Demonstration of Operator Algebras and Kron

Let M, D, F be matrix-based, diagonal-matrix-based, and function-based SciMLOperators respectively. Here are some examples of composing operators in order to build more complex objects and using their operations.

using SciMLOperators, LinearAlgebra
N = 4
function f(v, u, p, t)
    u .* v
end
function f(w, v, u, p, t)
    w .= u .* v
end

u = rand(4)
p = nothing # parameter struct
t = 0.0     # time

M = MatrixOperator(rand(N, N))
D = DiagonalOperator(rand(N))
F = FunctionOperator(f, zeros(N), zeros(N); u, p, t)

FunctionOperator(4 × 4)

Then, the following codes just work.

L1 = 2M + 3F + LinearAlgebra.I + rand(N, N)
L2 = D * F * M'
L3 = kron(M, D, F)
L4 = lu(M) \ D
L5 = [M; D]' * [M F; F D] * [F; D]

((((MatrixOperator(4 × 4) * MatrixOperator(4 × 4)) + (DiagonalOperator(4 × 4) * FunctionOperator(4 × 4))) * FunctionOperator(4 × 4)) + (((MatrixOperator(4 × 4) * FunctionOperator(4 × 4)) + (DiagonalOperator(4 × 4) * DiagonalOperator(4 × 4))) * DiagonalOperator(4 × 4)))

Each L# can be applied to AbstractVectors of appropriate sizes:

v = rand(N)
w = L1(v, u, p, t) # == L1 * v

v_kron = rand(N^3)
w_kron = L3(v_kron, u, p, t) # == L3 * v_kron

64-element reshape(transpose(::Matrix{Float64}), 64) with eltype Float64:
 0.39325763529442664
 0.5145433378693253
 0.10869509667197871
 0.3383269484256467
 0.06312637112771913
 0.08865300653768364
 0.05225938410910357
 0.07858856745418644
 0.04073051865149216
 0.03176433116201001
 ⋮
 0.06075135570174249
 0.02839199350346567
 0.022000753998853404
 0.013840222720650129
 0.02034165079034164
 0.3171844648687825
 0.46748767575476224
 0.20245424238432258
 0.2985035899478473

For mutating operator evaluations, call cache_operator to generate an in-place cache, so the operation is nonallocating.

α, β = rand(2)

# allocate cache
L2 = cache_operator(L2, u)
L4 = cache_operator(L4, u)

# allocation-free evaluation
L2(w, v, u, p, t) # == mul!(w, L2, v)
L4(w, v, u, p, t, α, β) # == mul!(w, L4, v, α, β)

4-element Vector{Float64}:
 -2.3054845638431973
  2.676988701325416
 -0.5011991211439749
  0.25653357270760674