Accelerators

src/Accelerators.jl

@@ -80,23 +80,25 @@ end
 """
 Performs state update with modified Nesterov momentum approach.
 The dependence of the momentum parameter for variable timestep can be found e.g. here "https://www.damtp.cam.ac.uk/user/hf323/M19-OPT/lecture5.pdf"
+This update takes the perspective that the EKP time step Δt = C/h where C constant, and h is the timestepping of a typical gradient method 
 """
 function accelerate!(
     ekp::EnsembleKalmanProcess{FT, IT, P, LRS, NesterovAccelerator{FT}},
     u::MA,
 ) where {FT <: AbstractFloat, IT <: Int, P <: Process, LRS <: LearningRateScheduler, MA <: AbstractMatrix}
     ## update "v" state:
-    #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_prev = length(ekp.Δt) == 1 ? ekp.Δt[end] : 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
-
+    # condition θ_prev^2 * (1 - θ) * h \leq h_prev * θ^2
+    b = θ_prev^2 * Δt / Δt_prev
+
+    θ_lowbd = (-b + sqrt(b^2 + 4*b)) / 2
+    θ = min((θ_lowbd + 1)/2, θ_lowbd + 1.0/length(ekp.Δt)^2) # can be unstable close to the boundary, so add a quadratic penalization
+
     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
@@ -109,6 +111,7 @@ end
 """
 State update method for UKI with Nesterov Accelerator.
 The dependence of the momentum parameter for variable timestep can be found e.g. here "https://www.damtp.cam.ac.uk/user/hf323/M19-OPT/lecture5.pdf"
+This update takes the perspective that the EKP time step Δt = C/h where C constant, and h is the timestepping of a typical gradient method 
 Performs identical update as with other methods, but requires reconstruction of mean and covariance of the accelerated positions prior to saving.
 """
 function accelerate!(
@@ -118,14 +121,15 @@ function accelerate!(
 
     #identical update stage as before
     ## update "v" state:
-    Δt_prev = length(uki.Δt) == 1 ? 1 : uki.Δt[end - 1]
+    Δt_prev = length(uki.Δt) == 1 ? uki.Δt[end] : 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
+    # condition θ_prev^2 * (1 - θ) * h \leq h_prev * θ^2
+    b = θ_prev^2 * Δt / Δt_prev
+    θ_lowbd = (-b + sqrt(b^2 + 4*b)) / 2
+    θ = min((θ_lowbd + 1)/2, θ_lowbd + 1.0/length(uki.Δt)^2)# can be unstable at the boundary, so add a quadratic penalization
 
     v = u .+ θ * (1 / θ_prev - 1) * (u .- uki.accelerator.u_prev)