
+∇₂_norm₂∞_est = √∇₂_norm₂∞_est²
+
+function ∇₂!(u₁, u₂, u)
+ @. @views begin
+ u₁[1:(end-1), :] = u[2:end, :] - u[1:(end-1), :]
+ u₁[end, :, :] = 0
+
+ u₂[:, 1:(end-1)] = u[:, 2:end] - u[:, 1:(end-1)]
+ u₂[:, end] = 0
+ end
+ return u₁, u₂
+end
+
+function ∇₂!(v, u)
+ ∇₂!(@view(v[1, :, :]), @view(v[2, :, :]), u)
+end
+
+@inline function ∇₂fold!(f!::Function, u, state)
+ g! = (state, pt) -> begin
+ (i, j) = pt
+ g = @inbounds [u[i+1, j]-u[i, j], u[i, j+1]-u[i, j]]
+ return f!(g, state, pt)
+ end
+ gr! = (state, pt) -> begin
+ (i, j) = pt
+ g = @inbounds [u[i+1, j]-u[i, j], 0.0]
+ return f!(g, state, pt)
+ end
+ gb! = (state, pt) -> begin
+ (i, j) = pt
+ g = @inbounds [0.0, u[i, j+1]-u[i, j]]
+ return f!(g, state, pt)
+ end
+ g0! = (state, pt) -> begin
+ return f!([0.0, 0.0], state, pt)
+ end
+ return imfold₂′!(g!, g!, gr!,
+ g!, g!, gr!,
+ gb!, gb!, g0!,
+ size(u, 1), size(u, 2), state)
+end
+
+function ∇₂ᵀ!(v, v₁, v₂)
+ @. @views begin
+ v[2:(end-1), :] = v₁[1:(end-2), :] - v₁[2:(end-1), :]
+ v[1, :] = -v₁[1, :]
+ v[end, :] = v₁[end-1, :]
+
+ v[:, 2:(end-1)] += v₂[:, 1:(end-2)] - v₂[:, 2:(end-1)]
+ v[:, 1] += -v₂[:, 1]
+ v[:, end] += v₂[:, end-1]
+ end
+ return v
+end
+
+function ∇₂ᵀ!(u, v)
+ ∇₂ᵀ!(u, @view(v[1, :, :]), @view(v[2, :, :]))
+end
+
+##################################################
+# 2D central differences (partial implementation)
+##################################################
+
+function ∇₂c!(v, u)
+ @. @views begin
+ v[1, 2:(end-1), :] = (u[3:end, :] - u[1:(end-2), :])/2
+ v[1, end, :] = (u[end, :] - u[end-1, :])/2
+ v[1, 1, :] = (u[2, :] - u[1, :])/2
+
+ v[2, :, 2:(end-1)] = (u[:, 3:end] - u[:, 1:(end-2)])/2
+ v[2, :, end] = (u[:, end] - u[:, end-1])/2
+ v[2, :, 1] = (u[:, 2] - u[:, 1])/2
+ end
+end
+
+#########################
+# 3D forward differences
+#########################
+
+function ∇₃!(u₁,u₂,u₃,u)
+ @. @views begin
+ u₁[1:(end-1), :, :] = u[2:end, :, :] - u[1:(end-1), :, :]
+ u₁[end, :, :] = 0
+
+ u₂[:, 1:(end-1), :] = u[:, 2:end, :] - u[:, 1:(end-1), :]
+ u₂[:, end, :] = 0
+
+ u₃[:, :, 1:(end-1)] = u[:, :, 2:end] - u[:, :, 1:(end-1)]
+ u₃[:, :, end] = 0
+ end
+ return u₁, u₂, u₃
+end
+
+function ∇₃ᵀ!(v,v₁,v₂,v₃)
+ @. @views begin
+ v[2:(end-1), :, :] = v₁[1:(end-2), :, :] - v₁[2:(end-1), :, :]
+ v[1, :, :] = -v₁[1, :, :]
+ v[end, :, :] = v₁[end-1, :, :]
+
+ v[:, 2:(end-1), :] += v₂[:, 1:(end-2), :] - v₂[:, 2:(end-1), :]
+ v[:, 1, :] += -v₂[:, 1, :]
+ v[:, end, :] += v₂[:, end-1, :]
+
+ v[:, :, 2:(end-1)] += v₃[:, :, 1:(end-2)] - v₃[:, :, 2:(end-1)]
+ v[:, :, 1] += -v₃[:, :, 1]
+ v[:, :, end] += v₃[:, :, end-1]
+ end
+ return v
+end
+
+###########################################
+# 3D forward differences for vector fields
+###########################################
+
+function vec∇₃!(u₁,u₂,u₃,u)
+ @. @views for j=1:size(u, 1)
+ ∇₃!(u₁[j, :, :, :],u₂[j, :, :, :],u₃[j, :, :, :],u[j, :, :, :])
+ end
+ return u₁, u₂, u₃
+end
+
+function vec∇₃ᵀ!(u,v₁,v₂,v₃)
+ @. @views for j=1:size(u, 1)
+ ∇₃ᵀ!(u[j, :, :, :],v₁[j, :, :, :],v₂[j, :, :, :],v₃[j, :, :, :])
+ end
+ return u
+end
+
+end # Module
