`Interp`

[oyamad]

Useful pieces of information about interpolation are stored in Interp:

struct Interp{N,TS<:VecOrMat,TM<:AbstractMatrix,TL<:Factorization}
    basis::Basis{N}
    S::TS
    Scoord::NTuple{N,Vector{Float64}}
    length::Int
    size::NTuple{N,Int}
    lb::NTuple{N,Float64}
    ub::NTuple{N,Float64}
    Phi::TM
    Phi_lu::TL
end

And ContinuousDP keeps an instance of Interp as one of its fields.

(1) Conceptually, this is not part of a problem, but part of a solution algorithm.

(2) One option is to include it in a "Solver" type, like

struct CDPCollocationSolver{Algo<:DPAlgorithm} <: AbstractCDPCollocationSolver
    interp::Interp
    tol
    max_iter
    options  # any options used in `solve`
end

solver = CDPCollocationSolver{PFI}(basis, tol, max_iter, ...)
res = solve(solver, cdp)

(3) Interp should not necessarily belong here. It seems to fit better in BasisMatrices.jl. Any thoughts @sglyon?

[sglyon]

I agree that it is a better fit for BasisMatrices.jl

There is an existing Interpoland type that does some of what your Interp does, but doesn't store everything as a field (e.g. it doesn't store properties like length and size and would have you just call a method to evaluate those when needed).

I'd be happy to iterate with you and extend that type to fill all your needs here.

[oyamad]

That would be nice. What I really need are basis, S, Phi, and Phi_lu (and maybe Scoord).

[sglyon]

I don't do this right now, but we could store S, and Scoord.

I do store a version of Phi evaluated in Tensor form. I'd like to see how you are using it -- maybe we don't need to store Phi (which is the Expanded form of what I currently store).

Similarly for Phi_lu, I have found that finding new coefficients based directly off of the Tensor form basis matrix is often faster than precomputing+reusing the LU decomposition of the Expanded BasisMatrix. Here's a 3d example:

julia> using BasisMatrices, BenchmarkTools

julia> b =  Basis(SplineParams(25, -1, 1, 1),SplineParams(20, -1, 2, 3), SplineParams(20, -3, 3, 3));

julia> Phi_tensor = BasisMatrix(b, Tensor, nodes(b)[2]);

julia> Phi = convert(Expanded, Phi_tensor);

julia> Phi_lu = lufact(Phi.vals[1]);

julia> y = rand(length(b));

julia> @benchmark Phi_lu \ y
BenchmarkTools.Trial:
  memory estimate:  756.50 KiB
  allocs estimate:  6
  --------------
  minimum time:     6.108 ms (0.00% GC)
  median time:      6.382 ms (0.00% GC)
  mean time:        6.528 ms (0.00% GC)
  maximum time:     8.466 ms (0.00% GC)
  --------------
  samples:          761
  evals/sample:     1

julia> @benchmark funfitxy(b, Phi_tensor, y)
BenchmarkTools.Trial:
  memory estimate:  2.90 MiB
  allocs estimate:  6579
  --------------
  minimum time:     1.610 ms (0.00% GC)
  median time:      1.772 ms (0.00% GC)
  mean time:        1.986 ms (8.00% GC)
  maximum time:     4.741 ms (35.18% GC)
  --------------
  samples:          2513
  evals/sample:     1

On the other hand, there are times when it is definitely faster to use the lu fact (same example, using just the first two dimensions):

julia> b =  Basis(SplineParams(25, -1, 1, 1),SplineParams(20, -1, 2, 3));

julia> Phi_tensor = BasisMatrix(b, Tensor, nodes(b)[2]);

julia> Phi = convert(Expanded, Phi_tensor);

julia> Phi_lu = lufact(Phi.vals[1]);

julia> y = rand(length(b));

julia> @benchmark Phi_lu \ y
BenchmarkTools.Trial:
  memory estimate:  34.69 KiB
  allocs estimate:  4
  --------------
  minimum time:     21.846 μs (0.00% GC)
  median time:      22.484 μs (0.00% GC)
  mean time:        23.865 μs (1.21% GC)
  maximum time:     977.398 μs (89.64% GC)
  --------------
  samples:          10000
  evals/sample:     1

julia> @benchmark funfitxy(b, Phi_tensor, y)
BenchmarkTools.Trial:
  memory estimate:  170.13 KiB
  allocs estimate:  346
  --------------
  minimum time:     91.221 μs (0.00% GC)
  median time:      99.814 μs (0.00% GC)
  mean time:        123.845 μs (8.72% GC)
  maximum time:     3.180 ms (58.89% GC)
  --------------
  samples:          10000
  evals/sample:     1

And once again using just the first dimension

julia> b =  Basis(SplineParams(25, -1, 1, 1));

julia> Phi_tensor = BasisMatrix(b, Tensor, nodes(b)[2]);

julia> Phi = convert(Expanded, Phi_tensor);

julia> Phi_lu = lufact(Phi.vals[1]);

julia> y = rand(length(b));

julia> @benchmark Phi_lu \ y
BenchmarkTools.Trial:
  memory estimate:  1.86 KiB
  allocs estimate:  4
  --------------
  minimum time:     933.016 ns (0.00% GC)
  median time:      1.157 μs (0.00% GC)
  mean time:        1.267 μs (6.23% GC)
  maximum time:     30.825 μs (90.51% GC)
  --------------
  samples:          10000
  evals/sample:     62

julia> @benchmark funfitxy(b, Phi_tensor, y)
BenchmarkTools.Trial:
  memory estimate:  42.60 KiB
  allocs estimate:  89
  --------------
  minimum time:     20.986 μs (0.00% GC)
  median time:      23.636 μs (0.00% GC)
  mean time:        30.564 μs (9.06% GC)
  maximum time:     3.731 ms (46.25% GC)
  --------------
  samples:          10000
  evals/sample:     1

We should probably be more scientific about these ideas, do some benchmarks to see when it becomes more efficient to use the Tensor form compared to precomputing the lu factorization. I would imagine that it is the size of the basis matrix and maybe also the sparsity pattern.

[oyamad]

Thanks! I have only used the Expanded form (which is "simplest conceptually"), and have to learn about the Tensor form.

Phi is used in evaluate_policy! to compute Phi - discount * EV, where EV is the matrix whose (i, j) entry is

E_e[ phi_j(g(s_i, X[s_i],e)) ]

(phi_j is the j-th basis function and s_i is the i-th node).