Parameter optimisation

In this how-to we fit two parameters of the coupled PO₄–POP model (τ_DIP, τ_POP) to PO₄ observations from WorldOceanAtlasTools.jl on the OCCA circulation. Restricting the fit to two parameters lets us also sweep the cost function in a 2D grid around the optimiser trajectory and visualise the loss landscape directly.

We use the SciML Optimization.jl frontend, with F1Method.jl supplying the cached steady-state gradient and Hessian, and share an UMFPACK factorisation between F1Method's adjoint solve and AIBECS's inner Newton step (warm-start trick) so the sweep stays cheap.

The API surface we touch:

@flattenable @bounds @logscaled @prior — declare which parameters are optimised and supply their priors
p2λ / λ2p — transform a parameter struct to/from an unconstrained flat vector
f_and_∇ₓf — build the volume-weighted mismatch objective and its gradient w.r.t. the state vector x
F1Method — gradient and Hessian (Pasquier et al., 2019)
Optimization.jl — the SciML optimisation frontend, wrapping many optimizers (we will use NewtonTrustRegion).
LinearSolve.jl — the SciML linear-solver frontend (we will use the default sparse UMFPACK), and share the factorisation between F1Method and AIBECS.

Re-build the P-model on OCCA

We reuse the source/sink terms from the P-model tutorial verbatim, just swapping the circulation to OCCA so the resolution stays moderate (~100k wet boxes × 2 tracers; OCIM2 is ~200k wet boxes for comparison).

julia

using AIBECS
using JLD2 # required by `OCCA.load`
grd, T_OCIM = OCCA.load()
T_DIP(p) = T_OCIM
T_POP(p) = transportoperator(grd, z -> w(z, p))

function w(z, p)
    @unpack w₀, w′ = p
    return @. w₀ + w′ * z
end

z = depthvec(grd)
function U(x, p)
    @unpack τ_DIP, k, z₀ = p
    return @. x / τ_DIP * x / (x + k) * (z ≤ z₀) * (x ≥ 0)
end

function R(x, p)
    @unpack τ_POP = p
    return x / τ_POP
end

function G_DIP(DIP, POP, p)
    @unpack DIP_geo, τ_geo = p
    return @. -$U(DIP, p) + $R(POP, p) + (DIP_geo - DIP) / τ_geo
end
function G_POP(DIP, POP, p)
    @unpack τ_geo = p
    return @. $U(DIP, p) - $R(POP, p) - POP / τ_geo
end

G_POP (generic function with 1 method)

Optimisable parameter struct

We layer the full optimisation metadata stack on top of the tutorial's PmodelParameters. The leftmost macro consumes the leftmost column (canonical example at src/Parameters.jl). Only τ_DIP and τ_POP are optimised and marked as flattenable. So we only pass a 2-element vector λ to the optimiser.

julia

import AIBECS: @initial_value, initial_value
import AIBECS: @units, units
import AIBECS: @flattenable, flattenable
import AIBECS: @bounds, bounds
import AIBECS: @logscaled, logscaled
import AIBECS: prior
using Unitful: m, d, s, yr, kyr, Myr, mol, mmol, μmol, μM
using Distributions
using Bijectors    # activates the `AIBECSBijectorsExt` extension (p2λ, λ2p)
using DataFrames   # activates the `AIBECSDataFramesExt` extension (AIBECS.table)

const ∞ = Inf

@initial_value @units @flattenable @bounds @logscaled struct OptParams{U} <: AbstractParameters{U}
    w₀::U      |   5.0  | m/d      | false | (0, ∞) | true
    w′::U      |  0.035 | m/d/m    | false | (0, ∞) | true
    τ_DIP::U   | 100.0  | d        | true  | (0, ∞) | true
    k::U       |  10.0  | μmol/m^3 | false | (0, ∞) | true
    z₀::U      |  80.0  | m        | false | (0, ∞) | false
    τ_POP::U   |  30.0  | d        | true  | (0, ∞) | true
    τ_geo::U   |   1.0  | kyr      | false | (0, ∞) | true
    DIP_geo::U |   2.12 | mmol/m^3 | false | (0, ∞) | false
end

initial_value (generic function with 17 methods)

We apply a penalty to the parameters. For this we derive priors from (lb, ub) = bounds(T, s) and the initial value, matching the convention in test/parameters.jl and GNOM: positive half-line gets a LogNormal, the real line a wide Normal, and finite-interval parameters get a LogitNormal stretched onto (lb, ub). Here we will only use the LogNormal priors but this gives you an idea of how to set up different priors if you want to.

Note

Under the hood AIBECS will perform a change of variables based on Turing.jl's Bijectors.jl extension to transform the constrained parameter space (0, ∞) into an unconstrained one, which is what the optimiser operates in. This is why we can use LogNormal and LogitNormal priors even though the optimiser only sees unconstrained vectors.

julia

function prior(::Type{T}, s::Symbol) where {T <: OptParams}
    flattenable(T, s) || return nothing
    lb, ub = bounds(T, s)
    if (lb, ub) == (0, ∞)
        return LogNormal(log(initial_value(T, s)), 1.0)
    elseif (lb, ub) == (-∞, ∞)
        return Normal(initial_value(T, s), 10.0)
    else
        m = initial_value(T, s)
        f = (m - lb) / (ub - lb)
        return LocationScale(lb, ub - lb, LogitNormal(log(f / (1 - f)), 1.0))
    end
end

p₀ = OptParams()

Main.OptParams{Float64}
  w₀ = 5.0 (m d⁻¹)
  w′ = 0.035 (d⁻¹)
  τ_DIP = 100.0 (d)
  k = 10.0 (μmol m⁻³)
  z₀ = 80.0 (m)
  τ_POP = 30.0 (d)
  τ_geo = 1.0 (kyr)
  DIP_geo = 2.12 (mmol m⁻³)

Two fields (τ_DIP, τ_POP) carry flattenable = true; the rest are held fixed at their initial value. The metadata table:

julia

AIBECS.table(p₀)

8×7 DataFrame

Row	Symbol	Value	Initial value	Unit	Bounds	Logscaled	Optimizable
	Symbol	Float64	Float64	FreeUnit…	Tuple…	Bool	Bool
1	w₀	5.0	5.0	m d⁻¹	(0, Inf)	true	false
2	w′	0.035	0.035	d⁻¹	(0, Inf)	true	false
3	τ_DIP	100.0	100.0	d	(0, Inf)	true	true
4	k	10.0	10.0	μmol m⁻³	(0, Inf)	true	false
5	z₀	80.0	80.0	m	(0, Inf)	false	false
6	τ_POP	30.0	30.0	d	(0, Inf)	true	true
7	τ_geo	1.0	1.0	kyr	(0, Inf)	true	false
8	DIP_geo	2.12	2.12	mmol m⁻³	(0, Inf)	false	false

State function in parameter-type form

Using the AIBECSFunction(Ts, Gs, nb, ::Type{P}) form lets the SciML solvers feed in a flat parameter vector λ; AIBECS will call λ2p(OptParams, λ) internally to reconstruct the struct.

julia

nb = sum(iswet(grd))
F = AIBECSFunction((T_DIP, T_POP), (G_DIP, G_POP), nb, OptParams)
x0 = p₀.DIP_geo * ones(2nb)

169322-element Vector{Float64}:
 2.12
 2.12
 2.12
 2.12
 2.12
 2.12
 2.12
 2.12
 2.12
 2.12
 ⋮
 2.12
 2.12
 2.12
 2.12
 2.12
 2.12
 2.12
 2.12
 2.12

Load WOA PO₄ observations

WorldOceanAtlasTools.fit_to_grid returns the mean and variance of WOA PO₄ regridded onto our grd as 3D fields. There is a subtle unit gotcha: the WOA 2018 PO₄ file's units attribute is "micromoles_per_kilogram", and WorldOceanAtlasTools.convert_to_SI_unit! does χ_3D .*= ustrip(upreferred(1.0u"μmol/kg")), which gives 1.0e-6 mol/kg — kg is already SI, so upreferred does not convert to mol/m³. The returned array is therefore in mol/kg (mean ≈ 1.6e-6), a factor of ~1000 smaller than the AIBECS model state. We close the gap with a nominal seawater density before anything downstream consumes these arrays.

julia

using WorldOceanAtlasTools
μDIPobs3D, σ²DIPobs3D = WorldOceanAtlasTools.fit_to_grid(grd, "PO₄")
iwet = iswet(grd)

ρ_factor = ustrip(upreferred(1.035u"kg/L"))   # = 1035.0 kg/m³
μDIPobs  = μDIPobs3D[iwet]  .* ρ_factor       # mol/kg → mol/m³
σ²DIPobs = σ²DIPobs3D[iwet] .* ρ_factor^2     # (mol/kg)² → (mol/m³)²
v = volumevec(grd)

84661-element Vector{Float64}:
 4.1259441424340015e11
 8.858714556466658e11
 9.658719592542885e11
 1.0446956966607107e12
 1.122246633283809e12
 1.1984302852530718e12
 1.2731538345237942e12
 1.3463262419614636e12
 1.417858358258977e12
 1.487663032551257e12
 ⋮
 6.642031628003376e13
 5.1445109315594125e13
 4.172486815218424e13
 1.3409911172962852e13
 1.824228219687533e13
 2.7565991089214062e13
 3.717262516144412e13
 4.7318477973128164e13
 4.2196399175523055e13

Build the objective `(f, ∇ₓf)`

We weight the DIP misfit at 1, ignore POP (no direct observations), and add a small parameter-prior penalty.

julia

ωs = (1.0, 0.0)
ωp = 1.0e-4
μx  = (μDIPobs,  missing)
σ²x = (σ²DIPobs, missing)
f, ∇ₓf = f_and_∇ₓf(ωs, μx, σ²x, v, ωp, OptParams)

(AIBECS.var"#f#generate_f##1"{Main.OptParams, Tuple{Float64, Float64}, Tuple{Vector{Float64}, Missing}, Tuple{Vector{Float64}, Missing}, Vector{Float64}, Float64, AIBECS.var"#tracers#generate_f##0"{Int64, Int64}}((1.0, 0.0), ([0.001993383847718889, 0.0019914651177959006, 0.0019652102196352048, 0.0019188907593353226, 0.0018741692451726304, 0.0018593180609968575, 0.0018762683281708847, 0.0018866130577569658, 0.0018618487300791524, 0.0017936109269207174  …  0.001597349353171885, 0.0016139239530996844, 0.0016146148645877836, 0.0016014571666717527, 0.0015958896704982308, 0.0015948419428572936, 0.0015868619350592294, 0.0015825331862406298, 0.0015800279642719972, 0.001589979759638126], missing), ([4.7499243508165185e-9, 2.3806476583116648e-9, 1.644850324443933e-9, 2.474518014121221e-9, 2.8420192007964507e-9, 1.9923744404030257e-9, 8.150346589335414e-10, 2.974775216581886e-10, 4.021567996605766e-10, 8.840013268111294e-10  …  4.92697369868074e-10, 2.974775216581886e-10, 2.974775216581886e-10, 2.3496760609714243e-9, 2.446816223482753e-9, 2.974775216581886e-10, 8.053041266157682e-10, 8.39703761723315e-10, 1.2798255287465522e-9, 1.7406554311458743e-9], missing), [4.1259441424340015e11, 8.858714556466658e11, 9.658719592542885e11, 1.0446956966607107e12, 1.122246633283809e12, 1.1984302852530718e12, 1.2731538345237942e12, 1.3463262419614636e12, 1.417858358258977e12, 1.487663032551257e12  …  5.551473635870713e13, 6.642031628003376e13, 5.1445109315594125e13, 4.172486815218424e13, 1.3409911172962852e13, 1.824228219687533e13, 2.7565991089214062e13, 3.717262516144412e13, 4.7318477973128164e13, 4.2196399175523055e13], 0.0001, AIBECS.var"#tracers#generate_f##0"{Int64, Int64}(84661, 2)), AIBECS.var"#∇ₓf#generate_∇ₓf##1"{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Missing}, Tuple{Vector{Float64}, Missing}, Vector{Float64}, AIBECS.var"#tracers#generate_∇ₓf##0"{Int64, Int64}}((1.0, 0.0), ([0.001993383847718889, 0.0019914651177959006, 0.0019652102196352048, 0.0019188907593353226, 0.0018741692451726304, 0.0018593180609968575, 0.0018762683281708847, 0.0018866130577569658, 0.0018618487300791524, 0.0017936109269207174  …  0.001597349353171885, 0.0016139239530996844, 0.0016146148645877836, 0.0016014571666717527, 0.0015958896704982308, 0.0015948419428572936, 0.0015868619350592294, 0.0015825331862406298, 0.0015800279642719972, 0.001589979759638126], missing), ([4.7499243508165185e-9, 2.3806476583116648e-9, 1.644850324443933e-9, 2.474518014121221e-9, 2.8420192007964507e-9, 1.9923744404030257e-9, 8.150346589335414e-10, 2.974775216581886e-10, 4.021567996605766e-10, 8.840013268111294e-10  …  4.92697369868074e-10, 2.974775216581886e-10, 2.974775216581886e-10, 2.3496760609714243e-9, 2.446816223482753e-9, 2.974775216581886e-10, 8.053041266157682e-10, 8.39703761723315e-10, 1.2798255287465522e-9, 1.7406554311458743e-9], missing), [4.1259441424340015e11, 8.858714556466658e11, 9.658719592542885e11, 1.0446956966607107e12, 1.122246633283809e12, 1.1984302852530718e12, 1.2731538345237942e12, 1.3463262419614636e12, 1.417858358258977e12, 1.487663032551257e12  …  5.551473635870713e13, 6.642031628003376e13, 5.1445109315594125e13, 4.172486815218424e13, 1.3409911172962852e13, 1.824228219687533e13, 2.7565991089214062e13, 3.717262516144412e13, 4.7318477973128164e13, 4.2196399175523055e13], AIBECS.var"#tracers#generate_∇ₓf##0"{Int64, Int64}(84661, 2), Core.Box(AIBECS.var"#∇ₓf#generate_∇ₓf##1"{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Missing}, Tuple{Vector{Float64}, Missing}, Vector{Float64}, AIBECS.var"#tracers#generate_∇ₓf##0"{Int64, Int64}}(#= circular reference @-2 =#))))

F-1 memory cache + warm-started linear solve

F1Method.initialize_mem solves the steady state once and caches the factorised Jacobian; later objective / gradient / Hessian calls reuse it whenever the parameter vector has not moved. We route F1Method's linear algebra through LinearSolve.jl and share its UMFPACK factors with a sibling cache that AIBECS's inner Newton step uses (AIBECScache.cacheval = mem.linear_cache.cacheval) — this makes the cost-surface sweep below ~10× faster. CTKAlg's Shamanskii + Armijo logic refreshes the factors automatically when they go stale.

julia

using F1Method, Optimization, OptimizationOptimJL, ADTypes
using LinearSolve

solver_kwargs = (; maxItNewton = 15)

λ₀ = p2λ(p₀)
mem = F1Method.initialize_mem(F, ∇ₓf, x0, λ₀, CTKAlg();
    ad = AutoForwardDiff(),
    linsolve = UMFPACKFactorization(),
    solver_kwargs...)

AIBECScache = init(
    LinearProblem(F.jac(x0, λ2p(OptParams, λ₀)), similar(x0)),
    UMFPACKFactorization(),
)
AIBECScache.cacheval = mem.linear_cache.cacheval
AIBECScache.isfresh  = false

solver_kwargs = (; solver_kwargs..., linear_cache = AIBECScache)

optfn = F1Method.optimization_function(f, F, ∇ₓf, mem, CTKAlg(); solver_kwargs...)

SciMLBase.OptimizationFunction{true, SciMLBase.NoAD, F1MethodOptimizationExt.var"#5#6"{Base.Pairs{Symbol, Any, Nothing, @NamedTuple{maxItNewton::Int64, linear_cache::LinearSolve.LinearCache{SparseMatrixCSC{Float64, Int64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, LinearSolve.UMFPACKFactorization, SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}, SciMLOperators.IdentityOperator, SciMLOperators.IdentityOperator, Float64, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}, Bool, LinearSolve.LinearSolveAdjoint{Missing}}}}, AIBECS.var"#f#generate_f##1"{Main.OptParams, Tuple{Float64, Float64}, Tuple{Vector{Float64}, Missing}, Tuple{Vector{Float64}, Missing}, Vector{Float64}, Float64, AIBECS.var"#tracers#generate_f##0"{Int64, Int64}}, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#AIBECSFunction##1"{Main.OptParams, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}, AIBECS.var"#tracer#67"{Int64, Int64}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#81"{AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#AIBECSFunction##0"{Main.OptParams, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}, AIBECS.var"#tracer#67"{Int64, Int64}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#81"{AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, F1Method.F1Cache{LinearSolve.LinearCache{SparseMatrixCSC{Float64, Int64}, Matrix{Float64}, Matrix{Float64}, SciMLBase.NullParameters, LinearSolve.UMFPACKFactorization, SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}, SciMLOperators.IdentityOperator, SciMLOperators.IdentityOperator, Float64, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}, Bool, LinearSolve.LinearSolveAdjoint{Missing}}, Vector{Float64}, Matrix{Float64}, LinearAlgebra.Transpose{Float64, Vector{Float64}}, Vector{Float64}, ADTypes.AutoForwardDiff{nothing, Nothing}}, CTKAlg{LinearSolve.UMFPACKFactorization}}, F1MethodOptimizationExt.var"#7#8"{Base.Pairs{Symbol, Any, Nothing, @NamedTuple{maxItNewton::Int64, linear_cache::LinearSolve.LinearCache{SparseMatrixCSC{Float64, Int64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, LinearSolve.UMFPACKFactorization, SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}, SciMLOperators.IdentityOperator, SciMLOperators.IdentityOperator, Float64, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}, Bool, LinearSolve.LinearSolveAdjoint{Missing}}}}, AIBECS.var"#f#generate_f##1"{Main.OptParams, Tuple{Float64, Float64}, Tuple{Vector{Float64}, Missing}, Tuple{Vector{Float64}, Missing}, Vector{Float64}, Float64, AIBECS.var"#tracers#generate_f##0"{Int64, Int64}}, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#AIBECSFunction##1"{Main.OptParams, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}, AIBECS.var"#tracer#67"{Int64, Int64}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#81"{AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#AIBECSFunction##0"{Main.OptParams, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, 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SciMLOperators.IdentityOperator, Float64, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}, Bool, LinearSolve.LinearSolveAdjoint{Missing}}(sparse([1, 2, 9551, 9606, 84662, 1, 2, 3, 57, 9552  …  84657, 169318, 84658, 169319, 84659, 169320, 84660, 169321, 84661, 169322], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2  …  169318, 169318, 169319, 169319, 169320, 169320, 169321, 169321, 169322, 169322], [-3.4050287839078227e-7, -1.8030621712982388e-10, 2.202759122121374e-7, 1.669794081839543e-8, 1.1220839187426168e-7, 8.32821319857397e-8, -4.783272470058407e-7, 2.4345873581518986e-8, -2.346272528650749e-8, 3.294980961099172e-7  …  3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7], 169322, 169322), [-4.512560904168677e-12 8.935618833654465e-13; -6.652536669931426e-12 1.317312568668086e-12; … ; -0.0 -2.135436433303535e-13; -0.0 -1.6871098776291334e-13], [5.7489882607834106e-5 -2.308373390511117e-5; 5.646653132302884e-5 -3.1428196353351436e-5; … ; -4.603312782669057e-8 1.1570159339507784e-6; 5.471653491870314e-9 8.932325243468619e-7], SciMLBase.NullParameters(), LinearSolve.UMFPACKFactorization(true, true), SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}(SparseArrays.UMFPACK.Symbolic{Float64, Int64}(Ptr{Nothing}(0x0000000125718b20)), SparseArrays.UMFPACK.Numeric{Float64, Int64}(Ptr{Nothing}(0x000000030121df80)), 169322, 169322, [0, 5, 12, 19, 26, 33, 40, 47, 54, 61  …  818900, 818902, 818904, 818906, 818908, 818910, 818912, 818914, 818916, 818918], [0, 1, 9550, 9605, 84661, 0, 1, 2, 56, 9551  …  84656, 169317, 84657, 169318, 84658, 169319, 84659, 169320, 84660, 169321], [-3.4050287839078227e-7, -1.8030621712982388e-10, 2.202759122121374e-7, 1.669794081839543e-8, 1.1220839187426168e-7, 8.32821319857397e-8, -4.783272470058407e-7, 2.4345873581518986e-8, -2.346272528650749e-8, 3.294980961099172e-7  …  3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7], 0, SparseArrays.UMFPACK.UmfpackWS{Int64}([6167, 646, 2414, 2416, 6147, 6157, 6158, 675, 2402, 6137  …  100589, 98900, 100764, 97371, 98568, 102479, 2971, 98602, 3061, 98610], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  -3.634301428314729e-9, 6.08605475490754e-9, 3.126003909338102e-8, 4.993000031340455e-8, 4.535492077596722e-9, 4.902645702569492e-9, 1.897352028675765e-7, 3.2195576264689603e-9, 2.6818076806826084e-7, -1.3166840863933614e-6]), [1.0, 0.2, 0.2, 0.1, 32.0, 0.0, 0.7, 0.0, 1.0, 0.3, 1.0, 1.0, 0.9, 0.0, 10.0, 0.001, 1.0, 0.5, 0.0, 1.0], [0.0, 169322.0, 818918.0, 16.0, 4.0, 8.0, 8.0, 8.0, 0.0, 0.0  …  0.0, 0.0, -1.0, -1.0, 4.1972978e7, 0.01968699999997625, 0.01968699999997625, -1.0, -1.0, -1.0], ReentrantLock()), false, false, IdentityOperator(169322), IdentityOperator(169322), 1.4901161193847656e-8, 1.4901161193847656e-8, 338644, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}(SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.DebugLevel(), SciMLLogging.Silent(), SciMLLogging.InfoLevel(), SciMLLogging.Silent(), SciMLLogging.WarnLevel(), SciMLLogging.WarnLevel(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.WarnLevel(), SciMLLogging.WarnLevel(), SciMLLogging.WarnLevel()), LinearSolve.OperatorAssumptions{Bool}(true, LinearSolve.OperatorCondition.IllConditioned), LinearSolve.LinearSolveAdjoint{Missing}(missing)), [4.7241543264007345e-5, 6.616497742507608e-5, 0.0007607097140667825, 0.0002001818247681397, 0.00015228345495271658, 0.0001377887715644329, 0.00014231181003870498, 0.00013249304462571178, 0.00012386276033894944, 8.471960438304386e-5  …  1.6870928837112562e-7, 1.107674177977734e-7, 1.6619353326168306e-7, 2.6533564688704694e-7, 1.0705570309942015e-6, 9.402540764874025e-7, 6.90519304703392e-7, 6.642473304457541e-7, 5.535051235122763e-7, 4.372988802814714e-7], [5.7489882607834106e-5 -2.308373390511117e-5; 5.646653132302884e-5 -3.1428196353351436e-5; … ; -4.603312782669057e-8 1.1570159339507784e-6; 5.471653491870314e-9 8.932325243468619e-7], transpose([-2.7297252656153418e-5, -0.00011568624430491445, -0.00011421097914680908, -0.00011716805364084193, -0.00010979266846249825, -0.00016721064603409375, -0.00043737172418684654, -0.0012819265870092442, -0.0009894443952221357, -0.0004643803065352418  …  0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]), [4.605170185988092, 3.4011973816621555], [4.605170185988092, 3.4011973816621555], ADTypes.AutoForwardDiff()), CTKAlg{LinearSolve.UMFPACKFactorization}(LinearSolve.UMFPACKFactorization(true, true))), nothing, F1MethodOptimizationExt.var"#9#10"{Base.Pairs{Symbol, Any, Nothing, @NamedTuple{maxItNewton::Int64, linear_cache::LinearSolve.LinearCache{SparseMatrixCSC{Float64, Int64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, LinearSolve.UMFPACKFactorization, SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}, SciMLOperators.IdentityOperator, SciMLOperators.IdentityOperator, Float64, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}, Bool, LinearSolve.LinearSolveAdjoint{Missing}}}}, AIBECS.var"#f#generate_f##1"{Main.OptParams, Tuple{Float64, Float64}, Tuple{Vector{Float64}, Missing}, Tuple{Vector{Float64}, Missing}, Vector{Float64}, Float64, AIBECS.var"#tracers#generate_f##0"{Int64, Int64}}, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#AIBECSFunction##1"{Main.OptParams, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}, AIBECS.var"#tracer#67"{Int64, Int64}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#81"{AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#AIBECSFunction##0"{Main.OptParams, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}, AIBECS.var"#tracer#67"{Int64, Int64}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#81"{AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, AIBECS.var"#∇ₓf#generate_∇ₓf##1"{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Missing}, Tuple{Vector{Float64}, Missing}, Vector{Float64}, AIBECS.var"#tracers#generate_∇ₓf##0"{Int64, Int64}}, F1Method.F1Cache{LinearSolve.LinearCache{SparseMatrixCSC{Float64, Int64}, Matrix{Float64}, Matrix{Float64}, SciMLBase.NullParameters, LinearSolve.UMFPACKFactorization, SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}, SciMLOperators.IdentityOperator, SciMLOperators.IdentityOperator, Float64, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}, Bool, LinearSolve.LinearSolveAdjoint{Missing}}, Vector{Float64}, Matrix{Float64}, LinearAlgebra.Transpose{Float64, Vector{Float64}}, Vector{Float64}, ADTypes.AutoForwardDiff{nothing, Nothing}}, CTKAlg{LinearSolve.UMFPACKFactorization}}(Base.Pairs{Symbol, Any, Nothing, @NamedTuple{maxItNewton::Int64, linear_cache::LinearSolve.LinearCache{SparseMatrixCSC{Float64, Int64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, LinearSolve.UMFPACKFactorization, SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}, SciMLOperators.IdentityOperator, SciMLOperators.IdentityOperator, Float64, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}, Bool, LinearSolve.LinearSolveAdjoint{Missing}}}}(:maxItNewton => 15, :linear_cache => LinearSolve.LinearCache{SparseMatrixCSC{Float64, Int64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, LinearSolve.UMFPACKFactorization, SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}, SciMLOperators.IdentityOperator, SciMLOperators.IdentityOperator, Float64, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}, Bool, LinearSolve.LinearSolveAdjoint{Missing}}(sparse([1, 2, 9551, 9606, 84662, 1, 2, 3, 57, 9552  …  84657, 169318, 84658, 169319, 84659, 169320, 84660, 169321, 84661, 169322], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2  …  169318, 169318, 169319, 169319, 169320, 169320, 169321, 169321, 169322, 169322], [-3.440352272546861e-7, -1.8030621712982388e-10, 2.202759122121374e-7, 1.669794081839543e-8, 1.157407407381655e-7, 8.32821319857397e-8, -4.803223973230515e-7, 2.4345873581518986e-8, -2.346272528650749e-8, 3.294980961099172e-7  …  3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7], 169322, 169322), [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], SciMLBase.NullParameters(), LinearSolve.UMFPACKFactorization(true, true), SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}(SparseArrays.UMFPACK.Symbolic{Float64, Int64}(Ptr{Nothing}(0x0000000125718b20)), SparseArrays.UMFPACK.Numeric{Float64, Int64}(Ptr{Nothing}(0x000000030121df80)), 169322, 169322, [0, 5, 12, 19, 26, 33, 40, 47, 54, 61  …  818900, 818902, 818904, 818906, 818908, 818910, 818912, 818914, 818916, 818918], [0, 1, 9550, 9605, 84661, 0, 1, 2, 56, 9551  …  84656, 169317, 84657, 169318, 84658, 169319, 84659, 169320, 84660, 169321], [-3.4050287839078227e-7, -1.8030621712982388e-10, 2.202759122121374e-7, 1.669794081839543e-8, 1.1220839187426168e-7, 8.32821319857397e-8, -4.783272470058407e-7, 2.4345873581518986e-8, -2.346272528650749e-8, 3.294980961099172e-7  …  3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7], 0, SparseArrays.UMFPACK.UmfpackWS{Int64}([6167, 646, 2414, 2416, 6147, 6157, 6158, 675, 2402, 6137  …  100589, 98900, 100764, 97371, 98568, 102479, 2971, 98602, 3061, 98610], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  -3.634301428314729e-9, 6.08605475490754e-9, 3.126003909338102e-8, 4.993000031340455e-8, 4.535492077596722e-9, 4.902645702569492e-9, 1.897352028675765e-7, 3.2195576264689603e-9, 2.6818076806826084e-7, -1.3166840863933614e-6]), [1.0, 0.2, 0.2, 0.1, 32.0, 0.0, 0.7, 0.0, 1.0, 0.3, 1.0, 1.0, 0.9, 0.0, 10.0, 0.001, 1.0, 0.5, 0.0, 1.0], [0.0, 169322.0, 818918.0, 16.0, 4.0, 8.0, 8.0, 8.0, 0.0, 0.0  …  0.0, 0.0, -1.0, -1.0, 4.1972978e7, 0.01968699999997625, 0.01968699999997625, -1.0, -1.0, -1.0], ReentrantLock()), false, false, IdentityOperator(169322), IdentityOperator(169322), 1.4901161193847656e-8, 1.4901161193847656e-8, 169322, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}(SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.DebugLevel(), SciMLLogging.Silent(), SciMLLogging.InfoLevel(), SciMLLogging.Silent(), SciMLLogging.WarnLevel(), SciMLLogging.WarnLevel(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.WarnLevel(), SciMLLogging.WarnLevel(), SciMLLogging.WarnLevel()), LinearSolve.OperatorAssumptions{Bool}(true, LinearSolve.OperatorCondition.IllConditioned), LinearSolve.LinearSolveAdjoint{Missing}(missing))), AIBECS.var"#f#generate_f##1"{Main.OptParams, Tuple{Float64, Float64}, Tuple{Vector{Float64}, Missing}, Tuple{Vector{Float64}, Missing}, Vector{Float64}, Float64, AIBECS.var"#tracers#generate_f##0"{Int64, Int64}}((1.0, 0.0), ([0.001993383847718889, 0.0019914651177959006, 0.0019652102196352048, 0.0019188907593353226, 0.0018741692451726304, 0.0018593180609968575, 0.0018762683281708847, 0.0018866130577569658, 0.0018618487300791524, 0.0017936109269207174  …  0.001597349353171885, 0.0016139239530996844, 0.0016146148645877836, 0.0016014571666717527, 0.0015958896704982308, 0.0015948419428572936, 0.0015868619350592294, 0.0015825331862406298, 0.0015800279642719972, 0.001589979759638126], missing), ([4.7499243508165185e-9, 2.3806476583116648e-9, 1.644850324443933e-9, 2.474518014121221e-9, 2.8420192007964507e-9, 1.9923744404030257e-9, 8.150346589335414e-10, 2.974775216581886e-10, 4.021567996605766e-10, 8.840013268111294e-10  …  4.92697369868074e-10, 2.974775216581886e-10, 2.974775216581886e-10, 2.3496760609714243e-9, 2.446816223482753e-9, 2.974775216581886e-10, 8.053041266157682e-10, 8.39703761723315e-10, 1.2798255287465522e-9, 1.7406554311458743e-9], missing), [4.1259441424340015e11, 8.858714556466658e11, 9.658719592542885e11, 1.0446956966607107e12, 1.122246633283809e12, 1.1984302852530718e12, 1.2731538345237942e12, 1.3463262419614636e12, 1.417858358258977e12, 1.487663032551257e12  …  5.551473635870713e13, 6.642031628003376e13, 5.1445109315594125e13, 4.172486815218424e13, 1.3409911172962852e13, 1.824228219687533e13, 2.7565991089214062e13, 3.717262516144412e13, 4.7318477973128164e13, 4.2196399175523055e13], 0.0001, AIBECS.var"#tracers#generate_f##0"{Int64, Int64}(84661, 2)), SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#AIBECSFunction##1"{Main.OptParams, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, 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typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}(AIBECS.var"#f#AIBECSFunction##1"{Main.OptParams, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}, AIBECS.var"#tracer#67"{Int64, Int64}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#81"{AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}}(SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}, AIBECS.var"#tracer#67"{Int64, Int64}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#81"{AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}(AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}, AIBECS.var"#tracer#67"{Int64, Int64}}((Main.T_DIP, Main.T_POP), 1:2, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}((Main.G_DIP, Main.G_POP), AIBECS.var"#tracers#66"{Int64, Int64}(84661, 2)), AIBECS.var"#tracer#67"{Int64, Int64}(84661, 2)), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, AIBECS.var"#jac#81"{AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}(AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}((Main.T_DIP, Main.T_POP), 2, 1:2), AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}((Main.G_DIP, Main.G_POP), 84661, 2)), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, nothing, nothing, nothing)), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, AIBECS.var"#jac#AIBECSFunction##0"{Main.OptParams, SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}, AIBECS.var"#tracer#67"{Int64, Int64}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#81"{AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}}(SciMLBase.ODEFunction{false, SciMLBase.FullSpecialize, AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}, AIBECS.var"#tracer#67"{Int64, Int64}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, AIBECS.var"#jac#81"{AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}(AIBECS.var"#f#71"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, UnitRange{Int64}, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}, AIBECS.var"#tracer#67"{Int64, Int64}}((Main.T_DIP, Main.T_POP), 1:2, AIBECS.var"#G#68"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, AIBECS.var"#tracers#66"{Int64, Int64}}((Main.G_DIP, Main.G_POP), AIBECS.var"#tracers#66"{Int64, Int64}(84661, 2)), AIBECS.var"#tracer#67"{Int64, Int64}(84661, 2)), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, AIBECS.var"#jac#81"{AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}, AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}}(AIBECS.var"#T#76"{Tuple{typeof(Main.T_DIP), typeof(Main.T_POP)}, Int64, UnitRange{Int64}}((Main.T_DIP, Main.T_POP), 2, 1:2), AIBECS.var"#72#73"{Tuple{typeof(Main.G_DIP), typeof(Main.G_POP)}, Int64, Int64}((Main.G_DIP, Main.G_POP), 84661, 2)), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, nothing, nothing, nothing)), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, nothing, nothing, nothing), AIBECS.var"#∇ₓf#generate_∇ₓf##1"{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Missing}, Tuple{Vector{Float64}, Missing}, Vector{Float64}, AIBECS.var"#tracers#generate_∇ₓf##0"{Int64, Int64}}((1.0, 0.0), ([0.001993383847718889, 0.0019914651177959006, 0.0019652102196352048, 0.0019188907593353226, 0.0018741692451726304, 0.0018593180609968575, 0.0018762683281708847, 0.0018866130577569658, 0.0018618487300791524, 0.0017936109269207174  …  0.001597349353171885, 0.0016139239530996844, 0.0016146148645877836, 0.0016014571666717527, 0.0015958896704982308, 0.0015948419428572936, 0.0015868619350592294, 0.0015825331862406298, 0.0015800279642719972, 0.001589979759638126], missing), ([4.7499243508165185e-9, 2.3806476583116648e-9, 1.644850324443933e-9, 2.474518014121221e-9, 2.8420192007964507e-9, 1.9923744404030257e-9, 8.150346589335414e-10, 2.974775216581886e-10, 4.021567996605766e-10, 8.840013268111294e-10  …  4.92697369868074e-10, 2.974775216581886e-10, 2.974775216581886e-10, 2.3496760609714243e-9, 2.446816223482753e-9, 2.974775216581886e-10, 8.053041266157682e-10, 8.39703761723315e-10, 1.2798255287465522e-9, 1.7406554311458743e-9], missing), [4.1259441424340015e11, 8.858714556466658e11, 9.658719592542885e11, 1.0446956966607107e12, 1.122246633283809e12, 1.1984302852530718e12, 1.2731538345237942e12, 1.3463262419614636e12, 1.417858358258977e12, 1.487663032551257e12  …  5.551473635870713e13, 6.642031628003376e13, 5.1445109315594125e13, 4.172486815218424e13, 1.3409911172962852e13, 1.824228219687533e13, 2.7565991089214062e13, 3.717262516144412e13, 4.7318477973128164e13, 4.2196399175523055e13], AIBECS.var"#tracers#generate_∇ₓf##0"{Int64, Int64}(84661, 2), Core.Box(AIBECS.var"#∇ₓf#generate_∇ₓf##1"{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Missing}, Tuple{Vector{Float64}, Missing}, Vector{Float64}, AIBECS.var"#tracers#generate_∇ₓf##0"{Int64, Int64}}(#= circular reference @-2 =#))), F1Method.F1Cache{LinearSolve.LinearCache{SparseMatrixCSC{Float64, Int64}, Matrix{Float64}, Matrix{Float64}, SciMLBase.NullParameters, LinearSolve.UMFPACKFactorization, SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}, SciMLOperators.IdentityOperator, SciMLOperators.IdentityOperator, Float64, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}, Bool, LinearSolve.LinearSolveAdjoint{Missing}}, Vector{Float64}, Matrix{Float64}, LinearAlgebra.Transpose{Float64, Vector{Float64}}, Vector{Float64}, ADTypes.AutoForwardDiff{nothing, Nothing}}(LinearSolve.LinearCache{SparseMatrixCSC{Float64, Int64}, Matrix{Float64}, Matrix{Float64}, SciMLBase.NullParameters, LinearSolve.UMFPACKFactorization, SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}, SciMLOperators.IdentityOperator, SciMLOperators.IdentityOperator, Float64, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}, Bool, LinearSolve.LinearSolveAdjoint{Missing}}(sparse([1, 2, 9551, 9606, 84662, 1, 2, 3, 57, 9552  …  84657, 169318, 84658, 169319, 84659, 169320, 84660, 169321, 84661, 169322], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2  …  169318, 169318, 169319, 169319, 169320, 169320, 169321, 169321, 169322, 169322], [-3.4050287839078227e-7, -1.8030621712982388e-10, 2.202759122121374e-7, 1.669794081839543e-8, 1.1220839187426168e-7, 8.32821319857397e-8, -4.783272470058407e-7, 2.4345873581518986e-8, -2.346272528650749e-8, 3.294980961099172e-7  …  3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7], 169322, 169322), [-4.512560904168677e-12 8.935618833654465e-13; -6.652536669931426e-12 1.317312568668086e-12; … ; -0.0 -2.135436433303535e-13; -0.0 -1.6871098776291334e-13], [5.7489882607834106e-5 -2.308373390511117e-5; 5.646653132302884e-5 -3.1428196353351436e-5; … ; -4.603312782669057e-8 1.1570159339507784e-6; 5.471653491870314e-9 8.932325243468619e-7], SciMLBase.NullParameters(), LinearSolve.UMFPACKFactorization(true, true), SparseArrays.UMFPACK.UmfpackLU{Float64, Int64}(SparseArrays.UMFPACK.Symbolic{Float64, Int64}(Ptr{Nothing}(0x0000000125718b20)), SparseArrays.UMFPACK.Numeric{Float64, Int64}(Ptr{Nothing}(0x000000030121df80)), 169322, 169322, [0, 5, 12, 19, 26, 33, 40, 47, 54, 61  …  818900, 818902, 818904, 818906, 818908, 818910, 818912, 818914, 818916, 818918], [0, 1, 9550, 9605, 84661, 0, 1, 2, 56, 9551  …  84656, 169317, 84657, 169318, 84658, 169319, 84659, 169320, 84660, 169321], [-3.4050287839078227e-7, -1.8030621712982388e-10, 2.202759122121374e-7, 1.669794081839543e-8, 1.1220839187426168e-7, 8.32821319857397e-8, -4.783272470058407e-7, 2.4345873581518986e-8, -2.346272528650749e-8, 3.294980961099172e-7  …  3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7, 3.858024691358024e-7, -3.8583415722361643e-7], 0, SparseArrays.UMFPACK.UmfpackWS{Int64}([6167, 646, 2414, 2416, 6147, 6157, 6158, 675, 2402, 6137  …  100589, 98900, 100764, 97371, 98568, 102479, 2971, 98602, 3061, 98610], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  -3.634301428314729e-9, 6.08605475490754e-9, 3.126003909338102e-8, 4.993000031340455e-8, 4.535492077596722e-9, 4.902645702569492e-9, 1.897352028675765e-7, 3.2195576264689603e-9, 2.6818076806826084e-7, -1.3166840863933614e-6]), [1.0, 0.2, 0.2, 0.1, 32.0, 0.0, 0.7, 0.0, 1.0, 0.3, 1.0, 1.0, 0.9, 0.0, 10.0, 0.001, 1.0, 0.5, 0.0, 1.0], [0.0, 169322.0, 818918.0, 16.0, 4.0, 8.0, 8.0, 8.0, 0.0, 0.0  …  0.0, 0.0, -1.0, -1.0, 4.1972978e7, 0.01968699999997625, 0.01968699999997625, -1.0, -1.0, -1.0], ReentrantLock()), false, false, IdentityOperator(169322), IdentityOperator(169322), 1.4901161193847656e-8, 1.4901161193847656e-8, 338644, LinearSolve.LinearVerbosity{SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.DebugLevel, SciMLLogging.Silent, SciMLLogging.InfoLevel, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.Silent, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel, SciMLLogging.WarnLevel}(SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.DebugLevel(), SciMLLogging.Silent(), SciMLLogging.InfoLevel(), SciMLLogging.Silent(), SciMLLogging.WarnLevel(), SciMLLogging.WarnLevel(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.Silent(), SciMLLogging.WarnLevel(), SciMLLogging.WarnLevel(), SciMLLogging.WarnLevel()), LinearSolve.OperatorAssumptions{Bool}(true, LinearSolve.OperatorCondition.IllConditioned), LinearSolve.LinearSolveAdjoint{Missing}(missing)), [4.7241543264007345e-5, 6.616497742507608e-5, 0.0007607097140667825, 0.0002001818247681397, 0.00015228345495271658, 0.0001377887715644329, 0.00014231181003870498, 0.00013249304462571178, 0.00012386276033894944, 8.471960438304386e-5  …  1.6870928837112562e-7, 1.107674177977734e-7, 1.6619353326168306e-7, 2.6533564688704694e-7, 1.0705570309942015e-6, 9.402540764874025e-7, 6.90519304703392e-7, 6.642473304457541e-7, 5.535051235122763e-7, 4.372988802814714e-7], [5.7489882607834106e-5 -2.308373390511117e-5; 5.646653132302884e-5 -3.1428196353351436e-5; … ; -4.603312782669057e-8 1.1570159339507784e-6; 5.471653491870314e-9 8.932325243468619e-7], transpose([-2.7297252656153418e-5, -0.00011568624430491445, -0.00011421097914680908, -0.00011716805364084193, -0.00010979266846249825, -0.00016721064603409375, -0.00043737172418684654, -0.0012819265870092442, -0.0009894443952221357, -0.0004643803065352418  …  0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]), [4.605170185988092, 3.4011973816621555], [4.605170185988092, 3.4011973816621555], ADTypes.AutoForwardDiff()), CTKAlg{LinearSolve.UMFPACKFactorization}(LinearSolve.UMFPACKFactorization(true, true))), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED_NO_TIME, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing)

Run the optimiser

maxiters = 20 is plenty to watch the outer Newton trust-region trace drive the gradient norm down by several decades. We attach a callback that records the parameter vector at each iteration so the diagnostics figure below can overlay the Newton trajectory on the cost-surface contourf.

julia

λ_trajectory = Vector{Vector{Float64}}()
push!(λ_trajectory, copy(λ₀))
trajectory_cb = function (state, _)
    push!(λ_trajectory, copy(state.u))
    return false
end

prob = OptimizationProblem(optfn, copy(λ₀))
results = solve(prob, NewtonTrustRegion();
    maxiters = 20, callback = trajectory_cb, show_trace = true)
p_opt = λ2p(OptParams, results.u)

Main.OptParams{Float64}
  w₀ = 5.0 (m d⁻¹)
  w′ = 0.035 (d⁻¹)
  τ_DIP = 398.61539377922907 (d)
  k = 10.0 (μmol m⁻³)
  z₀ = 80.0 (m)
  τ_POP = 12.169557505218696 (d)
  τ_geo = 1.0 (kyr)
  DIP_geo = 2.12 (mmol m⁻³)

Cost-function sweep around the trajectory

Evaluate f(x_ss, λ) on a 10×10 grid in λ-space centred on the trajectory (padded by ±0.2). Each cell solves the steady state from the same cold-start x0, warm-started by the shared UMFPACK cache from the previous section. Takes a couple of minutes.

julia

n_grid = 10
pad    = 0.2
λ1_lo, λ1_hi = extrema(λ[1] for λ in λ_trajectory) .+ (-pad, pad)
λ2_lo, λ2_hi = extrema(λ[2] for λ in λ_trajectory) .+ (-pad, pad)
λ1_vec = range(λ1_lo, λ1_hi; length = n_grid)
λ2_vec = range(λ2_lo, λ2_hi; length = n_grid)

F_grid = Matrix{Float64}(undef, n_grid, n_grid)
for j in 1:n_grid, i in 1:n_grid
    λ = [λ1_vec[i], λ2_vec[j]]
    sol = solve(SteadyStateProblem(F, x0, λ), CTKAlg(); solver_kwargs...)
    F_grid[i, j] = f(sol.u, λ)
end

┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 2.1483971561731135e-14
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 2.4991252418133468e-15
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 3.784922900580025e-15
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 7.486948351552839e-11
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 1.3309906925042203e-14
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 2.8351611483166536e-12
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 5.532024368145776e-15
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 1.9248858459254306e-14
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 3.1690160794423613e-12
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 2.5973607520649588e-14
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 3.219302917198701e-12
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 3.312083398850626e-14
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 3.1786108611166313e-12
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 4.38177660736921e-14
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 1.0736993472968038e-13
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 1.8001670317891646e-11
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 1.6546650051179805e-12
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164
┌ Warning: CTKAlg did not converge to a successful retcode.
│   retcode = ReturnCode.MaxIters = 4
│   residual_inf_norm = 1.1063176324372192e-13
│   abstol = 0.0
│   reltol = 3.168808781402895e-14
│   maxItNewton = 15
└ @ AIBECS ~/work/AIBECS.jl/AIBECS.jl/src/overload_solve.jl:164

Diagnostics — setup

We use CairoMakie (see also the Makie how-to) for plotting.

julia

using CairoMakie
using OceanBasins
using KernelDensity
CairoMakie.activate!(type = "png")

prob_opt = SteadyStateProblem(F, x0, p_opt)
DIPopt, _ = state_to_tracers(solve(prob_opt, CTKAlg(); solver_kwargs...).u, grd)

DIPopt_μM = ustrip.(μM, DIPopt  .* (mol/m^3))
μWOA_μM   = ustrip.(μM, μDIPobs .* (mol/m^3))
diff_μM   = DIPopt_μM .- μWOA_μM

depth = ustrip.(grd.depth)
lat   = ustrip.(grd.lat)
lon   = ustrip.(grd.lon)
maxdepth_round = ceil(maximum(depth) / 1000) * 1000

dip_levels  = 0:0.25:3.0
dip_cmap    = cgrad(:viridis, length(dip_levels) - 1; categorical = true)
diff_levels = -1.0:0.2:1.0
diff_cmap   = cgrad(:RdBu, length(diff_levels) - 1; categorical = true, rev = true)

latticks = (-90:30:90, ["90°S", "60°S", "30°S", "0°", "30°N", "60°N", "90°N"])
lonticks = (0:60:360, ["0°", "60°E", "120°E", "180°", "120°W", "60°W", "0°"])
gcdepthticks = (0:1000:maxdepth_round, string.(Int.(0:1000:maxdepth_round)))
plus_minus_ticks = (-1:0.5:1, ["−1", "−0.5", "0", "+0.5", "+1"])

(-1.0:0.5:1.0, ["−1", "−0.5", "0", "+0.5", "+1"])

Basin masks, each extended to the pole via its Southern Ocean sector.

julia

OCEANS = oceanpolygons()
lat_grd, lon_grd = latvec(grd), lonvec(grd)
mPAC = ispacific2(lat_grd, lon_grd, OCEANS)  .| isSOpacific(lat_grd, lon_grd, OCEANS)
mATL = isatlantic2(lat_grd, lon_grd, OCEANS) .| isSOatlantic(lat_grd, lon_grd, OCEANS)
mIND = isindian2(lat_grd, lon_grd, OCEANS)   .| isSOindian(lat_grd, lon_grd, OCEANS)
basin_specs = [
    (name = "Pacific",  mask = mPAC, color = :dodgerblue),
    (name = "Atlantic", mask = mATL, color = :firebrick),
    (name = "Indian",   mask = mIND, color = :darkorange),
]

3-element Vector{@NamedTuple{name::String, mask::BitVector, color::Symbol}}:
 (name = "Pacific", mask = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0  …  0, 0, 0, 0, 0, 0, 0, 0, 0, 0], color = :dodgerblue)
 (name = "Atlantic", mask = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1  …  1, 1, 1, 1, 1, 1, 1, 1, 1, 1], color = :firebrick)
 (name = "Indian", mask = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0  …  0, 0, 0, 0, 0, 0, 0, 0, 0, 0], color = :darkorange)

Optimised vs WOA scatter

Each wet box at (WOA, optimised), coloured by its mass-weighted density quantile under a volume-weighted bivariate KDE. Walking the sorted cumulative density once gives both the iso-mass contour levels and the per-point colours.

julia

bandwidth = 0.25 .* KernelDensity.default_bandwidth((μWOA_μM, DIPopt_μM))
d_kde = kde((μWOA_μM, DIPopt_μM); bandwidth, weights = v ./ sum(v))
ik = InterpKDE(d_kde)

dx, dy = step(d_kde.x), step(d_kde.y)
ρ_asc = sort(vec(d_kde.density))
cum_asc = cumsum(ρ_asc) .* (dx * dy)
cum_asc ./= cum_asc[end]
mass_qs = 0.1:0.1:0.9
levels = [ρ_asc[searchsortedfirst(cum_asc, q)] for q in mass_qs]
point_q = [(i = searchsortedlast(ρ_asc, d);
            i == 0 ? 0.0 : cum_asc[i]) for d in ik.itp.(μWOA_μM, DIPopt_μM)]

fig = Figure()
ax = Axis(fig[1, 1];
    xlabel = "WOA PO₄ (μM)", ylabel = "optimised PO₄ (μM)",
    xgridvisible = false, ygridvisible = false,
    aspect = DataAspect(), limits = (0, 3, 0, 3))
sc = scatter!(ax, μWOA_μM, DIPopt_μM;
    markersize = 4, color = point_q, colormap = :plasma, colorrange = (0, 1))
contour!(ax, d_kde.x, d_kde.y, d_kde.density;
    levels = levels, color = (:black, 0.5), linewidth = 1.2)
lines!(ax, [0, 3], [0, 3]; color = :red, linestyle = :dash)
Colorbar(fig[1, 2], sc; label = "density quantile", ticks = mass_qs)
fig

Per-basin depth profiles — model vs WOA

julia

fig = Figure(size = (900, 350))
for (c, basin) in enumerate(basin_specs)
    ax = Axis(fig[1, c];
        xgridvisible = false, ygridvisible = false,
        xlabel = "PO₄ (μM)",
        ylabel = c == 1 ? "depth (m)" : "",
        yreversed = true, yticks = gcdepthticks,
        title = basin.name,
        limits = (nothing, (0, maxdepth_round)))
    prof_model = AIBECS.horizontalmean(DIPopt_μM, grd, basin.mask)
    prof_obs   = AIBECS.horizontalmean(μWOA_μM,   grd, basin.mask)
    lines!(ax, prof_model, depth;
        color = basin.color, linewidth = 2, label = "model")
    lines!(ax, prof_obs, depth;
        color = basin.color, linewidth = 2, linestyle = :dash, label = "WOA")
    c == 1 && axislegend(ax; position = :rb, framevisible = false)
    c > 1 && hideydecorations!(ax;
        ticklabels = true, label = true, ticks = false, grid = false)
end
fig

Per-basin zonal means — model / WOA / model − WOA

julia

row_specs = [
    (label = "model",         field = DIPopt_μM, levels = dip_levels,  cmap = dip_cmap),
    (label = "WOA",           field = μWOA_μM,   levels = dip_levels,  cmap = dip_cmap),
    (label = "model − WOA",   field = diff_μM,   levels = diff_levels, cmap = diff_cmap),
]
zm_grid = [
    [AIBECS.zonalmean(spec.field, grd, basin.mask) for basin in basin_specs]
    for spec in row_specs
]
function wetlatrange(zm, pad = 5)
    haswet = [any(!isnan, view(zm, i, :)) for i in eachindex(lat)]
    wetidx = findall(haswet)
    isempty(wetidx) && return (-90.0, 90.0)
    return (max(-90, lat[first(wetidx)] - pad),
            min( 90, lat[last(wetidx)]  + pad))
end
xlim_per_basin = [wetlatrange(zm_grid[1][c]) for c in eachindex(basin_specs)]

# Documenter `@example` blocks run at module scope, where bare
# `co_v = co` inside a `for` would shadow the outer binding; `Ref` sidesteps it.
co_v = Ref{Any}()
co_d = Ref{Any}()

fig = Figure(size = (1400, 900))
for (r, spec) in enumerate(row_specs), (c, basin) in enumerate(basin_specs)
    xlim = xlim_per_basin[c]
    ax = Axis(fig[r, c];
        backgroundcolor = :lightgray,
        xgridvisible = false, ygridvisible = false,
        xticksmirrored = true, yticksmirrored = true,
        xticks = latticks, yticks = gcdepthticks,
        yreversed = true,
        ylabel = c == 1 ? "depth (m)" : "",
        title = r == 1 ? basin.name : "",
        limits = (xlim[1], xlim[2], 0, maxdepth_round))
    co = contourf!(ax, lat, depth, zm_grid[r][c];
        levels = spec.levels, colormap = spec.cmap, nan_color = :lightgray,
        extendlow = spec.cmap[1], extendhigh = spec.cmap[end])
    r == 1 && c == 1 && (co_v[] = co)
    r == 3 && c == 1 && (co_d[] = co)
    c > 1 && hideydecorations!(ax; ticklabels = true, label = true, ticks = false, grid = false)
    r < 3 && hidexdecorations!(ax; ticklabels = true, label = true, ticks = false, grid = false)
end
Label(fig[1, 0], "model";        rotation = π/2, font = :bold, fontsize = 16, tellheight = false)
Label(fig[2, 0], "WOA";          rotation = π/2, font = :bold, fontsize = 16, tellheight = false)
Label(fig[3, 0], "model − WOA";  rotation = π/2, font = :bold, fontsize = 16, tellheight = false)
Colorbar(fig[1:2, length(basin_specs) + 1], co_v[];
    label = "PO₄ (μM)", ticks = 0:1:3)
Colorbar(fig[3, length(basin_specs) + 1], co_d[];
    label = "model − WOA (μM)", ticks = plus_minus_ticks)
colsize!(fig.layout, 0, Auto(0.15))
for (c, xlim) in enumerate(xlim_per_basin)
    colsize!(fig.layout, c, Auto(xlim[2] - xlim[1]))
end
colsize!(fig.layout, length(basin_specs) + 1, Auto(0.25))
fig

Horizontal slices at 1000 m — model / WOA / diff

julia

sl_opt  = AIBECS.horizontalslice(DIPopt_μM, grd; depth = 1000)
sl_obs  = AIBECS.horizontalslice(μWOA_μM,   grd; depth = 1000)
sl_diff = sl_opt .- sl_obs

function map_axis(grid_pos; title = "")
    Axis(grid_pos;
        backgroundcolor = :lightgray,
        xgridvisible = false, ygridvisible = false,
        xticks = lonticks, yticks = latticks,
        title = title, limits = (0, 360, -90, 90))
end

fig = Figure(size = (1400, 500))
ax_m = map_axis(fig[1, 1]; title = "optimised")
co_m = contourf!(ax_m, lon, lat, sl_opt';
    levels = dip_levels, colormap = dip_cmap, nan_color = :lightgray,
    extendlow = dip_cmap[1], extendhigh = dip_cmap[end])
ax_o = map_axis(fig[1, 2]; title = "WOA")
contourf!(ax_o, lon, lat, sl_obs';
    levels = dip_levels, colormap = dip_cmap, nan_color = :lightgray,
    extendlow = dip_cmap[1], extendhigh = dip_cmap[end])
ax_d = map_axis(fig[1, 3]; title = "optimised − WOA")
co_dd = contourf!(ax_d, lon, lat, sl_diff';
    levels = diff_levels, colormap = diff_cmap, nan_color = :lightgray,
    extendlow = diff_cmap[1], extendhigh = diff_cmap[end])
linkyaxes!(ax_m, ax_o, ax_d)
hideydecorations!(ax_o; ticklabels = true, label = false, ticks = false, grid = false)
hideydecorations!(ax_d; ticklabels = true, label = false, ticks = false, grid = false)
cb_v = Colorbar(fig[2, 1:2], co_m; vertical = false, flipaxis = false,
    label = "PO₄ @ 1000 m (μM)", ticks = 0:1:3)
cb_v.width = Relative(0.7)
cb_diff = Colorbar(fig[2, 3], co_dd; vertical = false, flipaxis = false,
    label = "optimised − WOA (μM)", ticks = plus_minus_ticks)
cb_diff.width = Relative(0.85)
fig

Side-by-side parameter tables, initial vs optimised

julia

AIBECS.table(p₀)

8×7 DataFrame

Row	Symbol	Value	Initial value	Unit	Bounds	Logscaled	Optimizable
	Symbol	Float64	Float64	FreeUnit…	Tuple…	Bool	Bool
1	w₀	5.0	5.0	m d⁻¹	(0, Inf)	true	false
2	w′	0.035	0.035	d⁻¹	(0, Inf)	true	false
3	τ_DIP	100.0	100.0	d	(0, Inf)	true	true
4	k	10.0	10.0	μmol m⁻³	(0, Inf)	true	false
5	z₀	80.0	80.0	m	(0, Inf)	false	false
6	τ_POP	30.0	30.0	d	(0, Inf)	true	true
7	τ_geo	1.0	1.0	kyr	(0, Inf)	true	false
8	DIP_geo	2.12	2.12	mmol m⁻³	(0, Inf)	false	false

julia

AIBECS.table(p_opt)

8×7 DataFrame

Row	Symbol	Value	Initial value	Unit	Bounds	Logscaled	Optimizable
	Symbol	Float64	Float64	FreeUnit…	Tuple…	Bool	Bool
1	w₀	5.0	5.0	m d⁻¹	(0, Inf)	true	false
2	w′	0.035	0.035	d⁻¹	(0, Inf)	true	false
3	τ_DIP	398.615	100.0	d	(0, Inf)	true	true
4	k	10.0	10.0	μmol m⁻³	(0, Inf)	true	false
5	z₀	80.0	80.0	m	(0, Inf)	false	false
6	τ_POP	12.1696	30.0	d	(0, Inf)	true	true
7	τ_geo	1.0	1.0	kyr	(0, Inf)	true	false
8	DIP_geo	2.12	2.12	mmol m⁻³	(0, Inf)	false	false

Extra: cost-surface sweep + Newton trajectory

Mostly diagnostic. Once you've converged, the optimal parameters are the point — but the sweep + trajectory is useful when something goes wrong. Convert the λ-space sweep grid and trajectory to physical (τ_DIP, τ_POP) via λ2p, then a single contourf! + scatterlines! does it.

julia

τDIP_grid = [λ2p(OptParams, [λ1, λ₀[2]]).τ_DIP for λ1 in λ1_vec]
τPOP_grid = [λ2p(OptParams, [λ₀[1], λ2]).τ_POP for λ2 in λ2_vec]
τDIP_traj = [λ2p(OptParams, λ).τ_DIP for λ in λ_trajectory]
τPOP_traj = [λ2p(OptParams, λ).τ_POP for λ in λ_trajectory]

fig = Figure()
ax = Axis(fig[1, 1];
    xlabel = "τ_DIP (d)", ylabel = "τ_POP (d)",
    xscale = log10, yscale = log10,
    title = "√f(p) cost surface + Newton trajectory")
co = contourf!(ax, τDIP_grid, τPOP_grid, sqrt.(F_grid);
    levels = 20, colormap = cgrad(:BrBG; rev = true))
scatterlines!(ax, τDIP_traj, τPOP_traj;
    color = :red, linewidth = 2, linestyle = :dash, marker = :circle, markersize = 4)
scatter!(ax, [τDIP_traj[end]], [τPOP_traj[end]];
    marker = :star5, markersize = 22, color = :yellow,
    strokecolor = :black, strokewidth = 1.5)
annotation!(ax, 70, -40, τDIP_traj[1], τPOP_traj[1];
    text = "first iterate", path = Ann.Paths.Corner())
annotation!(ax, -70, +40, τDIP_traj[end], τPOP_traj[end];
    text = "optimal solution", path = Ann.Paths.Corner(), color = :white)
Colorbar(fig[1, 2], co; label = "√f(p)")
fig

Where to next

This example only fits two parameters of a single-tracer mismatch objective on a moderate-resolution circulation; the multi-tracer, multi-observation fit lives in GNOM. See the roadmap for alternative parameter-estimation backends (Turing, ImplicitDifferentiation, SciMLSensitivity), the table-form f_and_∇ₓf(ωs, ωp, grd, obs, ::Type{P}) for fitting against WorldOceanAtlasTools.observations(...) tables, and other directions where contributions are welcome.

This page was generated using Literate.jl.

Parameter optimisation ​

Re-build the P-model on OCCA ​

Optimisable parameter struct ​

State function in parameter-type form ​

Load WOA PO₄ observations ​

Build the objective (f, ∇ₓf) ​

F-1 memory cache + warm-started linear solve ​

Run the optimiser ​

Cost-function sweep around the trajectory ​

Diagnostics — setup ​

Optimised vs WOA scatter ​

Per-basin depth profiles — model vs WOA ​

Per-basin zonal means — model / WOA / model − WOA ​

Horizontal slices at 1000 m — model / WOA / diff ​

Side-by-side parameter tables, initial vs optimised ​

Extra: cost-surface sweep + Newton trajectory ​

Where to next ​

Parameter optimisation

Re-build the P-model on OCCA

Optimisable parameter struct

State function in parameter-type form

Load WOA PO₄ observations

Build the objective `(f, ∇ₓf)`

F-1 memory cache + warm-started linear solve

Run the optimiser

Cost-function sweep around the trajectory

Diagnostics — setup

Optimised vs WOA scatter

Per-basin depth profiles — model vs WOA

Per-basin zonal means — model / WOA / model − WOA

Horizontal slices at 1000 m — model / WOA / diff

Side-by-side parameter tables, initial vs optimised

Extra: cost-surface sweep + Newton trajectory

Where to next