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Working version of Multiple Shooting 🎉
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| Original file line number | Diff line number | Diff line change |
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| # TODO: incorporate `initial_guess` similar to MIRK methods | ||
| function SciMLBase.__solve(prob::BVProblem, alg::MultipleShooting; odesolve_kwargs = (;), | ||
| nlsolve_kwargs = (;), kwargs...) | ||
| @unpack f, bc, tspan = prob | ||
| bcresid_prototype = prob.f.bcresid_prototype === nothing ? similar(prob.u0) : | ||
| prob.f.bcresid_prototype | ||
| N, u0_size, nshoots, iip = length(prob.u0), size(prob.u0), alg.nshoots, isinplace(prob) | ||
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| @views function loss!(resid::ArrayPartition, us, p, cur_nshoots, nodes) | ||
| ts_ = Vector{Vector{typeof(first(tspan))}}(undef, cur_nshoots) | ||
| us_ = Vector{Vector{typeof(us)}}(undef, cur_nshoots) | ||
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| resid_bc, resid_nodes = resid.x[1], resid.x[2] | ||
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| for i in 1:cur_nshoots | ||
| local odeprob = ODEProblem{iip}(f, | ||
| reshape(us[((i - 1) * N + 1):(i * N)], u0_size), (nodes[i], nodes[i + 1]), | ||
| p) | ||
| sol = solve(odeprob, alg.ode_alg; odesolve_kwargs..., kwargs..., | ||
| save_end = true, save_everystep = false) | ||
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| ts_[i] = sol.t | ||
| us_[i] = sol.u | ||
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| resid_nodes[((i - 1) * N + 1):(i * N)] .= vec(us[(i * N + 1):((i + 1) * N)]) .- | ||
| vec(sol.u[end]) | ||
| end | ||
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| _ts = foldl(vcat, ts_) | ||
| _us = foldl(vcat, us_) | ||
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| # Boundary conditions | ||
| # Builds an ODESolution object to keep the framework for bc(,,) consistent | ||
| odeprob = ODEProblem{iip}(f, reshape(us[1:N], u0_size), tspan, p) | ||
| total_solution = SciMLBase.build_solution(odeprob, nothing, _ts, _us) | ||
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| if iip | ||
| eval_bc_residual!(resid_bc, prob.problem_type, bc, total_solution, p) | ||
| else | ||
| resid_bc .= eval_bc_residual(prob.problem_type, bc, total_solution, p) | ||
| end | ||
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| return resid | ||
| end | ||
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| # This gets all the nshoots except the final SingleShooting case | ||
| all_nshoots = get_all_nshoots(alg) | ||
| u_at_nodes, nodes = nothing, nothing | ||
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| for (i, cur_nshoot) in enumerate(all_nshoots) | ||
| if i == 1 | ||
| nodes, u_at_nodes = multiple_shooting_initialize(prob, alg; odesolve_kwargs, | ||
| kwargs...) | ||
| else | ||
| nodes, u_at_nodes = multiple_shooting_initialize(u_at_nodes, prob, alg, nodes, | ||
| cur_nshoot, all_nshoots[i - 1]; odesolve_kwargs, kwargs...) | ||
| end | ||
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| resid_prototype = ArrayPartition(bcresid_prototype, | ||
| similar(u_at_nodes, cur_nshoot * N)) | ||
| loss_function! = NonlinearFunction{true}((args...) -> loss!(args..., | ||
| cur_nshoot, nodes); resid_prototype) | ||
| nlprob = NonlinearProblem(loss_function!, u_at_nodes, prob.p) | ||
| sol_nlsolve = solve(nlprob, alg.nlsolve; nlsolve_kwargs..., kwargs...) | ||
| u_at_nodes = sol_nlsolve.u | ||
| end | ||
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| single_shooting_prob = remake(prob; u0 = reshape(u_at_nodes[1:N], u0_size)) | ||
| return SciMLBase.__solve(single_shooting_prob, Shooting(alg.ode_alg; alg.nlsolve); | ||
| odesolve_kwargs, nlsolve_kwargs, kwargs...) | ||
| end | ||
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| function multiple_shooting_initialize(prob, alg::MultipleShooting; odesolve_kwargs = (;), | ||
| kwargs...) | ||
| @unpack f, bc, u0, tspan, p = prob | ||
| @unpack ode_alg, nshoots = alg | ||
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| N = length(u0) | ||
| nodes = range(tspan[1], tspan[2]; length = nshoots + 1) | ||
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| # Ensures type stability in case the parameters are dual numbers | ||
| if !(typeof(p) <: SciMLBase.NullParameters) | ||
| if !isconcretetype(eltype(p)) | ||
| @warn "Type inference will fail if eltype(p) is not a concrete type" | ||
| end | ||
| u_at_nodes = similar(u0, promote_type(eltype(u0), eltype(p)), (nshoots + 1) * N) | ||
| else | ||
| u_at_nodes = similar(u0, (nshoots + 1) * N) | ||
| end | ||
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| # Assumes no initial guess for now | ||
| start_prob = ODEProblem{isinplace(prob)}(f, u0, tspan, p) | ||
| sol = solve(start_prob, ode_alg; odesolve_kwargs..., kwargs..., saveat = nodes) | ||
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| if SciMLBase.successful_retcode(sol) | ||
| u_at_nodes[1:N] .= sol.u[1] | ||
| for i in 2:(nshoots + 1) | ||
| u_at_nodes[(N + (i - 2) * N) .+ (1:N)] .= sol.u[i] | ||
| end | ||
| else | ||
| @warn "Initialization using odesolve failed. Initializing using 0s. It is \ | ||
| recommended to provide an `initial_guess` function in this case." | ||
| fill!(u_at_nodes, 0) | ||
| end | ||
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| return nodes, u_at_nodes | ||
| end | ||
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| @views @inline function multiple_shooting_initialize(u_at_nodes_prev, prob, alg, | ||
| prev_nodes, nshoots, old_nshoots; odesolve_kwargs = (;), kwargs...) | ||
| @unpack f, bc, u0, tspan, p = prob | ||
| nodes = range(tspan[1], tspan[2]; length = nshoots + 1) | ||
| N = length(u0) | ||
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| u_at_nodes = similar(u_at_nodes_prev, N + nshoots * N) | ||
| u_at_nodes[1:N] .= u_at_nodes_prev[1:N] | ||
| u_at_nodes[(end - N + 1):end] .= u_at_nodes_prev[(end - N + 1):end] | ||
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| skipsize = old_nshoots / nshoots | ||
| for i in 2:nshoots | ||
| pos = skipsize * (i - 1) + 1 | ||
| idxs = (N + (i - 2) * N) .+ (1:N) | ||
| if isinteger(pos) | ||
| # If the current node is also a node of the finer grid | ||
| ind = trunc(Int, pos) | ||
| idxs_prev = (N + (ind - 2) * N .+ (1:N)) | ||
| u_at_nodes[idxs] .= u_at_nodes_prev[idxs_prev] | ||
| else | ||
| # If the current node is not a node of the finer grid simulate from closest | ||
| # previous node and take result from simulation | ||
| fpos = floor(Int, pos) | ||
| r = pos - fpos | ||
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| t0 = prev_nodes[fpos] | ||
| tf = prev_nodes[fpos + 1] | ||
| tstop = t0 + r * (tf - t0) | ||
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| idxs_prev = (N + (fpos - 2) * N .+ (1:N)) | ||
| ustart = u_at_nodes_prev[idxs_prev] | ||
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| odeprob = ODEProblem(f, ustart, (t0, tstop), p) | ||
| odesol = solve(odeprob, alg.ode_alg; odesolve_kwargs..., kwargs..., | ||
| saveat = (), save_end = true) | ||
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| u_at_nodes[idxs] .= odesol.u[end] | ||
| end | ||
| end | ||
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| return nodes, u_at_nodes | ||
| end | ||
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| @inline function get_all_nshoots(alg::MultipleShooting) | ||
| @unpack nshoots, grid_coarsening = alg | ||
| if grid_coarsening isa Bool | ||
| !grid_coarsening && return [nshoots] | ||
| update_fn = Base.Fix2(÷, 2) | ||
| elseif grid_coarsening isa Function | ||
| update_fn = grid_coarsening | ||
| else | ||
| grid_coarsening[1] == nshoots && return grid_coarsening | ||
| return vcat(nshoots, grid_coarsening) | ||
| end | ||
| nshoots_vec = Int[nshoots] | ||
| next = update_fn(nshoots) | ||
| while next > 1 | ||
| push!(nshoots_vec, next) | ||
| next = update_fn(last(nshoots_vec)) | ||
| end | ||
| @assert !(1 in nshoots_vec) | ||
| return nshoots_vec | ||
| end |
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