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Banded concretization of derivatives with boundary conditions gives dense matrices #182

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jagot opened this issue Sep 21, 2019 · 1 comment · May be fixed by #185
Open
4 tasks

Banded concretization of derivatives with boundary conditions gives dense matrices #182

jagot opened this issue Sep 21, 2019 · 1 comment · May be fixed by #185
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concretization Operators

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@jagot
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jagot commented Sep 21, 2019 •
edited by ChrisRackauckas

Consider the following example:

julia> L = 10.0
10.0

julia> N = 9
9

julia> δx = L/(N+1)
1.0

julia> Δ = CenteredDifference(2, 2, δx, N)
DerivativeOperator{Float64,1,false,Float64,StaticArrays.SArray{Tuple{3},Float64,1,3},StaticArrays.SArray{Tuple{0},StaticArrays.SArray{Tuple{4},Float64,1,4},1,0},Nothing,Nothing}(2, 2, 1.0, 9, 3, [1.0, -2.0, 1.0], 4, 0, StaticArrays.SArray{Tuple{4},Float64,1,4}[], StaticArrays.SArray{Tuple{4},Float64,1,4}[], nothing, nothing)

julia> BandedMatrix(Δ)
9×11 BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}:
 1.0  -2.0   1.0   0.0   0.0    ⋅     ⋅     ⋅     ⋅     ⋅    ⋅
 0.0   1.0  -2.0   1.0   0.0   0.0    ⋅     ⋅     ⋅     ⋅    ⋅
 0.0   0.0   1.0  -2.0   1.0   0.0   0.0    ⋅     ⋅     ⋅    ⋅
 0.0   0.0   0.0   1.0  -2.0   1.0   0.0   0.0    ⋅     ⋅    ⋅
 0.0   0.0   0.0   0.0   1.0  -2.0   1.0   0.0   0.0    ⋅    ⋅
  ⋅    0.0   0.0   0.0   0.0   1.0  -2.0   1.0   0.0   0.0   ⋅
  ⋅     ⋅    0.0   0.0   0.0   0.0   1.0  -2.0   1.0   0.0  0.0
  ⋅     ⋅     ⋅    0.0   0.0   0.0   0.0   1.0  -2.0   1.0  0.0
  ⋅     ⋅     ⋅     ⋅    0.0   0.0   0.0   0.0   1.0  -2.0  1.0

(Should not the bandwidths be smaller?)

If I now add BCs, I get a dense matrix upon concretization:

julia> Q = Dirichlet0BC(typeof(δx))
RobinBC{Float64,Array{Float64,1}}([-0.0, 0.0], 0.0, [-0.0, 0.0], 0.0)

julia> M,u = BandedMatrix(Δ*Q);


julia> M
9×9 Array{Float64,2}:
 -2.0   1.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0
  1.0  -2.0   1.0   0.0   0.0   0.0   0.0   0.0   0.0
  0.0   1.0  -2.0   1.0   0.0   0.0   0.0   0.0   0.0
  0.0   0.0   1.0  -2.0   1.0   0.0   0.0   0.0   0.0
  0.0   0.0   0.0   1.0  -2.0   1.0   0.0   0.0   0.0
  0.0   0.0   0.0   0.0   1.0  -2.0   1.0   0.0   0.0
  0.0   0.0   0.0   0.0   0.0   1.0  -2.0   1.0   0.0
  0.0   0.0   0.0   0.0   0.0   0.0   1.0  -2.0   1.0
  0.0   0.0   0.0   0.0   0.0   0.0   0.0   1.0  -2.0

This is because the BCs are concretized as dense matrices. Arguably, they should be BandedMatrices as well, however that function does not work either:

  • inbands_setindex! does not support ranges anymore, it seems,
  • some typos (Q_L vs Q_l)
  • BandedMatrix of a boundary should return a banded matrix. It is fine if PeriodicBC throws an error here.
  • sparse of a BC should return a BandedMatrix unless it's periodic, in which case it should be SparseMatrixCSC
  • sparse on a GhostDerivativeOperator should prefer BandedMatrix unless PeriodicBC
  • The bandwidths are too large
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[WIP] Correct concretization BCs and ghost derivative operators jagot/DiffEqOperators.jl
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