MPI prototype

Author: dlfivefifty

examples/BSplinesSphericalHarmonics.jl

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+SH = SphericalHarmonics()
+Spl = BSplines(0.0:0.01:1)
+S = SH ⊗ Spl
+
+λ = @. -(1:∞)*((1:∞) +1)
+L = BlockDiagonal(Diagonal.(Fill.(λ,1:2:∞))) # create a Block matrix with growing block sizes
+Δ_s = Mul(SH, L, inv(SH)) # the Laplace–Boltrami operator
+D_r = Mul(Spl, SymTridiagonal(...), inv(Spl))
+R = Mul(Spl, Diagonal(Spl.points), inv(Spl))
+
+Δ = I ⊗ (D_r*R^2*D_r) + Δ_s ⊗ I # creates Laplacian as a Mul(S, ..., inv(S)) operator
+
+# finite_dimensional case
+N = 100
+S_N = Spl * SH[:, Block.(1:N)]  # take only first N degree spherical harmonics
+
+Δ_N = Δ*S_N  # Knows that inv(S)*S_N === inv(S_N)
+
+# R²Δ_N is a lazy BandedBlockBandedMatrix, with diagonal blockbandwidths
+backend = BandedBlockBandedMatrix{Float64,MPIMatrix{Float64}}(undef, size(R²Δ_N), (0,0), bandwidths(D_r);
+                        workers = ...) # distribute different blocks based on workers
+
+MPI_Δ_N =  Mul(S_N, backend, inv(S_N))
+
+MPI_Δ_N .= Δ_N # populates MPI array, generating data on workers remotely
+
+
+f = LazyFun((x,y,z) -> cos(x*y)+z, S_N)  # not sure how constructors fit in yet...
+MPI_f = Mul(S_N, BlockVector{Float64,MPIVector{Float64}}(undef, blocklengths(backend, 2))
+
+MPI_f .= f # automatically generates data remotely, via MPI version of FastTransforms
+
+MPI_v = similar(MPI_f)
+MPI_v .= Mul(MPI_Δ_N, MPI_f) # applies the Laplacian and fills in missing information.