/
lenseflow.jl
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/
lenseflow.jl
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abstract type LenseFlowOp{I<:ODESolver,t₀,t₁,Φ} <: FlowOpWithAdjoint{I,t₀,t₁} end
# `L = LenseFlow(ϕ)` just creates a wrapper holding ϕ. Then when you do `L*f` or
# `cache(L,f)` we create a CachedLenseFlow object which holds all the
# precomputed quantities and preallocated memory needed to do the lense.
struct LenseFlow{I<:ODESolver,t₀,t₁,Φ<:Field{<:Any,<:S0}} <: LenseFlowOp{I,t₀,t₁,Φ}
ϕ::Φ
end
struct CachedLenseFlow{N,t₀,t₁,Φ<:Field,ŁΦ<:Field,ÐΦ<:Field,ŁF<:Field,ÐF<:Field,T} <: LenseFlowOp{RK4Solver{N},t₀,t₁,Φ}
# save ϕ to know when to trigger recaching
ϕ :: Ref{Φ}
# p and M⁻¹ quantities precomputed at every time step
p :: Dict{Float16,FieldOrOpVector{Diagonal{T,ŁΦ}}}
M⁻¹ :: Dict{Float16,FieldOrOpMatrix{Diagonal{T,ŁΦ}}}
# f type memory
memŁf :: ŁF
memÐf :: ÐF
memŁvf :: FieldVector{ŁF}
memÐvf :: FieldVector{ÐF}
# ϕ type memory
memŁϕ :: ŁΦ
memÐϕ :: ÐΦ
memŁvϕ :: FieldVector{ŁΦ}
memÐvϕ :: FieldVector{ÐΦ}
end
### constructors
LenseFlow(ϕ,n=7) = LenseFlow{RK4Solver{n}}(ϕ)
LenseFlow{I}(ϕ) where {I<:ODESolver} = LenseFlow{I,0,1}(ϕ)
LenseFlow{I,t₀,t₁}(ϕ) where {I,t₀,t₁} = LenseFlow{I,float(t₀),float(t₁),typeof(ϕ)}(ϕ)
### printing
show(io::IO, ::L) where {I,t₀,t₁,Φ,L<:LenseFlow{I,t₀,t₁,Φ}} = print(io, "$(L.name.name){$t₀→$t₁, $I}(ϕ::$Φ)")
show(io::IO, ::L) where {N,t₀,t₁,Φ,ŁF,L<:CachedLenseFlow{N,t₀,t₁,Φ,<:Any,<:Any,ŁF}} = print(io, "$(L.name.name){$t₀→$t₁, $(RK4Solver{N})}(ϕ::$Φ, Łf::$ŁF)")
string(::Type{RK4Solver{N}}) where {N} = "$N-step RK4"
# convenience for getting the actual ϕ map
getϕ(L::LenseFlow) = L.ϕ
getϕ(L::CachedLenseFlow) = L.ϕ[]
### caching
τ(t) = Float16(t)
cache(L::LenseFlow, f) = cache!(alloc_cache(L,f),L,f)
cache(cL::CachedLenseFlow, f) = cL
cache!(cL::CachedLenseFlow{N,t₀,t₁}, ϕ) where {N,t₀,t₁} = (cL.ϕ[]===ϕ) ? cL : cache!(cL,LenseFlow{RK4Solver{N},t₀,t₁}(ϕ),cL.memŁf)
(cL::CachedLenseFlow)(ϕ) = cache!(cL,ϕ)
function cache!(cL::CachedLenseFlow{N,t₀,t₁}, L::LenseFlow{RK4Solver{N},t₀,t₁}, f) where {N,t₀,t₁}
ts = range(t₀,t₁,length=2N+1)
∇ϕ,Hϕ = map(Ł, gradhess(L.ϕ))
T = eltype(L.ϕ)
for (t,τ) in zip(ts,τ.(ts))
@! cL.M⁻¹[τ] = pinv(Diagonal.(I + T(t)*Hϕ))
@! cL.p[τ] = cL.M⁻¹[τ]' * Diagonal.(∇ϕ)
end
cL.ϕ[] = L.ϕ
cL
end
function alloc_cache(L::LenseFlow{RK4Solver{N},t₀,t₁}, f) where {N,t₀,t₁}
ts = range(t₀,t₁,length=2N+1)
p, M⁻¹ = Dict(), Dict()
Łf,Ðf = Ł(f), Ð(f)
Łϕ,Ðϕ = Ł(L.ϕ),Ð(L.ϕ)
for (t,τ) in zip(ts,τ.(ts))
M⁻¹[τ] = Diagonal.(similar.(@SMatrix[Łϕ Łϕ; Łϕ Łϕ]))
p[τ] = Diagonal.(similar.(@SVector[Łϕ,Łϕ]))
end
CachedLenseFlow{N,t₀,t₁,typeof(L.ϕ),typeof(Łϕ),typeof(Ðϕ),typeof(Łf),typeof(Ðf),eltype(Łϕ)}(
Ref(L.ϕ), p, M⁻¹,
similar(Łf), similar(Ðf), similar.(@SVector[Łf,Łf]), similar.(@SVector[Ðf,Ðf]),
similar(Łϕ), similar(Ðϕ), similar.(@SVector[Łϕ,Łϕ]), similar.(@SVector[Ðϕ,Ðϕ]),
)
end
# the way these velocities work is that they unpack the preallocated fields
# stored in L.mem* into variables with more meaningful names, which are then
# used in a bunch of in-place (eg mul!, Ł!, etc...) functions. note the use of
# the @! macro, which just rewrites @! x = f(y) to x = f!(x,y) for easier
# reading.
function velocity(L::LenseFlowOp{<:RK4Solver}, f₀::Field)
function v!(v::Field, t::Real, f::Field)
Ðf, Ð∇f, Ł∇f = L.memÐf, L.memÐvf, L.memŁvf
p = L.p[τ(t)]
@! Ðf = Ð(f)
@! Ð∇f = ∇ᵢ * Ðf
@! Ł∇f = Ł(Ð∇f)
@! v = p' * Ł∇f
end
return (v!, Ł(f₀))
end
function velocityᴴ(L::LenseFlowOp{<:RK4Solver}, f₀::Field)
function v!(v::Field, t::Real, f::Field)
Łf, Łf_p, Ð_Łf_p = L.memŁf, L.memŁvf, L.memÐvf
p = L.p[τ(t)]
@! Łf = Ł(f)
@! Łf_p = p * Łf
@! Ð_Łf_p = Ð(Łf_p)
@! v = -∇ᵢ' * Ð_Łf_p
end
return (v!, Ð(f₀))
end
function negδvelocityᴴ(L::LenseFlowOp{<:RK4Solver}, (f₀, δf₀)::FieldTuple)
function v!((df_dt, dδf_dt, dδϕ_dt)::FieldTuple, t::Real, (f, δf, δϕ)::FieldTuple)
p = L.p[τ(t)]
M⁻¹ = L.M⁻¹[τ(t)]
# dδf/dt
Łδf, Łδf_p, Ð_Łδf_p = L.memŁf, L.memŁvf, L.memÐvf
@! Łδf = Ł(δf)
@! Łδf_p = p * Łδf
@! Ð_Łδf_p = Ð(Łδf_p)
@! dδf_dt = -∇ᵢ' * Ð_Łδf_p
# df/dt
Ðf, Ð∇f, Ł∇f = L.memÐf, L.memÐvf, L.memŁvf
@! Ðf = Ð(f)
@! Ð∇f = ∇ᵢ * Ðf
@! Ł∇f = Ł(Ð∇f)
@! df_dt = p' * Ł∇f
# dδϕ/dt
δfᵀ_∇f, M⁻¹_δfᵀ_∇f, Ð_M⁻¹_δfᵀ_∇f = L.memŁvϕ, L.memŁvϕ, L.memÐvϕ
@! δfᵀ_∇f = tuple_adjoint(Łδf) * Ł∇f
@! M⁻¹_δfᵀ_∇f = M⁻¹ * δfᵀ_∇f
@! Ð_M⁻¹_δfᵀ_∇f = Ð(M⁻¹_δfᵀ_∇f)
@! dδϕ_dt = -∇ⁱ' * Ð_M⁻¹_δfᵀ_∇f
memÐϕ = L.memÐϕ
for i=1:2, j=1:2
dδϕ_dt .+= (@! memÐϕ = ∇ⁱ[i]' * (@! memÐϕ = ∇ᵢ[j]' * (@! memÐϕ = Ð(@. L.memŁϕ = t * p[j].diag * M⁻¹_δfᵀ_∇f[i]))))
end
FieldTuple(df_dt, dδf_dt, dδϕ_dt)
end
return (v!, FieldTuple(Ł(f₀), Ð(δf₀), Ð(zero(getϕ(L)))))
end
# adapting storage
adapt_structure(storage, Lϕ::LenseFlow{I,t₀,t₁}) where {I<:ODESolver,t₀,t₁} = LenseFlow{I,t₀,t₁}(adapt(storage,Lϕ.ϕ))
adapt_structure(storage, Lϕ::CachedLenseFlow{N,t₀,t₁}) where {N,t₀,t₁} = cache(LenseFlow{RK4Solver{N},t₀,t₁}(adapt(storage,Lϕ.ϕ[])), adapt(storage,Lϕ.memŁf))