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FixedPointSolvers: Recommendations and Wrappers
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| # FixedPointAcceleration.jl | ||
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| This is a extension for importing solvers from FixedPointAcceleration.jl into the SciML | ||
| interface. Note that these solvers do not come by default, and thus one needs to install | ||
| the package before using these solvers: | ||
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| ```julia | ||
| using Pkg | ||
| Pkg.add("FixedPointAcceleration") | ||
| using FixedPointAcceleration, NonlinearSolve | ||
| ``` | ||
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| ## Solver API | ||
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| ```@docs | ||
| FixedPointAccelerationJL | ||
| ``` |
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| # SpeedMapping.jl | ||
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| This is a extension for importing solvers from SpeedMapping.jl into the SciML | ||
| interface. Note that these solvers do not come by default, and thus one needs to install | ||
| the package before using these solvers: | ||
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| ```julia | ||
| using Pkg | ||
| Pkg.add("SpeedMapping") | ||
| using SpeedMapping, NonlinearSolve | ||
| ``` | ||
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| ## Solver API | ||
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| ```@docs | ||
| SpeedMappingJL | ||
| ``` |
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| # Fixed Point Solvers | ||
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| Currently we don't have an API to directly specify Fixed Point Solvers. However, a Fixed | ||
| Point Problem can be trivially converted to a Root Finding Problem. Say we want to solve: | ||
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| ```math | ||
| f(u) = u | ||
| ``` | ||
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| This can be written as: | ||
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| ```math | ||
| g(u) = f(u) - u = 0 | ||
| ``` | ||
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| ``g(u) = 0`` is a root finding problem. Note that we can use any root finding | ||
| algorithm to solve this problem. However, this is often not the most efficient way to | ||
| solve a fixed point problem. We provide a few algorithms available via extensions that | ||
| are more efficient for fixed point problems. | ||
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| Note that even if you use one of the Fixed Point Solvers mentioned here, you must still | ||
| use the `NonlinearProblem` API to specify the problem, i.e., ``g(u) = 0``. | ||
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| ## Recommended Methods | ||
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| Using [native NonlinearSolve.jl methods](@ref nonlinearsystemsolvers) is the recommended | ||
| approach. For systems where constructing Jacobian Matrices are expensive, we recommend | ||
| using a Krylov Method with one of those solvers. | ||
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| ## Full List of Methods | ||
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| We are only listing the methods that natively solve fixed point problems. | ||
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| ### SpeedMapping.jl | ||
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| - `SpeedMappingJL()`: accelerates the convergence of a mapping to a fixed point by the | ||
| Alternating cyclic extrapolation algorithm (ACX). | ||
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| ### FixedPointAcceleration.jl | ||
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| - `FixedPointAccelerationJL()`: accelerates the convergence of a mapping to a fixed point | ||
| by the Anderson acceleration algorithm and a few other methods. |
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| module NonlinearSolveFixedPointAccelerationExt | ||
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| using NonlinearSolve, FixedPointAcceleration, DiffEqBase, SciMLBase | ||
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| function SciMLBase.__solve(prob::NonlinearProblem, alg::FixedPointAccelerationJL, args...; | ||
| abstol = nothing, maxiters = 1000, alias_u0::Bool = false, | ||
| show_trace::Val{PrintReports} = Val(false), termination_condition = nothing, | ||
| kwargs...) where {PrintReports} | ||
| @assert (termination_condition === | ||
| nothing)||(termination_condition isa AbsNormTerminationMode) "SpeedMappingJL does not support termination conditions!" | ||
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| u0 = NonlinearSolve.__maybe_unaliased(prob.u0, alias_u0) | ||
| u_size = size(u0) | ||
| T = eltype(u0) | ||
| iip = isinplace(prob) | ||
| p = prob.p | ||
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| if !iip && prob.u0 isa Number | ||
| # FixedPointAcceleration makes the scalar problem into a vector problem | ||
| f = (u) -> [prob.f(u[1], p) .+ u[1]] | ||
| elseif !iip && prob.u0 isa AbstractVector | ||
| f = (u) -> (prob.f(u, p) .+ u) | ||
| elseif !iip && prob.u0 isa AbstractArray | ||
| f = (u) -> vec(prob.f(reshape(u, u_size), p) .+ u) | ||
| elseif iip && prob.u0 isa AbstractVector | ||
| du = similar(u0) | ||
| f = (u) -> (prob.f(du, u, p); du .+ u) | ||
| else | ||
| du = similar(u0) | ||
| f = (u) -> (prob.f(du, reshape(u, u_size), p); vec(du) .+ u) | ||
| end | ||
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| tol = abstol === nothing ? real(oneunit(T)) * (eps(real(one(T))))^(4 // 5) : abstol | ||
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| sol = fixed_point(f, NonlinearSolve._vec(u0); Algorithm = alg.algorithm, | ||
| ConvergenceMetricThreshold = tol, MaxIter = maxiters, MaxM = alg.m, | ||
| ExtrapolationPeriod = alg.extrapolation_period, Dampening = alg.dampening, | ||
| PrintReports, ReplaceInvalids = alg.replace_invalids, | ||
| ConditionNumberThreshold = alg.condition_number_threshold, quiet_errors = true) | ||
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| res = prob.u0 isa Number ? first(sol.FixedPoint_) : sol.FixedPoint_ | ||
| if res === missing | ||
| resid = NonlinearSolve.evaluate_f(prob, u0) | ||
| res = u0 | ||
| converged = false | ||
| else | ||
| resid = NonlinearSolve.evaluate_f(prob, res) | ||
| converged = maximum(abs, resid) ≤ tol | ||
| end | ||
| return SciMLBase.build_solution(prob, alg, res, resid; | ||
| retcode = converged ? ReturnCode.Success : ReturnCode.Failure, | ||
| stats = SciMLBase.NLStats(sol.Iterations_, 0, 0, 0, sol.Iterations_), | ||
| original = sol) | ||
| end | ||
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| end |
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| module NonlinearSolveSpeedMappingExt | ||
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| using NonlinearSolve, SpeedMapping, DiffEqBase, SciMLBase | ||
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| function SciMLBase.__solve(prob::NonlinearProblem, alg::SpeedMappingJL, args...; | ||
| abstol = nothing, maxiters = 1000, alias_u0::Bool = false, | ||
| store_trace::Val{store_info} = Val(false), termination_condition = nothing, | ||
| kwargs...) where {store_info} | ||
| @assert (termination_condition === | ||
| nothing)||(termination_condition isa AbsNormTerminationMode) "SpeedMappingJL does not support termination conditions!" | ||
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| if typeof(prob.u0) <: Number | ||
| u0 = [prob.u0] | ||
| else | ||
| u0 = NonlinearSolve.__maybe_unaliased(prob.u0, alias_u0) | ||
| end | ||
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| T = eltype(u0) | ||
| iip = isinplace(prob) | ||
| p = prob.p | ||
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| if !iip && prob.u0 isa Number | ||
| m! = (du, u) -> (du .= prob.f(first(u), p) .+ first(u)) | ||
| elseif !iip | ||
| m! = (du, u) -> (du .= prob.f(u, p) .+ u) | ||
| else | ||
| m! = (du, u) -> (prob.f(du, u, p); du .+= u) | ||
| end | ||
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| tol = abstol === nothing ? real(oneunit(T)) * (eps(real(one(T))))^(4 // 5) : abstol | ||
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| sol = speedmapping(u0; m!, tol, Lp = Inf, maps_limit = maxiters, alg.orders, | ||
| alg.check_obj, store_info, alg.σ_min, alg.stabilize) | ||
| res = prob.u0 isa Number ? first(sol.minimizer) : sol.minimizer | ||
| resid = NonlinearSolve.evaluate_f(prob, sol.minimizer) | ||
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| return SciMLBase.build_solution(prob, alg, res, resid; | ||
| retcode = sol.converged ? ReturnCode.Success : ReturnCode.Failure, | ||
| stats = SciMLBase.NLStats(sol.maps, 0, 0, 0, sol.maps), original = sol) | ||
| end | ||
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| end |
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