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:
 0.04427358963431242
 0.027074897219948352
 0.016101499791376813
 0.15069276926136796
 0.006979012013527663
 0.008089164769699135
 0.0041665971320092275
 0.02837050374218955
 0.01666204652418488
 0.017782023591942756
 ⋮
 0.032473405208257616
 0.019454121424494875
 0.019091612006638777
 0.0024130996203839484
 0.04721748962977583
 0.15976865540138224
 0.10763926873020568
 0.06791540892566537
 0.47102904819504077

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}:
  0.8321704936205964
  3.110239285968808
 -5.911848925195614
 -2.3191439779598513