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sensitivity.jl
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sensitivity.jl
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@inbounds function residual(ts, y, p, u0, Tign, idt)
ng = length(ts)
pred = reshape(y, :, ng)
F = similar(pred) * p[1]
_ts = ts ./ idt
i = 1
@view(F[1:end - 1, i]) .= @views((pred[1:end - 1, i] .- u0))
@view(F[end, i]) .= @views((pred[end, i + 1] - pred[end, i]))
du = similar(pred[1:end - 1, 1]) * p[1]
for i = 2:ng - 1
@view(F[1:end - 1, i]) .=
@views((pred[1:end - 1, i] .- pred[1:end - 1, i - 1]) ./ (_ts[i] - _ts[i - 1])) .-
dudt!(du, @view(pred[1:end - 1, i]), p, 0.0) .* pred[end, i]
@view(F[end, i]) .=
@views(pred[end, i + 1] - pred[end, i])
end
i = ng
@view(F[1:end - 1, i]) .=
@views((pred[1:end - 1, i] .- pred[1:end - 1, i - 1]) ./ (_ts[i] - _ts[i - 1])) .-
dudt!(du, @view(pred[1:end - 1, i]), p, 0.0) .* pred[end, i]
@view(F[end, i]) .= @views((pred[end - 1, i] - Tign))
return vcat(F...)
end
function downsampling(ts, pred; dT=0.1, n_max=100, verbose=false)
ind_sample = [1]
Ts = pred[end, :]
_T = Ts[1]
_t = ts[1]
for i = 2:length(ts) - 1
if (abs(Ts[i] - _T) > dT) & (ts[i] - _t > ts[end] / n_max)
_T = Ts[i]
_t = ts[i]
push!(ind_sample, i)
end
end
push!(ind_sample, length(ts))
if verbose
println("\n original sample size $(length(ts))
downsampled size $(length(ind_sample)) \n")
end
ts = ts[ind_sample]
pred = pred[:, ind_sample]
return ts, pred
end
@inbounds function sensBVP(ts, pred, p)
local ng = length(ts)
Fp_ = zeros(ng * nu, np)
Fy_ = BandedMatrix(Zeros(ng * nu, ng * nu), (nu, nu))
return sensBVP!(Fy_, Fp_, ts, pred, p)
end
@inbounds function sensBVP!(Fy, Fp, ts, pred, p)
local idt = ts[end]
local Tign = pred[end, end]
local ng = length(ts)
Fy .*= 0.0
Fp .*= 0.0
i = 1
i_F = 1 + (i - 1) * nu:i * nu - 1
@view(Fy[i_F, i_F])[ind_diag] .= ones_nu
Fy[i * nu, i * nu] = -1.0
Fy[i * nu, (i + 1) * nu] = 1.0
du = similar(@view(pred[:, i]))
dts = @views(ts[2:end] .- ts[1:end - 1]) ./ idt
for i = 2:ng
u = @view(pred[:, i])
i_F = 1 + (i - 1) * nu:i * nu - 1
@view(Fy[i_F, i_F]) .= jacobian((du, x) -> dudt!(du, x, p, 0.0),
du, u)::Array{Float64,2} .* (-idt)
@view(Fy[i_F, i * nu]) .= - dudt!(du, u, p, 0.0)
@view(Fy[i_F, i_F])[ind_diag] .+= ones_nu ./ (dts[i - 1])
@view(Fy[i_F, i_F .- nu])[ind_diag] .+= ones_nu ./ (-dts[i - 1])
if i < ng
Fy[i * nu, i * nu] = -1.0
Fy[i * nu, (i + 1) * nu] = 1.0
else
Fy[i * nu, i * nu - 1] = 1.0
end
@view(Fp[i_F, :]) .= jacobian((du, x) -> dudt!(du, u, x, 0.0),
du, p)::Array{Float64,2} .* (-idt)
end
dydp = - @views(Fy[1:ng * nu, 1:ng * nu] \ Fp[1:ng * nu, :])
return @view(dydp[end, :]) ./ idt
end
function sensBVP_mthread(ts, pred, p)
idt = ts[end]
Tign = pred[end, end]
ng = length(ts)
Fy = BandedMatrix(Zeros(ng * nu, ng * nu), (nu, nu));
Fp = zeros(ng * nu, np)
i = 1
i_F = 1 + (i - 1) * nu:i * nu - 1
@view(Fy[i_F, i_F])[ind_diag] .= ones_nu
Fy[i * nu, i * nu] = -1.0
Fy[i * nu, (i + 1) * nu] = 1.0
dts = @views(ts[2:end] .- ts[1:end - 1]) ./ idt
@threads for i = 2:ng
u = @view(pred[:, i])
du = similar(u)
i_F = 1 + (i - 1) * nu:i * nu - 1
@view(Fp[i_F, :]) .= jacobian((du, x) -> dudt!(du, u, x, 0.0),
du, p)::Array{Float64,2} .* (-idt)
end
@threads for i = 2:ng
u = @view(pred[:, i])
du = similar(u)
i_F = 1 + (i - 1) * nu:i * nu - 1
@view(Fy[i_F, i_F]) .= jacobian((du, x) -> dudt!(du, x, p, 0.0),
du, u)::Array{Float64,2} .* (-idt)
end
for i = 2:ng
u = @view(pred[:, i])
du = similar(u)
i_F = 1 + (i - 1) * nu:i * nu - 1
@view(Fy[i_F, i * nu]) .= - dudt!(du, u, p, 0.0)
@view(Fy[i_F, i_F])[ind_diag] .+= ones_nu ./ (dts[i - 1])
@view(Fy[i_F, i_F .- nu])[ind_diag] .+= ones_nu ./ (-dts[i - 1])
if i < ng
Fy[i * nu, i * nu] = -1.0
Fy[i * nu, (i + 1) * nu] = 1.0
else
Fy[i * nu, i * nu - 1] = 1.0
end
end
dydp = - Fy \ sparse(Fp)
return @view(dydp[end, :]) ./ idt
end
# sensitivity calculated by BFSA method
# d\tau/dp = d\tau/dT * dT/dp
sensealg = ForwardDiffSensitivity()
function sensBFSA(phi, P, T0, p; dT=200, dTabort=600, tfinal=1.0)
ts, pred = get_Tcurve(phi, P, T0, p;
dT=dT, dTabort=dTabort, tfinal=tfinal);
idt = interpx(ts, pred[end,:], pred[end,1] + dT);
dTdidt = (pred[end,end] - pred[end,end - 1]) / (ts[end] - ts[end - 1]);
prob = make_prob(phi, P, T0, p; tfinal=idt)
function fsol_T(p, idt)
sol = solve(prob, Trapezoid(), p=p, saveat=[0,idt],
reltol=1e-6, abstol=1e-9, sensealg=sensealg);
return sol[end,end];
end
dTdp = ForwardDiff.gradient(x -> fsol_T(x, idt), p);
return dTdp ./ dTdidt ./ idt
end
# sensitivity calculated by Brute-force method
function sensBF_mthread(phi, P, T0, p; dT=200, dTabort=600, pdiff=5e-3)
idt = get_idt(phi, P, T0, p; dT=dT, dTabort=dTabort)
np = length(p);
sens = zeros(np);
@threads for i = 1:np
pp = deepcopy(p);
pp[i] += pdiff;
pidt = get_idt(phi, P, T0, pp; dT=dT, dTabort=dTabort)
sens[i] = (pidt - idt) / pdiff
end
return sens ./ idt;
end