Summary: This paper presents the design and implementation of a memory scalable parallel symbolic factorization algorithm for general sparse unsymmetric matrices. Our parallel algorithm uses a graph partitioning approach, applied to the graph of \(|A|+|A|^T\), to partition the matrix in such a way that is good for sparsity preservation as well as for parallel factorization. The partitioning yields a so-called separator tree which represents the dependencies among the computations.
We use the separator tree to distribute the input matrix over the processors using a block cyclic approach and a subtree to subprocessor mapping. The parallel algorithm performs a bottom-up traversal of the separator tree. With a combination of right-looking and left-looking partial factorizations, the algorithm obtains one column structure of \(L\) and one row structure of \(U\) at each step.
The algorithm is implemented in C and MPI. From a performance study on large matrices, we show that the parallel algorithm significantly reduces the memory requirement of the symbolic factorization step, as well as the overall memory requirement of the parallel solver. It also often reduces the runtime of the sequential algorithm, which is already relatively small. In general, the parallel algorithm prevents the symbolic factorization step from being a time or memory bottleneck of the parallel solver.