some tests on speeding up

project_partialbridge/lowrankricatti.jl

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+# implement Mena et al. for solving
+# (d P)/(d t) = A P(t) + P(t) A' + Q,   P(t_0) = P0
+# with a low rank approximation for Q
+
+using SparseArrays
+using SuiteSparse
+using LinearAlgebra
+using Expokit  # for matrix exponentials
+using BenchmarkTools
+
+# construct an example
+d = 800
+Q = sprand(d,d,.01) - I# Diagonal(randn(d))
+diagels = 0.4.^(1:d)
+A = sprand(d,d,0.1)#Diagonal(diagels)
+P0 = Matrix(sprand(d,d,0.1))# Diagonal(diagels)
+
+# we use Expokit, which gives the funciton expmv!
+# w = expmv!{T}( w::Vector{T}, t::Number, A, v::Vector{T}; kwargs...)
+# The function expmv! calculates w = exp(t*A)*v, where A is a matrix or any type that supports size, eltype and mul! and v is a dense vector by using Krylov subspace projections. The result is stored in w.
+
+# It is much faster for large systems
+v = rand(d)
+@time a1=expmv!(zeros(d),1,Q,v)
+@time a2=exp(Matrix(Q))*v
+a1-a2
+
+# Let's consider a few implementations
+matexpvec1 = function(A,U,τ)
+    exp(Matrix(τ * A)) * U
+end
+
+matexpvec2 = function(A,U,τ)
+    M = []
+    for j in 1:size(U,2)
+       push!(M, expmv!(zeros(size(U,1)),τ,A,U[:,j]))
+    end
+    M = hcat(M...)
+end
+
+# matexpvec3 = function(A,U)  # is wrong for some reason
+#     τ = 1.0
+#     for j in 1:size(U,2)
+#       expmv!(U[:,j],τ,A,U[:,j])
+#     end
+#     U
+# end
+
+matexpvec4 = function(A,U,τ)
+    hcat([  expmv!(U[:,j],τ,A,U[:,j]) for  j in 1:size(U,2)]...)
+end
+
+τ = 1.0
+R = A # or Q
+M0 = svd(P0)
+U = Matrix(M0.U[:,1:ld])
+
+@benchmark  matexpvec1(R,U,τ)
+@benchmark  matexpvec2(R,U,τ)
+@benchmark  matexpvec4(R,U,τ)
+
+maximum(matexpvec4(R,U,τ)-matexpvec1(R,U,τ))
+
+
+function lowrankricatti(P0, A, Q, t, ld)
+# P0 is the value of P at time t[1]; t is an increasing sequence of times,
+# ld is the dimension of the low rank approximation
+# output: at each time t a low rank approximation of P(t), where the couple (U,S) is such that P = U*S*U'
+
+    # step 1
+    M0 = svd(P0)
+    global U = Matrix(M0.U[:,1:ld])
+    global S = Diagonal(M0.S[1:ld])
+
+    Ulowrank = [U]
+    Slowrank = [S]
+    for i in 1:length(t)-1
+        τ = t[i+1] - t[i]
+        if false
+            matexp = exp(Matrix(τ * A))
+            # step 3
+            M = matexp * U
+        else
+            M = matexpvec4(A,U,τ)
+        end
+        Uᴬcompact, R = qr(M)
+        Uᴬ = Matrix(Uᴬcompact)
+        # step 4
+        Sᴬ = R * S * R'
+        # step 5
+        K =  Uᴬ * Sᴬ + (τ * Q) * Uᴬ
+        Ucompact, Ŝ = qr(K)
+        U = Matrix(Ucompact)
+        Ŝ = Ŝ - U' * (τ * Q) * Uᴬ
+        L = Ŝ * Uᴬ' +  U' * (τ * Q)
+        S = L * U
+
+        push!(Ulowrank,U)
+        push!(Slowrank,Diagonal(S))
+        # ??? should S be diagonal or not: either convert it, or make Slowrank an array of full matrices
+        #push!(Slowrank,S)
+    end
+
+    Ulowrank, Slowrank
+end
+
+
+# Example:
+ld = 10  # low rank dimension
+τ = 0.01; t0 =0.5; tend=0.8
+t = t0:τ:tend
+
+@time U,S = lowrankricatti(P0,A, Q, t,ld)
+P = U[end]*S[end]*U[end]'
+
+# compute exact solution
+λ = lyap(Matrix(Q), -Matrix(A)) + P0
+Φ = exp(Matrix(Q)*(tend- t0))
+Pexact = Φ*λ*Φ' - λ + P0
+
+# assess difference
+dif = P - Pexact
+
+print(maximum(abs.(dif)))
+dif