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Matrix/vector representation

HYPRE.jl defines the structs HYPREMatrix and HYPREVector representing HYPREs datastructures. Specifically it uses the IJ System Interface which can be used for general sparse matrices.

HYPREMatrix and HYPREVector can be constructed either by assembling directly, or by first assembling into a Julia datastructure and the converting it. These various methods are outlined in the following sections:

Direct assembly (multi-/single-process)

Creating HYPREMatrix and/or HYPREVector directly is possible by first creating an assembler which is used to add all individual contributions to the matrix/vector. The required steps are:

  1. Create a new matrix and/or vector using the constructor.
  2. Create an assembler and initialize the assembling procedure using HYPRE.start_assemble!.
  3. Assemble all non-zero contributions (e.g. element matrix/vector in a finite element simulation) using HYPRE.assemble!.
  4. Finalize the assembly using HYPRE.finish_assemble!.

After these steps the matrix and vector are ready to pass to the solver. In case of multiple consecutive solves with the same sparsity pattern (e.g. multiple Newton steps, multiple time steps, ...) it is possible to reuse the same matrix by simply skipping the first step above.

Example pseudocode

# MPI communicator
comm = MPI.COMM_WORLD # MPI.COMM_SELF for single-process setups

# Create empty matrix and vector -- this process owns rows ilower to iupper
A = HYPREMatrix(comm, ilower, iupper)
b = HYPREVector(comm, ilower, iupper)

# Create assembler
assembler = HYPRE.start_assemble!(A, b)

# Assemble contributions from all elements owned by this process
for element in owned_elements
    Ae, be = compute_element_contribution(...)
    global_indices = get_global_indices(...)
    HYPRE.assemble!(assembler, global_indices, Ae, be)
end

# Finalize the assembly
A, b = HYPRE.finish_assemble!(assembler)

Create from PartitionedArrays.jl (multi-process)

HYPRE.jl integrates seemlessly with PSparseMatrix and PVector from the PartitionedArrays.jl package. These can be passed directly to solve and solve!. Internally this will construct a HYPREMatrix and HYPREVectors and then convert the solution back to a PVector.

The HYPREMatrix constructor support both SparseMatrixCSC and SparseMatrixCSR as storage backends for the PSparseMatrix. However, since HYPREs internal storage is also CSR based it can be slightly more resource efficient to use SparseMatrixCSR.

The constructors also support both PartitionedArrays.jl backends: When using the MPI backend the communicator of the PSparseMatrix/PVector is used also for the HYPREMatrix/HYPREVector, and when using the Sequential backend it is assumed to be a single-process setup, and the MPI.COMM_SELF communicator is used.

Example pseudocode

# Assemble linear system (see documentation for PartitionedArrays)
A = PSparseMatrix(...)
b = PVector(...)

# Solve with zero initial guess
x = solve(solver, A, b)

# Inplace solve with x as initial guess
x = PVector(...)
solve!(solver, x, A, b)

It is also possible to construct the arrays explicitly. This can save some resources when performing multiple consecutive solves (multiple time steps, Newton iterations, etc). To copy data back and forth between PSparseMatrix/PVector and HYPREMatrix/HYPREVector use the copy! function.

Example pseudocode

A = PSparseMatrix(...)
x = PVector(...)
b = PVector(...)

# Construct the HYPRE arrays
A_h = HYPREMatrix(A)
x_h = HYPREVector(x)
b_h = HYPREVector(b)

# Solve
solve!(solver, x_h, A_h, b_h)

# Copy solution back to x
copy!(x, x_h)

Create from SparseMatrixCSC / SparseMatrixCSR (single-process)

HYPRE.jl also support working directly with SparseMatrixCSC (from the SparseArrays.jl standard library) and SparseMatrixCSR (from the SparseMatricesCSR.jl package). This makes it possible to use solvers and preconditioners even for single-process problems. When using these type of spars matrices it is assumed that the right hand side and solution vectors are regular Julia Vectors.

Just like when using the PartitionedArrays.jl package, it is possible to pass sparse matrices directly to solve and solve!, but it is also possible to create HYPREMatrix and HYPREVector explicitly, possibly saving some resources when doing multiple consecutive linear solves (see previous section).

Example pseudocode

A = SparseMatrixCSC(...)
x = Vector(...)
b = Vector(...)

# Solve with zero initial guess
x = solve(solver, A, b)

# Inplace solve with x as initial guess
x = zeros(length(b))
solve!(solver, x, A, b)