|
| 1 | +""" |
| 2 | + SimpleDFSane(; σ_min::Real = 1e-10, σ_max::Real = 1e10, σ_1::Real = 1.0, |
| 3 | + M::Union{Int, Val} = Val(10), γ::Real = 1e-4, τ_min::Real = 0.1, τ_max::Real = 0.5, |
| 4 | + nexp::Int = 2, η_strategy::Function = (f_1, k, x, F) -> f_1 ./ k^2) |
1 | 5 |
|
| 6 | +A low-overhead implementation of the df-sane method for solving large-scale nonlinear |
| 7 | +systems of equations. For in depth information about all the parameters and the algorithm, |
| 8 | +see [la2006spectral](@citet). |
| 9 | +
|
| 10 | +### Keyword Arguments |
| 11 | +
|
| 12 | + - `σ_min`: the minimum value of the spectral coefficient `σ_k` which is related to the |
| 13 | + step size in the algorithm. Defaults to `1e-10`. |
| 14 | + - `σ_max`: the maximum value of the spectral coefficient `σ_k` which is related to the |
| 15 | + step size in the algorithm. Defaults to `1e10`. |
| 16 | + - `σ_1`: the initial value of the spectral coefficient `σ_k` which is related to the step |
| 17 | + size in the algorithm.. Defaults to `1.0`. |
| 18 | + - `M`: The monotonicity of the algorithm is determined by a this positive integer. |
| 19 | + A value of 1 for `M` would result in strict monotonicity in the decrease of the L2-norm |
| 20 | + of the function `f`. However, higher values allow for more flexibility in this |
| 21 | + reduction. Despite this, the algorithm still ensures global convergence through the use |
| 22 | + of a non-monotone line-search algorithm that adheres to the Grippo-Lampariello-Lucidi |
| 23 | + condition. Values in the range of 5 to 20 are usually sufficient, but some cases may call |
| 24 | + for a higher value of `M`. The default setting is 10. |
| 25 | + - `γ`: a parameter that influences if a proposed step will be accepted. Higher value of |
| 26 | + `γ` will make the algorithm more restrictive in accepting steps. Defaults to `1e-4`. |
| 27 | + - `τ_min`: if a step is rejected the new step size will get multiplied by factor, and this |
| 28 | + parameter is the minimum value of that factor. Defaults to `0.1`. |
| 29 | + - `τ_max`: if a step is rejected the new step size will get multiplied by factor, and this |
| 30 | + parameter is the maximum value of that factor. Defaults to `0.5`. |
| 31 | + - `nexp`: the exponent of the loss, i.e. ``f_k=||F(x_k)||^{nexp}``. The paper uses |
| 32 | + `nexp ∈ {1,2}`. Defaults to `2`. |
| 33 | + - `η_strategy`: function to determine the parameter `η_k`, which enables growth |
| 34 | + of ``||F||^2``. Called as `η_k = η_strategy(f_1, k, x, F)` with `f_1` initialized as |
| 35 | + ``f_1=||F(x_1)||^{nexp}``, `k` is the iteration number, `x` is the current `x`-value and |
| 36 | + `F` the current residual. Should satisfy ``η_k > 0`` and ``∑ₖ ηₖ < ∞``. Defaults to |
| 37 | + ``||F||^2 / k^2``. |
| 38 | +""" |
| 39 | +@concrete struct SimpleDFSane <: AbstractSimpleNonlinearSolveAlgorithm |
| 40 | + σ_min |
| 41 | + σ_max |
| 42 | + σ_1 |
| 43 | + γ |
| 44 | + τ_min |
| 45 | + τ_max |
| 46 | + nexp::Int |
| 47 | + η_strategy |
| 48 | + M <: Val |
| 49 | +end |
| 50 | + |
| 51 | +# XXX[breaking]: we should change the names to not have unicode |
| 52 | +function SimpleDFSane(; σ_min::Real = 1e-10, σ_max::Real = 1e10, σ_1::Real = 1.0, |
| 53 | + M::Union{Int, Val} = Val(10), γ::Real = 1e-4, τ_min::Real = 0.1, τ_max::Real = 0.5, |
| 54 | + nexp::Int = 2, η_strategy::F = (f_1, k, x, F) -> f_1 ./ k^2) where {F} |
| 55 | + M = M isa Int ? Val(M) : M |
| 56 | + return SimpleDFSane(σ_min, σ_max, σ_1, γ, τ_min, τ_max, nexp, η_strategy, M) |
| 57 | +end |
| 58 | + |
| 59 | +function SciMLBase.__solve(prob::ImmutableNonlinearProblem, alg::SimpleDFSane, args...; |
| 60 | + abstol = nothing, reltol = nothing, maxiters = 1000, alias_u0 = false, |
| 61 | + termination_condition = nothing, kwargs...) |
| 62 | + x = Utils.maybe_unaliased(prob.u0, alias_u0) |
| 63 | + fx = Utils.get_fx(prob, x) |
| 64 | + fx = Utils.eval_f(prob, fx, x) |
| 65 | + T = promote_type(eltype(fx), eltype(x)) |
| 66 | + |
| 67 | + σ_min = T(alg.σ_min) |
| 68 | + σ_max = T(alg.σ_max) |
| 69 | + σ_k = T(alg.σ_1) |
| 70 | + |
| 71 | + (; nexp, η_strategy, M) = alg |
| 72 | + γ = T(alg.γ) |
| 73 | + τ_min = T(alg.τ_min) |
| 74 | + τ_max = T(alg.τ_max) |
| 75 | + |
| 76 | + abstol, reltol, tc_cache = NonlinearSolveBase.init_termination_cache( |
| 77 | + prob, abstol, reltol, fx, x, termination_condition, Val(:simple)) |
| 78 | + |
| 79 | + fx_norm = L2_NORM(fx)^nexp |
| 80 | + α_1 = one(T) |
| 81 | + f_1 = fx_norm |
| 82 | + |
| 83 | + history_f_k = dfsane_history_vec(x, fx_norm, alg.M) |
| 84 | + |
| 85 | + # Generate the cache |
| 86 | + @bb x_cache = similar(x) |
| 87 | + @bb d = copy(x) |
| 88 | + @bb xo = copy(x) |
| 89 | + @bb δx = copy(x) |
| 90 | + @bb δf = copy(fx) |
| 91 | + |
| 92 | + k = 0 |
| 93 | + while k < maxiters |
| 94 | + # Spectral parameter range check |
| 95 | + σ_k = sign(σ_k) * clamp(abs(σ_k), σ_min, σ_max) |
| 96 | + |
| 97 | + # Line search direction |
| 98 | + @bb @. d = -σ_k * fx |
| 99 | + |
| 100 | + η = η_strategy(f_1, k + 1, x, fx) |
| 101 | + f_bar = maximum(history_f_k) |
| 102 | + α_p = α_1 |
| 103 | + α_m = α_1 |
| 104 | + |
| 105 | + @bb @. x_cache = x + α_p * d |
| 106 | + |
| 107 | + fx = Utils.eval_f(prob, fx, x_cache) |
| 108 | + fx_norm_new = L2_NORM(fx)^nexp |
| 109 | + |
| 110 | + while k < maxiters |
| 111 | + (fx_norm_new ≤ (f_bar + η - γ * α_p^2 * fx_norm)) && break |
| 112 | + |
| 113 | + α_tp = α_p^2 * fx_norm / (fx_norm_new + (T(2) * α_p - T(1)) * fx_norm) |
| 114 | + @bb @. x_cache = x - α_m * d |
| 115 | + |
| 116 | + fx = Utils.eval_f(prob, fx, x_cache) |
| 117 | + fx_norm_new = L2_NORM(fx)^nexp |
| 118 | + |
| 119 | + (fx_norm_new ≤ (f_bar + η - γ * α_m^2 * fx_norm)) && break |
| 120 | + |
| 121 | + α_tm = α_m^2 * fx_norm / (fx_norm_new + (T(2) * α_m - T(1)) * fx_norm) |
| 122 | + α_p = clamp(α_tp, τ_min * α_p, τ_max * α_p) |
| 123 | + α_m = clamp(α_tm, τ_min * α_m, τ_max * α_m) |
| 124 | + @bb @. x_cache = x + α_p * d |
| 125 | + |
| 126 | + fx = Utils.eval_f(prob, fx, x_cache) |
| 127 | + fx_norm_new = L2_NORM(fx)^nexp |
| 128 | + |
| 129 | + k += 1 |
| 130 | + end |
| 131 | + |
| 132 | + @bb copyto!(x, x_cache) |
| 133 | + |
| 134 | + solved, retcode, fx_sol, x_sol = Utils.check_termination(tc_cache, fx, x, xo, prob) |
| 135 | + solved && return SciMLBase.build_solution(prob, alg, x_sol, fx_sol; retcode) |
| 136 | + |
| 137 | + # Update spectral parameter |
| 138 | + @bb @. δx = x - xo |
| 139 | + @bb @. δf = fx - δf |
| 140 | + |
| 141 | + σ_k = dot(δx, δx) / dot(δx, δf) |
| 142 | + |
| 143 | + # Take step |
| 144 | + @bb copyto!(xo, x) |
| 145 | + @bb copyto!(δf, fx) |
| 146 | + fx_norm = fx_norm_new |
| 147 | + |
| 148 | + # Store function value |
| 149 | + idx = mod1(k, SciMLBase._unwrap_val(alg.M)) |
| 150 | + if history_f_k isa SVector |
| 151 | + history_f_k = Base.setindex(history_f_k, fx_norm_new, idx) |
| 152 | + elseif history_f_k isa NTuple |
| 153 | + @reset history_f_k[idx] = fx_norm_new |
| 154 | + else |
| 155 | + history_f_k[idx] = fx_norm_new |
| 156 | + end |
| 157 | + k += 1 |
| 158 | + end |
| 159 | + |
| 160 | + return SciMLBase.build_solution(prob, alg, x, fx; retcode = ReturnCode.MaxIters) |
| 161 | +end |
| 162 | + |
| 163 | +function dfsane_history_vec(x::StaticArray, fx_norm, ::Val{M}) where {M} |
| 164 | + return ones(SVector{M, eltype(x)}) .* fx_norm |
| 165 | +end |
| 166 | + |
| 167 | +@generated function dfsane_history_vec(x, fx_norm, ::Val{M}) where {M} |
| 168 | + M ≥ 11 && return :(fill(fx_norm, M)) # Julia can't specialize here |
| 169 | + return :(ntuple(Returns(fx_norm), $(M))) |
| 170 | +end |
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