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geothermal_2D_ps_inv.jl
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geothermal_2D_ps_inv.jl
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using Printf, LinearAlgebra
using CairoMakie
using Enzyme
using Optim
using ParallelStencil
using ParallelStencil.FiniteDifferences2D
@init_parallel_stencil(Threads, Float64, 2)
# @init_parallel_stencil(CUDA, Float64, 2)
@views avx(A) = 0.5 .* (A[1:end-1, :] .+ A[2:end, :])
@views avy(A) = 0.5 .* (A[:, 1:end-1] .+ A[:, 2:end])
@views maxloc(A) = max.(A[2:end-1, 2:end-1], max.(max.(A[1:end-2, 2:end-1], A[3:end, 2:end-1]),
max.(A[2:end-1, 1:end-2], A[2:end-1, 3:end])))
@parallel function smooth_d!(A2, A)
@inn(A2) = @inn(A) + 0.2 * (@d2_xi(A) + @d2_yi(A))
return
end
function smooth!(A2, A; nsm=1)
for _ ∈ 1:nsm
@parallel smooth_d!(A2, A)
A, A2 = A2, A
end
return
end
@parallel function residual_fluxes!(Rqx, Rqy, ???)
@inn_x(Rqx) = ???
@inn_y(Rqy) = ???
return
end
@parallel function residual_pressure!(RPf, ???)
@all(RPf) = ???
return
end
@parallel function update_fluxes!(qx, qy, ???)
@inn_x(qx) = ???
@inn_y(qy) = ???
return
end
@parallel function update_pressure!(Pf, ???)
@all(Pf) = ???
return
end
@views function forward_solve!(logK, fields, scalars, iter_params; visu=nothing)
(;Pf, qx, qy, Qf, RPf, Rqx, Rqy, K) = fields
(;nx, ny, dx, dy) = scalars
(;cfl, re, vdτ, ly, ϵtol, maxiter, ncheck, K_max) = iter_params
isnothing(visu) || ((;qx_c, qy_c, qM, fig, plt, st) = visu)
K .= exp.(logK)
# approximate diagonal (Jacobi) preconditioner
K_max .= K; K_max[2:end-1, 2:end-1] .= maxloc(K); K_max[:, [1, end]] .= K_max[:, [2, end-1]]
# iterative loop
iters_evo = Float64[]; errs_evo = Float64[]
err = 2ϵtol; iter = 1
while err >= ϵtol && iter <= maxiter
@parallel residual_fluxes!(Rqx, Rqy, ???)
@parallel update_fluxes!(qx, qy, ???)
@parallel residual_pressure!(RPf, ???)
@parallel update_pressure!(Pf, ???)
if iter % ncheck == 0
err = maximum(abs.(RPf))
push!(iters_evo, iter/nx); push!(errs_evo, err)
@printf(" #iter/nx=%.1f, max(err)=%1.3e\n", iter/nx, err)
if !isnothing(visu)
qx_c .= Array(avx(qx)); qy_c .= Array(avy(qy)); qM .= sqrt.(qx_c.^2 .+ qy_c.^2)
qx_c ./= qM; qy_c ./= qM
plt.fld.Pf[3] = Array(Pf)
plt.fld.K[3] = Array(log10.(K))
plt.fld.qM[3] = qM
plt.fld.ar[3] = qx_c[1:st:end, 1:st:end]
plt.fld.ar[4] = qy_c[1:st:end, 1:st:end]
plt.err[1] = Point2.(iters_evo, errs_evo)
display(fig)
end
end
iter += 1
end
return
end
@views function adjoint_solve!(logK, fwd_params, adj_params, loss_params)
# unpack forward
(;Pf, qx, qy, Qf, RPf, Rqx, Rqy, K) = fwd_params.fields
(;nx, ny, dx, dy) = fwd_params.scalars
# unpack adjoint
(;P̄f, q̄x, q̄y, R̄Pf, R̄qx, R̄qy, Ψ_qx, Ψ_qy, Ψ_Pf) = adj_params.fields
(;∂J_∂Pf) = loss_params.fields
(;cfl, re_a, vdτ, ly, ϵtol, maxiter, ncheck, K_max) = adj_params.iter_params
# iterative loop
iters_evo = Float64[]; errs_evo = Float64[]
err = 2ϵtol; iter = 1
while err >= ϵtol && iter <= maxiter
R̄qx .= Ψ_qx
R̄qy .= Ψ_qy
P̄f .= .-∂J_∂Pf
q̄x .= 0.0
q̄y .= 0.0
@parallel ∇=(???) residual_fluxes!(Rqx, Rqy, ???)
P̄f[[1, end], :] .= 0.0; P̄f[:, [1, end]] .= 0.0
@parallel update_pressure!(Ψ_Pf, P̄f, K_max, vdτ, ly, re_a)
R̄Pf .= Ψ_Pf
@parallel ∇=(???) residual_pressure!(RPf, ???)
@parallel update_fluxes!(Ψ_qx, Ψ_qy, q̄x, q̄y, cfl, nx, ny, re_a)
if iter % ncheck == 0
err = maximum(abs.(P̄f))
push!(iters_evo, iter/nx); push!(errs_evo, err)
@printf(" #iter/nx=%.1f, max(err)=%1.6e\n", iter/nx, err)
end
iter += 1
end
return
end
@views function loss(logK, fwd_params, loss_params; kwargs...)
(;Pf_obs) = loss_params.fields
(;ixobs, iyobs) = loss_params.scalars
@info "Forward solve"
???
Pf = fwd_params.fields.Pf
return 0.5*sum((Pf[ixobs, iyobs] .- Pf_obs).^2)
end
function ∇loss!(logK̄, logK, fwd_params, adj_params, loss_params; reg=nothing, kwargs...)
# unpack
(;R̄qx, R̄qy, Ψ_qx, Ψ_qy) = adj_params.fields
(;Pf, qx, qy, Rqx, Rqy, K) = fwd_params.fields
(;dx, dy) = fwd_params.scalars
(;Pf_obs, ∂J_∂Pf) = loss_params.fields
(;ixobs, iyobs) = loss_params.scalars
@info "Forward solve"
forward_solve!(logK, fwd_params...; kwargs...)
# set tangent
∂J_∂Pf[ixobs, iyobs] .= Pf[ixobs, iyobs] .- Pf_obs
@info "Adjoint solve"
???
# evaluate gradient dJ_dK
R̄qx .= ???
R̄qy .= ???
logK̄ .= 0.0
@parallel ∇=(???) ???
# Tikhonov regularisation (smoothing)
if !isnothing(reg)
(;nsm, Tmp) = reg
Tmp .= logK̄; smooth!(logK̄, Tmp; nsm)
end
logK̄ .*= K # convert to dJ_dlogK by chain rule
return
end
@views function main()
# physics
lx, ly = 2.0, 1.0 # domain extend
k0_μ = 1.0 # background permeability / fluid viscosity
kb_μ = 1e-6 # barrier permeability / fluid viscosity
Q_in = 1.0 # injection flux
b_w = 0.02lx # barrier width
b_b = 0.3ly # barrier bottom location
b_t = 0.8ly # barrier top location
# observations
xobs_rng = LinRange(-lx / 6, lx / 6, 8)
yobs_rng = LinRange(0.25ly, 0.85ly , 8)
# numerics
ny = 127
nx = ceil(Int, (ny + 1) * lx / ly) - 1
cfl = 1 / 2.1
ϵtol = 1e-6
maxiter = 30nx
ncheck = 2nx
re = 0.8π # fwd re
st = ceil(Int, nx / 30)
# preprocessing
re_a = 2re # adjoint re
dx, dy = lx / nx, ly / ny
xc, yc = LinRange(-lx / 2 + dx / 2, lx / 2 - dx / 2, nx), LinRange(dy / 2, ly - dy / 2, ny)
vdτ = cfl * min(dx, dy)
ixobs = floor.(Int, (xobs_rng .- xc[1]) ./ dx) .+ 1
iyobs = floor.(Int, (yobs_rng .- yc[1]) ./ dy) .+ 1
# init
Pf = @zeros(nx, ny)
RPf = @zeros(nx, ny)
qx = @zeros(nx + 1, ny)
Rqx = @zeros(nx + 1, ny)
qy = @zeros(nx, ny + 1)
Rqy = @zeros(nx, ny + 1)
Qf = @zeros(nx, ny)
K = k0_μ .* @ones(nx, ny)
logK = @zeros(nx, ny)
Tmp = @zeros(nx, ny)
# init adjoint storage
Ψ_qx = @zeros(nx + 1, ny)
q̄x = @zeros(nx + 1, ny)
R̄qx = @zeros(nx + 1, ny)
Ψ_qy = @zeros(nx, ny + 1)
q̄y = @zeros(nx, ny + 1)
R̄qy = @zeros(nx, ny + 1)
Ψ_Pf = @zeros(nx, ny)
P̄f = @zeros(nx, ny)
R̄Pf = @zeros(nx, ny)
∂J_∂Pf = @zeros(nx, ny)
# set low permeability barrier location
K[ceil(Int, (lx/2-b_w)/dx):ceil(Int, (lx/2+b_w)/dx), ceil(Int, b_b/dy):ceil(Int, b_t/dy)] .= kb_μ
logK .= log.(K)
K_max = copy(K)
# set wells location
x_iw, x_ew, y_w = ceil.(Int, (lx / 5 / dx, 4lx / 5 / dx, 0.45ly / dy))
Qf[x_iw:x_iw, y_w:y_w] .= Q_in / dx / dy # injection
Qf[x_ew:x_ew, y_w:y_w] .= -Q_in / dx / dy # extraction
# init visu
iters_evo = Float64[]; errs_evo = Float64[]
qM, qx_c, qy_c = zeros(nx, ny), zeros(nx, ny), zeros(nx, ny)
fig = Figure(resolution=(2500, 1200), fontsize=32)
ax = ( Pf = Axis(fig[1, 1][1, 1]; aspect=DataAspect(), title="Pf"),
K = Axis(fig[1, 2][1, 1]; aspect=DataAspect(), title="log10(K)"),
qM = Axis(fig[2, 1][1, 1]; aspect=DataAspect(), title="|q|"),
err = Axis(fig[2, 2]; yscale=log10, title="Convergence", xlabel="# iter/nx", ylabel="error"), )
plt = (fld = ( Pf = heatmap!(ax.Pf, xc, yc, Array(Pf); colormap=:turbo, colorrange=(-1,1)),
K = heatmap!(ax.K , xc, yc, Array(log10.(K)); colormap=:turbo, colorrange=(-6,0)),
xobs = scatter!(ax.K , vec(Point2.(xobs_rng, yobs_rng')); color=:white),
qM = heatmap!(ax.qM, xc, yc, qM; colormap=:turbo, colorrange=(0,30)),
ar = arrows!(ax.Pf, xc[1:st:end], yc[1:st:end], qx_c[1:st:end, 1:st:end], qy_c[1:st:end, 1:st:end]; lengthscale=0.05, arrowsize=15), ),
err = scatterlines!(ax.err, Point2.(iters_evo, errs_evo), linewidth=4), )
Colorbar(fig[1, 1][1, 2], plt.fld.Pf)
Colorbar(fig[1, 2][1, 2], plt.fld.K)
Colorbar(fig[2, 1][1, 2], plt.fld.qM)
# action
fwd_params = (
fields = (;Pf, qx, qy, Qf, RPf, Rqx, Rqy, K),
scalars = (;nx, ny, dx, dy),
iter_params = (;cfl, re, vdτ, ly, ϵtol, maxiter, ncheck, K_max),
)
fwd_visu = (;qx_c, qy_c, qM, fig, plt, st)
@info "Synthetic solve"
???
# store true data
Pf_obs = ???
adj_params = (
fields = (;P̄f, q̄x, q̄y, R̄Pf, R̄qx, R̄qy, Ψ_qx, Ψ_qy, Ψ_Pf),
iter_params = (;cfl, re_a, vdτ, ly, ϵtol, maxiter, ncheck, K_max),
)
loss_params = (
fields = (;Pf_obs, ∂J_∂Pf),
scalars = (;ixobs, iyobs),
)
reg = (;nsm=50, Tmp)
# loss functions
J(_logK) = loss(_logK, fwd_params, loss_params)
∇J!(_logK̄, _logK) = ∇loss!(_logK̄, _logK, fwd_params, adj_params, loss_params; reg)
@info "Inversion for K"
# initial guess
K .= k0_μ
logK .= log.(K)
# Optim
opt = Optim.Options(
f_tol = 1e-2,
g_tol = 1e-6,
iterations = 20,
store_trace=true, show_trace=true,
)
result = ???
K .= exp.(Optim.minimizer(result))
@show result
# visu
errs_evo = Optim.f_trace(result)
errs_evo ./= errs_evo[1]
iters_evo = 1:length(errs_evo)
qx_c .= Array(avx(qx)); qy_c .= Array(avy(qy)); qM .= sqrt.(qx_c.^2 .+ qy_c.^2)
qx_c ./= qM; qy_c ./= qM
plt.fld.Pf[3] = Array(Pf)
plt.fld.K[3] = Array(log10.(K))
plt.fld.qM[3] = qM
plt.fld.ar[3] = qx_c[1:st:end, 1:st:end]
plt.fld.ar[4] = qy_c[1:st:end, 1:st:end]
plt.err[1] = Point2.(iters_evo, errs_evo)
display(fig)
return
end
main()