LinearAlgebraMPI.jl

Author: S. Loisel

Distributed sparse matrix and vector operations using MPI for Julia. This package provides efficient parallel linear algebra operations across multiple MPI ranks.

Features

• Distributed sparse matrices (SparseMatrixMPI{T}) with row-partitioning across MPI ranks
• Distributed dense vectors (VectorMPI{T}) with flexible partitioning
• Matrix-matrix multiplication (A * B) with memoized communication plans
• Matrix-vector multiplication (A * x, mul!(y, A, x))
• Sparse direct solvers: LU and LDLT factorization using MUMPS
• Lazy transpose with optimized multiplication rules
• Matrix addition/subtraction (A + B, A - B)
• Vector operations: norms, reductions, arithmetic with automatic partition alignment
• Support for both Float64 and ComplexF64 element types

Installation

using Pkg
Pkg.add("LinearAlgebraMPI")

Quick Start

using MPI
MPI.Init()

using LinearAlgebraMPI
using SparseArrays

# Create a sparse matrix (must be identical on all ranks)
A_global = sprand(1000, 1000, 0.01)
A = SparseMatrixMPI{Float64}(A_global)

# Create a vector
x_global = rand(1000)
x = VectorMPI(x_global)

# Matrix-vector multiplication
y = A * x

# Matrix-matrix multiplication
B_global = sprand(1000, 500, 0.01)
B = SparseMatrixMPI{Float64}(B_global)
C = A * B

# Transpose operations
At = transpose(A)
D = At * B  # Materializes transpose as needed

# Solve linear systems
using LinearAlgebra
A_sym = A + transpose(A) + 10I  # Make symmetric positive definite
A_sym_dist = SparseMatrixMPI{Float64}(A_sym)
F = ldlt(A_sym_dist)  # LDLT factorization
x_sol = solve(F, y)   # Solve A_sym * x_sol = y

Running with MPI

mpiexec -n 4 julia your_script.jl

Documentation

For detailed documentation, see the stable docs or dev docs.