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interpolation.jl
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interpolation.jl
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# Naive implementations of multiply_dimensionwise used to demonstrate the functionality
# without performance optimizations and for testing correctness of the optimized versions
# implemented below.
function multiply_dimensionwise_naive(matrix::AbstractMatrix,
data_in::AbstractArray{<:Any, 2})
size_out = size(matrix, 1)
size_in = size(matrix, 2)
n_vars = size(data_in, 1)
data_out = zeros(promote_type(eltype(data_in), eltype(matrix)), n_vars, size_out)
for i in 1:size_out
for ii in 1:size_in
for v in 1:n_vars
data_out[v, i] += matrix[i, ii] * data_in[v, ii]
end
end
end
return data_out
end
function multiply_dimensionwise_naive(matrix::AbstractMatrix,
data_in::AbstractArray{<:Any, 3})
size_out = size(matrix, 1)
size_in = size(matrix, 2)
n_vars = size(data_in, 1)
data_out = zeros(promote_type(eltype(data_in), eltype(matrix)), n_vars, size_out,
size_out)
for j in 1:size_out, i in 1:size_out
for jj in 1:size_in, ii in 1:size_in
for v in 1:n_vars
data_out[v, i, j] += matrix[i, ii] * matrix[j, jj] * data_in[v, ii, jj]
end
end
end
return data_out
end
function multiply_dimensionwise_naive(matrix::AbstractMatrix,
data_in::AbstractArray{<:Any, 4})
size_out = size(matrix, 1)
size_in = size(matrix, 2)
n_vars = size(data_in, 1)
data_out = zeros(promote_type(eltype(data_in), eltype(matrix)), n_vars, size_out,
size_out, size_out)
for k in 1:size_out, j in 1:size_out, i in 1:size_out
for kk in 1:size_in, jj in 1:size_in, ii in 1:size_in
for v in 1:n_vars
data_out[v, i, j, k] += matrix[i, ii] * matrix[j, jj] * matrix[k, kk] *
data_in[v, ii, jj, kk]
end
end
end
return data_out
end
"""
multiply_dimensionwise(matrix::AbstractMatrix, data_in::AbstractArray{<:Any, NDIMS+1})
Multiply the array `data_in` by `matrix` in each coordinate direction, where `data_in`
is assumed to have the first coordinate for the number of variables and the remaining coordinates
are multiplied by `matrix`.
"""
function multiply_dimensionwise(matrix::AbstractMatrix, data_in::AbstractArray{<:Any, 2})
# 1D
# optimized version of multiply_dimensionwise_naive
size_out = size(matrix, 1)
n_vars = size(data_in, 1)
data_out = zeros(promote_type(eltype(data_in), eltype(matrix)), n_vars, size_out)
multiply_dimensionwise!(data_out, matrix, data_in)
return data_out
end
function multiply_dimensionwise(matrix::AbstractMatrix, data_in::AbstractArray{<:Any, 3})
# 2D
# optimized version of multiply_dimensionwise_naive
size_out = size(matrix, 1)
n_vars = size(data_in, 1)
data_out = zeros(promote_type(eltype(data_in), eltype(matrix)), n_vars, size_out,
size_out)
multiply_dimensionwise!(data_out, matrix, data_in)
return data_out
end
function multiply_dimensionwise(matrix::AbstractMatrix, data_in::AbstractArray{<:Any, 4})
# 3D
# optimized version of multiply_dimensionwise_naive
size_out = size(matrix, 1)
n_vars = size(data_in, 1)
data_out = zeros(promote_type(eltype(data_in), eltype(matrix)), n_vars, size_out,
size_out, size_out)
multiply_dimensionwise!(data_out, matrix, data_in)
return data_out
end
# In the following, there are several optimized in-place versions of multiply_dimensionwise.
# These may make use of advanced optimization features such as the macro `@tullio` from Tullio.jl,
# which basically uses an Einstein summation convention syntax.
# Another possibility is `@turbo` from LoopVectorization.jl. The runtime performance could be even
# optimized further by using `@turbo inline=true for` instead of `@turbo for`, but that comes at the
# cost of increased latency, at least on some systems...
# 1D version
function multiply_dimensionwise!(data_out::AbstractArray{<:Any, 2}, matrix::AbstractMatrix,
data_in::AbstractArray{<:Any, 2})
# @tullio threads=false data_out[v, i] = matrix[i, ii] * data_in[v, ii]
@turbo for i in axes(data_out, 2), v in axes(data_out, 1)
res = zero(eltype(data_out))
for ii in axes(matrix, 2)
res += matrix[i, ii] * data_in[v, ii]
end
data_out[v, i] = res
end
return nothing
end
# 1D version for scalars
# Instead of having a leading dimension of size 1 in `data_out, data_in`, this leading dimension
# of size unity is dropped, resulting in one dimension less than in `multiply_dimensionwise!`.
function multiply_scalar_dimensionwise!(data_out::AbstractArray{<:Any, 1},
matrix::AbstractMatrix,
data_in::AbstractArray{<:Any, 1})
# @tullio threads=false data_out[i] = matrix[i, ii] * data_in[ii]
@turbo for i in axes(data_out, 1)
res = zero(eltype(data_out))
for ii in axes(matrix, 2)
res += matrix[i, ii] * data_in[ii]
end
data_out[i] = res
end
return nothing
end
# 1D version, apply matrixJ to data_inJ
function multiply_dimensionwise!(data_out::AbstractArray{<:Any, 2}, matrix1::AbstractMatrix,
data_in1::AbstractArray{<:Any, 2}, matrix2::AbstractMatrix,
data_in2::AbstractArray{<:Any, 2})
# @tullio threads=false data_out[v, i] = matrix1[i, ii] * data_in1[v, ii] + matrix2[i, ii] * data_in2[v, ii]
# TODO: LoopVectorization upgrade
# We would like to use `@turbo` for the outermost loop possibly fuse both inner
# loops, but that does currently not work because of limitations of
# LoopVectorizationjl. However, Chris Elrod is planning to address this in
# the future, cf. https://github.com/JuliaSIMD/LoopVectorization.jl/issues/230#issuecomment-810632972
@turbo for i in axes(data_out, 2), v in axes(data_out, 1)
res = zero(eltype(data_out))
for ii in axes(matrix1, 2)
res += matrix1[i, ii] * data_in1[v, ii]
end
data_out[v, i] = res
end
@turbo for i in axes(data_out, 2), v in axes(data_out, 1)
res = zero(eltype(data_out))
for ii in axes(matrix2, 2)
res += matrix2[i, ii] * data_in2[v, ii]
end
data_out[v, i] += res
end
return nothing
end
# 2D version
function multiply_dimensionwise!(data_out::AbstractArray{<:Any, 3}, matrix::AbstractMatrix,
data_in::AbstractArray{<:Any, 3},
tmp1 = zeros(eltype(data_out), size(data_out, 1),
size(matrix, 1), size(matrix, 2)))
# Interpolate in x-direction
# @tullio threads=false tmp1[v, i, j] = matrix[i, ii] * data_in[v, ii, j]
@turbo for j in axes(tmp1, 3), i in axes(tmp1, 2), v in axes(tmp1, 1)
res = zero(eltype(tmp1))
for ii in axes(matrix, 2)
res += matrix[i, ii] * data_in[v, ii, j]
end
tmp1[v, i, j] = res
end
# Interpolate in y-direction
# @tullio threads=false data_out[v, i, j] = matrix[j, jj] * tmp1[v, i, jj]
@turbo for j in axes(data_out, 3), i in axes(data_out, 2), v in axes(data_out, 1)
res = zero(eltype(data_out))
for jj in axes(matrix, 2)
res += matrix[j, jj] * tmp1[v, i, jj]
end
data_out[v, i, j] = res
end
return nothing
end
# 2D version for scalars
# Instead of having a leading dimension of size 1 in `data_out, data_in`, this leading dimension
# of size unity is dropped, resulting in one dimension less than in `multiply_dimensionwise!`.
function multiply_scalar_dimensionwise!(data_out::AbstractArray{<:Any, 2},
matrix::AbstractMatrix,
data_in::AbstractArray{<:Any, 2},
tmp1 = zeros(eltype(data_out), size(matrix, 1),
size(matrix, 2)))
# Interpolate in x-direction
# @tullio threads=false tmp1[i, j] = matrix[i, ii] * data_in[ii, j]
@turbo for j in axes(tmp1, 2), i in axes(tmp1, 1)
res = zero(eltype(tmp1))
for ii in axes(matrix, 2)
res += matrix[i, ii] * data_in[ii, j]
end
tmp1[i, j] = res
end
# Interpolate in y-direction
# @tullio threads=false data_out[i, j] = matrix[j, jj] * tmp1[i, jj]
@turbo for j in axes(data_out, 2), i in axes(data_out, 1)
res = zero(eltype(data_out))
for jj in axes(matrix, 2)
res += matrix[j, jj] * tmp1[i, jj]
end
data_out[i, j] = res
end
return nothing
end
# 2D version, apply matrixJ to dimension J of data_in
function multiply_dimensionwise!(data_out::AbstractArray{<:Any, 3},
matrix1::AbstractMatrix, matrix2::AbstractMatrix,
data_in::AbstractArray{<:Any, 3},
tmp1 = zeros(eltype(data_out), size(data_out, 1),
size(matrix1, 1), size(matrix1, 2)))
# Interpolate in x-direction
# @tullio threads=false tmp1[v, i, j] = matrix1[i, ii] * data_in[v, ii, j]
@turbo for j in axes(tmp1, 3), i in axes(tmp1, 2), v in axes(tmp1, 1)
res = zero(eltype(tmp1))
for ii in axes(matrix1, 2)
res += matrix1[i, ii] * data_in[v, ii, j]
end
tmp1[v, i, j] = res
end
# Interpolate in y-direction
# @tullio threads=false data_out[v, i, j] = matrix2[j, jj] * tmp1[v, i, jj]
@turbo for j in axes(data_out, 3), i in axes(data_out, 2), v in axes(data_out, 1)
res = zero(eltype(data_out))
for jj in axes(matrix2, 2)
res += matrix2[j, jj] * tmp1[v, i, jj]
end
data_out[v, i, j] = res
end
return nothing
end
# 2D version, apply matrixJ to dimension J of data_in and add the result to data_out
function add_multiply_dimensionwise!(data_out::AbstractArray{<:Any, 3},
matrix1::AbstractMatrix, matrix2::AbstractMatrix,
data_in::AbstractArray{<:Any, 3},
tmp1 = zeros(eltype(data_out), size(data_out, 1),
size(matrix1, 1), size(matrix1, 2)))
# Interpolate in x-direction
# @tullio threads=false tmp1[v, i, j] = matrix1[i, ii] * data_in[v, ii, j]
@turbo for j in axes(tmp1, 3), i in axes(tmp1, 2), v in axes(tmp1, 1)
res = zero(eltype(tmp1))
for ii in axes(matrix1, 2)
res += matrix1[i, ii] * data_in[v, ii, j]
end
tmp1[v, i, j] = res
end
# Interpolate in y-direction
# @tullio threads=false data_out[v, i, j] += matrix2[j, jj] * tmp1[v, i, jj]
@turbo for j in axes(data_out, 3), i in axes(data_out, 2), v in axes(data_out, 1)
res = zero(eltype(data_out))
for jj in axes(matrix2, 2)
res += matrix2[j, jj] * tmp1[v, i, jj]
end
data_out[v, i, j] += res
end
return nothing
end
# 3D version
function multiply_dimensionwise!(data_out::AbstractArray{<:Any, 4}, matrix::AbstractMatrix,
data_in::AbstractArray{<:Any, 4},
tmp1 = zeros(eltype(data_out), size(data_out, 1),
size(matrix, 1), size(matrix, 2),
size(matrix, 2)),
tmp2 = zeros(eltype(data_out), size(data_out, 1),
size(matrix, 1), size(matrix, 1),
size(matrix, 2)))
# Interpolate in x-direction
# @tullio threads=false tmp1[v, i, j, k] = matrix[i, ii] * data_in[v, ii, j, k]
@turbo for k in axes(tmp1, 4), j in axes(tmp1, 3), i in axes(tmp1, 2),
v in axes(tmp1, 1)
res = zero(eltype(tmp1))
for ii in axes(matrix, 2)
res += matrix[i, ii] * data_in[v, ii, j, k]
end
tmp1[v, i, j, k] = res
end
# Interpolate in y-direction
# @tullio threads=false tmp2[v, i, j, k] = matrix[j, jj] * tmp1[v, i, jj, k]
@turbo for k in axes(tmp2, 4), j in axes(tmp2, 3), i in axes(tmp2, 2),
v in axes(tmp2, 1)
res = zero(eltype(tmp2))
for jj in axes(matrix, 2)
res += matrix[j, jj] * tmp1[v, i, jj, k]
end
tmp2[v, i, j, k] = res
end
# Interpolate in z-direction
# @tullio threads=false data_out[v, i, j, k] = matrix[k, kk] * tmp2[v, i, j, kk]
@turbo for k in axes(data_out, 4), j in axes(data_out, 3), i in axes(data_out, 2),
v in axes(data_out, 1)
res = zero(eltype(data_out))
for kk in axes(matrix, 2)
res += matrix[k, kk] * tmp2[v, i, j, kk]
end
data_out[v, i, j, k] = res
end
return nothing
end
# 3D version for scalars
# Instead of having a leading dimension of size 1 in `data_out, data_in`, this leading dimension
# of size unity is dropped, resulting in one dimension less than in `multiply_dimensionwise!`.
function multiply_scalar_dimensionwise!(data_out::AbstractArray{<:Any, 3},
matrix::AbstractMatrix,
data_in::AbstractArray{<:Any, 3},
tmp1 = zeros(eltype(data_out), size(matrix, 1),
size(matrix, 2), size(matrix, 2)),
tmp2 = zeros(eltype(data_out), size(matrix, 1),
size(matrix, 1), size(matrix, 2)))
# Interpolate in x-direction
# @tullio threads=false tmp1[i, j, k] = matrix[i, ii] * data_in[ii, j, k]
@turbo for k in axes(tmp1, 3), j in axes(tmp1, 2), i in axes(tmp1, 1)
res = zero(eltype(tmp1))
for ii in axes(matrix, 2)
res += matrix[i, ii] * data_in[ii, j, k]
end
tmp1[i, j, k] = res
end
# Interpolate in y-direction
# @tullio threads=false tmp2[i, j, k] = matrix[j, jj] * tmp1[i, jj, k]
@turbo for k in axes(tmp2, 3), j in axes(tmp2, 2), i in axes(tmp2, 1)
res = zero(eltype(tmp2))
for jj in axes(matrix, 2)
res += matrix[j, jj] * tmp1[i, jj, k]
end
tmp2[i, j, k] = res
end
# Interpolate in z-direction
# @tullio threads=false data_out[i, j, k] = matrix[k, kk] * tmp2[i, j, kk]
@turbo for k in axes(data_out, 3), j in axes(data_out, 2), i in axes(data_out, 1)
res = zero(eltype(data_out))
for kk in axes(matrix, 2)
res += matrix[k, kk] * tmp2[i, j, kk]
end
data_out[i, j, k] = res
end
return nothing
end
# 3D version, apply matrixJ to dimension J of data_in
function multiply_dimensionwise!(data_out::AbstractArray{<:Any, 4},
matrix1::AbstractMatrix, matrix2::AbstractMatrix,
matrix3::AbstractMatrix,
data_in::AbstractArray{<:Any, 4},
tmp1 = zeros(eltype(data_out), size(data_out, 1),
size(matrix1, 1), size(matrix1, 2),
size(matrix1, 2)),
tmp2 = zeros(eltype(data_out), size(data_out, 1),
size(matrix1, 1), size(matrix1, 1),
size(matrix1, 2)))
# Interpolate in x-direction
# @tullio threads=false tmp1[v, i, j, k] = matrix1[i, ii] * data_in[v, ii, j, k]
@turbo for k in axes(tmp1, 4), j in axes(tmp1, 3), i in axes(tmp1, 2),
v in axes(tmp1, 1)
res = zero(eltype(tmp1))
for ii in axes(matrix1, 2)
res += matrix1[i, ii] * data_in[v, ii, j, k]
end
tmp1[v, i, j, k] = res
end
# Interpolate in y-direction
# @tullio threads=false tmp2[v, i, j, k] = matrix2[j, jj] * tmp1[v, i, jj, k]
@turbo for k in axes(tmp2, 4), j in axes(tmp2, 3), i in axes(tmp2, 2),
v in axes(tmp2, 1)
res = zero(eltype(tmp1))
for jj in axes(matrix2, 2)
res += matrix2[j, jj] * tmp1[v, i, jj, k]
end
tmp2[v, i, j, k] = res
end
# Interpolate in z-direction
# @tullio threads=false data_out[v, i, j, k] = matrix3[k, kk] * tmp2[v, i, j, kk]
@turbo for k in axes(data_out, 4), j in axes(data_out, 3), i in axes(data_out, 2),
v in axes(data_out, 1)
res = zero(eltype(data_out))
for kk in axes(matrix3, 2)
res += matrix3[k, kk] * tmp2[v, i, j, kk]
end
data_out[v, i, j, k] = res
end
return nothing
end
# 3D version, apply matrixJ to dimension J of data_in and add the result to data_out
function add_multiply_dimensionwise!(data_out::AbstractArray{<:Any, 4},
matrix1::AbstractMatrix, matrix2::AbstractMatrix,
matrix3::AbstractMatrix,
data_in::AbstractArray{<:Any, 4},
tmp1 = zeros(eltype(data_out), size(data_out, 1),
size(matrix1, 1), size(matrix1, 2),
size(matrix1, 2)),
tmp2 = zeros(eltype(data_out), size(data_out, 1),
size(matrix1, 1), size(matrix1, 1),
size(matrix1, 2)))
# Interpolate in x-direction
# @tullio threads=false tmp1[v, i, j, k] = matrix1[i, ii] * data_in[v, ii, j, k]
@turbo for k in axes(tmp1, 4), j in axes(tmp1, 3), i in axes(tmp1, 2),
v in axes(tmp1, 1)
res = zero(eltype(tmp1))
for ii in axes(matrix1, 2)
res += matrix1[i, ii] * data_in[v, ii, j, k]
end
tmp1[v, i, j, k] = res
end
# Interpolate in y-direction
# @tullio threads=false tmp2[v, i, j, k] = matrix2[j, jj] * tmp1[v, i, jj, k]
@turbo for k in axes(tmp2, 4), j in axes(tmp2, 3), i in axes(tmp2, 2),
v in axes(tmp2, 1)
res = zero(eltype(tmp1))
for jj in axes(matrix2, 2)
res += matrix2[j, jj] * tmp1[v, i, jj, k]
end
tmp2[v, i, j, k] = res
end
# Interpolate in z-direction
# @tullio threads=false data_out[v, i, j, k] += matrix3[k, kk] * tmp2[v, i, j, kk]
@turbo for k in axes(data_out, 4), j in axes(data_out, 3), i in axes(data_out, 2),
v in axes(data_out, 1)
res = zero(eltype(data_out))
for kk in axes(matrix3, 2)
res += matrix3[k, kk] * tmp2[v, i, j, kk]
end
data_out[v, i, j, k] += res
end
return nothing
end