Volume Projection Matrix

The idea of "projection" stems from finite volume application. Assume the user has solved a FV problem on a coarse geometry VG and wants to project this solution onto the refined geometry VG2 as an initial guess for a solver of the FV problem on VG2. Have a look at the following code.

    using SparseArrays
    VG = VoronoiGeometry(VoronoiNodes(rand(2,10)),cuboid(2,periodic = [1]),integrator=HighVoronoi.VI_POLYGON)
    VG2 = refine(VG,VoronoiNodes(0.2*rand(2,4)))
    rows, cols, vals = interactionmatrix(VG2,VG)
    A = sparse(rows,cols,vals)
    println(A*ones(Float64,10)) # this will print out a vector or 14 values `1.0` (mass conservation)

Then A is the matrix that projects vectors on VG to vectors on VG2. More precisely, let VG be a partition in cells with masses $m_i$ and let VG2 a partition in cells with masses $\tilde m_j$. Then, if $u$ is the vector of data on VG and $\tilde u$ is the data on VG2 with $\tilde u = A \cdot u$ then

\[\sum_j \tilde m_j \tilde u_j = \sum_i m_i u_i\,.\]

Optional Parameters

• check_compatibility=true: This enforces that the geometries are verified for their compatibility.
• tolerance = 1.0E-12,: This two points in VG and VG2 have a distance of less than tolerance they are considered identical
• hits_per_cell = 1000: The method is based on a sampling procedure. This parameter controlls how many samples are taken from each cell on average.
• bounding_box=Boundary(): It is mathematicaly not reasonable to apply the method for unbounded domains. However, if you anyway wish to do so, you have to provide a bounded bounding_box.

Conditions on VG and VG2

VG and VG2 may be completely unrelated, the methods works anyway as long as the domains are bounded (or an additional bound is provided) and identical, including periodicity.

This works

Example 1

    VG = VoronoiGeometry(VoronoiNodes(rand(2,10)),cuboid(2,periodic = [1]),integrator=HighVoronoi.VI_GEOMETRY)
    VG2 = VoronoiGeometry(VoronoiNodes(rand(2,20)),cuboid(2,periodic = [1]),integrator=HighVoronoi.VI_POLYGON)
    rows, cols, vals = interactionmatrix(VG2,VG)
    rows2, cols2, vals2 = interactionmatrix(VG,VG2) # the inverse projection

Example 2

    VG = VoronoiGeometry(VoronoiNodes(rand(2,10)),cuboid(2,periodic = []))
    VG2 = VoronoiGeometry(VoronoiNodes(rand(2,20)),cuboid(2,periodic = []))
    rows, cols, vals = interactionmatrix(VG2,VG)
    rows2, cols2, vals2 = interactionmatrix(VG,VG2) # the inverse projection

Example 5

    VG = VoronoiGeometry(VoronoiNodes(rand(2,10)))
    VG2 = VoronoiGeometry(VoronoiNodes(2*rand(2,20)))
    rows, cols, vals = interactionmatrix(VG2,VG,bounding_box=cuboid(2))

This does not works

Example 4

The following breaks because the periodicities are not compatible

    VG = VoronoiGeometry(VoronoiNodes(rand(2,10)),cuboid(2,periodic = [1]),integrator=HighVoronoi.VI_GEOMETRY)
    VG2 = VoronoiGeometry(VoronoiNodes(rand(2,20)),cuboid(2,periodic = []),integrator=HighVoronoi.VI_POLYGON)
    rows, cols, vals = interactionmatrix(VG2,VG)

Example 5

The following breaks because the dimensions of the domain are different.

    VG = VoronoiGeometry(VoronoiNodes(rand(2,10)),cuboid(2,periodic = []))
    VG2 = VoronoiGeometry(VoronoiNodes(2*rand(2,20)),cuboid(2,periodic = [],dimensions=2*ones(Float64,dim)))
    rows, cols, vals = interactionmatrix(VG2,VG)

Example 6

The following breaks because the dimension is unbounded.

    VG = VoronoiGeometry(VoronoiNodes(rand(2,10)))
    VG2 = VoronoiGeometry(VoronoiNodes(2*rand(2,20)))
    rows, cols, vals = interactionmatrix(VG2,VG)