modified accelerator for variable timestep

simplify expression

format

rm typo in UKI that causes warning

needs a sqrt

remove r

src/Accelerators.jl

@@ -19,12 +19,12 @@ Stores a previous state value u_prev for computational purposes (note this is di
 $(TYPEDFIELDS)
 """
 mutable struct NesterovAccelerator{FT <: AbstractFloat} <: Accelerator
-    r::FT
-    u_prev::Any
+    u_prev::Array{FT}
+    θ_prev::FT
 end
 
-function NesterovAccelerator(r = 3.0, initial = Float64[])
-    return NesterovAccelerator(r, initial)
+function NesterovAccelerator(initial = Float64[])
+    return NesterovAccelerator(initial, 1.0)
 end
 
 
@@ -54,11 +54,20 @@ function accelerate!(
     u::MA,
 ) where {FT <: AbstractFloat, IT <: Int, P <: Process, LRS <: LearningRateScheduler, MA <: AbstractMatrix}
     ## update "v" state:
-    k = get_N_iterations(ekp) + 2
-    v = u .+ (1 - ekp.accelerator.r / k) * (u .- ekp.accelerator.u_prev)
+    #v = u .+ 2 / (get_N_iterations(ekp) + 2) * (u .- ekp.accelerator.u_prev)
+    Δt_prev = length(ekp.Δt) == 1 ? 1 : ekp.Δt[end - 1]
+    Δt = ekp.Δt[end]
+    θ_prev = ekp.accelerator.θ_prev
+
+    # condition θ_prev^2 * (1 - θ) * Δt \leq Δt_prev * θ^2
+    a = sqrt(θ_prev^2 * Δt / Δt_prev)
+    θ = (-a + sqrt(a^2 + 4)) / 2
+
+    v = u .+ θ * (1 / θ_prev - 1) * (u .- ekp.accelerator.u_prev)
 
     ## update "u" state: 
     ekp.accelerator.u_prev = u
+    ekp.accelerator.θ_prev = θ
 
     ## push "v" state to EKP object
     push!(ekp.u, DataContainer(v, data_are_columns = true))
@@ -76,11 +85,20 @@ function accelerate!(
 
     #identical update stage as before
     ## update "v" state:
-    k = get_N_iterations(uki) + 2
-    v = u .+ (1 - uki.accelerator.r / k) * (u .- uki.accelerator.u_prev)
+    Δt_prev = length(uki.Δt) == 1 ? 1 : uki.Δt[end - 1]
+    Δt = uki.Δt[end]
+    θ_prev = uki.accelerator.θ_prev
+
+
+    # condition θ_prev^2 * (1 - θ) * Δt \leq Δt_prev * θ^2
+    a = sqrt(θ_prev^2 * Δt / Δt_prev)
+    θ = (-a + sqrt(a^2 + 4)) / 2
+
+    v = u .+ θ * (1 / θ_prev - 1) * (u .- uki.accelerator.u_prev)
 
     ## update "u" state: 
     uki.accelerator.u_prev = u
+    uki.accelerator.θ_prev = θ
 
     ## push "v" state to UKI object
     push!(uki.u, DataContainer(v, data_are_columns = true))

src/UnscentedKalmanInversion.jl

@@ -551,10 +551,7 @@ end
 
 UKI prediction step : generate sigma points.
 """
-function update_ensemble_prediction!(
-    process::Unscented,
-    Δt::FT,
-) where {FT <: AbstractFloat, AV <: AbstractVector, AM <: AbstractMatrix}
+function update_ensemble_prediction!(process::Unscented, Δt::FT) where {FT <: AbstractFloat}
 
     process.iter += 1
     # update evolution covariance matrix

test/EnsembleKalmanProcess/runtests.jl

@@ -82,6 +82,8 @@ end
 @testset "Accelerators" begin
     # Get an inverse problem
     y_obs, G, Γy, _ = inv_problems[end - 2] # additive noise inv problem (deterministic map)
+    inv_sqrt_Γy = sqrt(inv(Γy))
+
     rng = Random.MersenneTwister(rng_seed)
     N_ens_tmp = 5
     initial_ensemble = EKP.construct_initial_ensemble(rng, prior, N_ens_tmp)
@@ -105,20 +107,32 @@ end
     @test typeof(eksobj_noacc_specified.accelerator) <: DefaultAccelerator
 
     ## test NesterovAccelerators satisfy desired ICs
-    @test ekiobj.accelerator.r ≈ 3.0
     @test ekiobj.accelerator.u_prev == initial_ensemble
-    @test eksobj.accelerator.r ≈ 3.0
+    @test ekiobj.accelerator.θ_prev == 1.0
     @test eksobj.accelerator.u_prev == initial_ensemble
+    @test eksobj.accelerator.θ_prev == 1.0
 
     ## test method convergence
     # Note: this test only requires that the final ensemble is an improvement on the initial ensemble,
     # NOT that the accelerated processes are more effective than the default, as this is not guaranteed.
     # Specific cost values are printed to give an idea of acceleration.
-    processes = [Inversion(), TransformInversion(inv(Γy)), Unscented(prior; impose_prior = true), Sampler(prior)]
-    schedulers = [repeat([DefaultScheduler(0.1)], 3)..., EKSStableScheduler()]
+    processes = [
+        Inversion(),
+        TransformInversion(inv(Γy)),
+        Unscented(prior; impose_prior = true),
+        Inversion(),
+        TransformInversion(inv(Γy)),
+        Unscented(prior; impose_prior = true),
+        Sampler(prior),
+    ]
+    schedulers = [
+        repeat([DefaultScheduler(0.1)], 3)...,
+        repeat([DataMisfitController(terminate_at = 100)], 3)...,
+        EKSStableScheduler(),
+    ]
     for (process, scheduler) in zip(processes, schedulers)
         accelerators = [DefaultAccelerator(), NesterovAccelerator()]
-        N_iters = [5, 5, 5, 5]
+        N_iters = [20, 20]
         init_means = []
         final_means = []
 
@@ -160,9 +174,8 @@ end
             push!(init_means, vec(mean(get_u_prior(ekpobj), dims = 2)))
             push!(final_means, vec(mean(get_u_final(ekpobj), dims = 2)))
 
-            inv_sqrt_Γy = sqrt(inv(Γy))
             cost_initial =
-                norm(inv_sqrt_Γy * (y_obs .- G(transform_unconstrained_to_constrained(prior, init_means[end]))))
+                norm(inv_sqrt_Γy * (y_obs .- G(transform_unconstrained_to_constrained(prior, initial_ensemble))))
             cost_final =
                 norm(inv_sqrt_Γy * (y_obs .- G(transform_unconstrained_to_constrained(prior, final_means[end]))))
             @info "Convergence:" cost_initial cost_final