Merge pull request #24 from c-merchant/master

3D-VAR Functional Experiment Sketch and Updated Docs

Project.toml

@@ -9,6 +9,7 @@ Distributed = "8ba89e20-285c-5b6f-9357-94700520ee1b"
 Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 DocumenterTools = "35a29f4d-8980-5a13-9543-d66fff28ecb8"
+FileIO = "5789e2e9-d7fb-5bc7-8068-2c6fae9b9549"
 ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
 HDF5 = "f67ccb44-e63f-5c2f-98bd-6dc0ccc4ba2f"
 JLD = "4138dd39-2aa7-5051-a626-17a0bb65d9c8"
@@ -17,9 +18,12 @@ LineSearches = "d3d80556-e9d4-5f37-9878-2ab0fcc64255"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 LsqFit = "2fda8390-95c7-5789-9bda-21331edee243"
 Optim = "429524aa-4258-5aef-a3af-852621145aeb"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
+Revise = "295af30f-e4ad-537b-8983-00126c2a3abe"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
+StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
 StatsFuns = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
 
 [compat]

docs/src/submodules/experiments/D3VARExps.md

@@ -0,0 +1,22 @@
+# D3VARExps
+
+This module defines methods for experiments with classical variational data assimilation with
+3D-VAR.  Primal cost functions are defined, with their implicit differentiation
+performed with automatic differentiation with [JuliaDiff](https://github.com/JuliaDiff)
+methods. Development of gradient-based optimization schemes using automatic
+differentiation is ongoing, with future development planned to integrate variational
+benchmark experiments.
+
+The basic 3D-VAR cost function API is defined as follows
+```{julia}
+    D3_var_cost(x::VecA(T), obs::VecA(T), x_background::VecA(T), state_cov::CovM(T),
+    obs_cov::CovM(T), kwargs::StepKwargs) where T <: Real
+```
+where the control variable `x` is optimized, with fixed hyper-parameters defined in a
+wrapping function passed to auto-differentiation.
+
+## Methods
+
+```@autodocs
+Modules = [DataAssimilationBenchmarks.D3VARExps]
+```

docs/src/submodules/experiments/VarAnalysisExperimentDriver.md

@@ -0,0 +1,8 @@
+# VarAnalysisExperimentDriver
+
+
+## Methods
+
+```@autodocs
+Modules = [DataAssimilationBenchmarks.VarAnalysisExperimentDriver]
+```

src/experiments/D3VARExps.jl

@@ -0,0 +1,315 @@
+##############################################################################################
+module D3VARExps
+##############################################################################################
+# imports and exports
+using Random, Distributions, LinearAlgebra, StatsBase, Statistics, Measures
+using JLD2, HDF5, Plots
+using ..DataAssimilationBenchmarks, ..ObsOperators, ..DeSolvers, ..XdVAR
+##############################################################################################
+# Main 3DVAR experiments
+##############################################################################################
+"""
+    D3_var_filter_analysis_simple()
+"""
+function D3_var_filter_analysis_simple()
+    # time the experiment
+    t1 = time()
+
+    # Define experiment parameters
+    # number of cycles in experiment
+    nanl = 40
+    diffusion = 0.0
+    tanl = 0.05
+    γ = [8.0]
+
+    # define the observation operator HARD-CODED in this line
+    H_obs = alternating_obs_operator
+
+    # set the integration step size for the ensemble at 0.01 - we are assuming SDE
+    h = 0.01
+
+    # define derivative parameter
+    dx_params = Dict{String, Vector{Float64}}("F" => [8.0])
+
+    # define the dynamical model derivative for this experiment - we are assuming 
+    # Lorenz-96 model
+    dx_dt = L96.dx_dt
+
+    # define integration method
+    step_model! = rk4_step!
+
+    # number of discrete forecast steps
+    f_steps = convert(Int64, tanl / h)
+
+    # set seed
+    seed = 234
+    Random.seed!(seed)
+
+    # define the initialization
+    # observation noise
+    v = rand(Normal(0, 1), 40)
+    # define the initial observation range and truth reference solution
+    x_b = zeros(40)
+    x_t = x_b + v
+
+    # define kwargs for the analysis method
+    # and the underlying dynamical model
+    kwargs = Dict{String,Any}(
+                              "dx_dt" => dx_dt,
+                              "f_steps" => f_steps,
+                              "step_model" => step_model!,
+                              "dx_params" => dx_params,
+                              "h" => h,
+                              "diffusion" => diffusion,
+                              "gamma" => γ,
+                             )
+
+    # create storage for the forecast and analysis statistics
+    fore_rmse = Vector{Float64}(undef, nanl)
+    filt_rmse = Vector{Float64}(undef, nanl)
+
+    for i in 1:nanl
+        #print("Iteration: ")
+        #display(i)
+        for j in 1:f_steps
+            # M(x^b)
+            step_model!(x_b, 0.0, kwargs)
+            # M(x^t)
+            step_model!(x_t, 0.0, kwargs)
+        end
+
+        # multivariate - rand(MvNormal(zeros(40), I))
+        w = rand(Normal(0, 1), 40)
+        obs = x_t + w
+
+        state_cov = I
+        obs_cov = I
+
+        # optimized cost function input and value
+        x_opt = XdVAR.D3_var_NewtonOp(x_b, obs, x_b, state_cov, H_obs, obs_cov, kwargs)
+
+        # compare model forecast and truth twin via RMSE
+        rmse_forecast = sqrt(msd(x_b, x_t))
+        fore_rmse[i] = rmse_forecast
+
+        # compare optimal forecast and truth twin via RMSE
+        rmse_filter = sqrt(msd(x_opt, x_t))
+        filt_rmse[i] = rmse_filter
+
+        # reinitializing x_b and x_t for next cycle
+        x_b = x_opt
+    end
+
+    data = Dict{String,Any}(
+                            "seed" => seed,
+                            "diffusion" => diffusion,
+                            "dx_params" => dx_params,
+                            "gamma" => γ,
+                            "tanl" => tanl,
+                            "nanl" => nanl,
+                            "h" =>  h,
+                            "fore_rmse" => fore_rmse,
+                            "filt_rmse" => filt_rmse
+                           )
+
+    path = pkgdir(DataAssimilationBenchmarks) * "/src/data/time_series/"
+    name = "L96_3DVAR_time_series_seed_" * lpad(seed, 4, "0") *
+           "_diff_" * rpad(diffusion, 5, "0") *
+           #"_F_" * lpad(, 4, "0") *
+           "_tanl_" * rpad(tanl, 4, "0") *
+           "_nanl_" * lpad(nanl, 5, "0") *
+           "_h_" * rpad(h, 5, "0") *
+           ".jld2"
+
+    save(path * name, data)
+
+    # output time
+    print("Runtime " * string(round((time() - t1)  / 60.0, digits=4))  * " minutes\n")
+
+    # make plot
+    path = pkgdir(DataAssimilationBenchmarks) * "/src/analysis/var_exp/"
+    t = 1:nanl
+    plot(t, fore_rmse, marker=(:circle,5), label = "Forecast", 
+        title="Update: Root-Mean-Square Error vs. Time", 
+        legend_position = :outertopright, 
+        margin=15mm, size=(800,500), dpi = 600)
+    plot!(t, filt_rmse, marker=(:circle,5), label = "Filter")
+    xlabel!("Time [Cycles]")
+    ylabel!("Root-Mean-Square Error [RMSE]")
+    savefig(path * "I_Update_SIMPLE")
+end
+
+
+##############################################################################################
+"""
+function D3_var_filter_analysis((time_series, γ, is_informed, tuning_factor, is_updated)::NamedTuple{
+    (:time_series,:γ,:is_informed,:tuning_factor,:is_updated),<:Tuple{String,
+        Float64,Bool,Float64,Bool}})
+    Plotting capabilities are commented out for parallel experiment.
+"""
+function D3_var_filter_analysis((time_series, γ, is_informed, tuning_factor, is_updated)::NamedTuple{
+                        (:time_series,:γ,:is_informed,:tuning_factor,:is_updated),<:Tuple{String,
+                            Float64,Bool,Float64,Bool}})
+
+    # time the experiment
+    t1 = time()
+
+    # load the path, timeseries, and associated parameters
+    path = pkgdir(DataAssimilationBenchmarks) * "/src/data/"
+    ts = load(path * time_series)::Dict{String,Any}
+    diffusion = ts["diffusion"]::Float64
+    dx_params = ts["dx_params"]::ParamDict(Float64)
+    tanl = ts["tanl"]::Float64
+    nanl = ts["nanl"]::Int64
+    # set the integration step size for the ensemble 
+    h = ts["h"]::Float64
+
+    # define the observation operator HARD-CODED in this line
+    H_obs = alternating_obs_operator
+
+    # define the dynamical model derivative for this experiment - we are assuming 
+    # Lorenz-96 model
+    dx_dt = L96.dx_dt
+
+    # define integration method
+    step_model! = rk4_step!
+
+    # number of discrete forecast steps
+    f_steps = convert(Int64, tanl / h)
+
+    # set seed
+    seed = ts["seed"]::Int64
+    Random.seed!(seed)
+
+    # define the initialization
+    o = ts["obs"]::Array{Float64, 2}
+    obs_un = 1
+    obs_cov = obs_un^2.0 * I
+
+    # define state covaraince based on input
+    if is_informed == true
+        c = cov(o, dims = 2)
+        state_cov = tuning_factor*Symmetric(c)
+    else
+        state_cov = tuning_factor*I
+    end
+
+    x_t = o[:,1]
+    # observation noise
+    v = rand(MvNormal(zeros(40), I))
+    # define the initial background state
+    x_b = x_t + v;
+
+    # define kwargs for the analysis method
+    # and the underlying dynamical model
+    kwargs = Dict{String,Any}(
+                              "dx_dt" => dx_dt,
+                              "f_steps" => f_steps,
+                              "step_model" => step_model!,
+                              "dx_params" => dx_params,
+                              "h" => h,
+                              "diffusion" => diffusion,
+                              "γ" => γ,
+                              "gamma" => γ,
+                              "obs_un" => obs_un,
+                              "obs_cov" => obs_cov,
+                              "state_cov" => state_cov
+                             )
+
+    # create storage for the forecast and analysis statistics
+    fore_rmse = Vector{Float64}(undef, nanl)
+    filt_rmse = Vector{Float64}(undef, nanl)
+
+    for i in 1:(nanl-1)
+        for j in 1:f_steps
+            # M(x^b)
+            step_model!(x_b, 0.0, kwargs)
+        end
+
+        w = rand(MvNormal(zeros(40), I))
+        obs = o[:, i+1] + w
+
+        # optimized cost function input and value 
+        x_opt = XdVAR.D3_var_NewtonOp(x_b, obs, x_b, state_cov, H_obs, obs_cov, kwargs)
+
+        # generate actual observation value
+        x_t = o[:, i+1]
+
+        # compare model forecast and filter via RMSE
+        rmse_forecast = sqrt(msd(x_b, x_t))
+        fore_rmse[i] = rmse_forecast
+        rmse_filter = sqrt(msd(x_opt, x_t))
+        filt_rmse[i] = rmse_filter
+
+        # reinitializing x_b for next cycle if updated
+        if is_updated == true
+            x_b = x_opt
+        end
+    end
+
+    data = Dict{String,Any}(
+                            "seed" => seed,
+                            "diffusion" => diffusion,
+                            "dx_params" => dx_params,
+                            "gamma" => γ,
+                            "γ" => γ,
+                            "tanl" => tanl,
+                            "nanl" => nanl,
+                            "h" =>  h,
+                            "fore_rmse" => fore_rmse,
+                            "filt_rmse" => filt_rmse
+                           )
+
+    if is_informed == true
+        inf = "true"
+    else
+        inf = "false"
+    end
+
+    if is_updated == true
+        upd = "true"
+    else
+        upd = "false"
+    end
+
+    path = pkgdir(DataAssimilationBenchmarks) * "/src/data/d3_var_exp/"
+    name = "D3_var_filter_analysis_" * "L96_time_series_seed_" * lpad(seed, 4, "0") *
+           "_gam_" * rpad(γ, 5, "0") *
+           "_Informed_" * lpad(inf, 4, "0") *
+           "_Updated_" * lpad(upd, 4, "0") *
+           "_Tuned_" * rpad(tuning_factor, 5, "0") *
+           ".jld2"
+
+    save(path * name, data)
+    # output time
+    print("Runtime " * string(round((time() - t1)  / 60.0, digits=4))  * " minutes\n")
+
+    #= path = pkgdir(DataAssimilationBenchmarks) * "/src/analysis/var_exp/"
+    name = "D3_var_filter_analysis_" * "L96_time_series_seed_" * lpad(seed, 4, "0") *
+           "_gam_" * rpad(γ, 5, "0") *
+           "_Informed_" * lpad(is_informed, 4, "0") *
+           "_Updated_" * rpad(is_informed, 4, "0") *
+           "_Tuned_" * lpad(tuning_factor, 5, "0")
+
+    # make plot
+    t = 1:nanl
+    fore_rmse_ra = Vector{Float64}(undef, nanl)
+    filt_rmse_ra = Vector{Float64}(undef, nanl)
+
+    for i in 1:nanl
+        fore_rmse_ra[i] = sum(fore_rmse[1:i])/i
+        filt_rmse_ra[i] = sum(filt_rmse[1:i])/i
+    end
+
+    plot(t, fore_rmse_ra, label = "Forecast", title="Average Analysis RMSE vs. Time")
+    plot!(t, filt_rmse_ra, label = "Filter")
+    plot!([0, 5000], [1, 1])
+    xlabel!("Time [Cycles]")
+    ylabel!("Average Analysis RMSE")
+    savefig(path * name)=#
+end
+
+##############################################################################################
+# end module
+
+end