Data traffic reduction schemes for Cholesky factorization on asynchronous multiprocessor systems

For multiprocessor systems with two level memory hierarchy; the communication requirements of parallel Cholesky factorization of dense and sparse symmetric, positive definite matrices are analyzed. The data traffic associated with computing the Chloesky factor of an <italic>n</italic>n ×<italic>n</italic> dense matrix using <italic>n</italic><supscrpt>α</supscrpt> processors, α ≤ 2, is shown to be &OHgr;(<italic>n</italic><supscrpt>2+α/2</supscrpt>), assuming that the computational load is uniformly distributed. For an <italic>n</italic>n ×<italic>n</italic> sparse matrix, representing a √<italic>n</italic> × √<italic>n</italic> regular grid graph, the corresponding data traffic is shown to be &OHgr;(<italic>n</italic><supscrpt>1+α/2</supscrpt>), α ≤ 1. Partitioning schemes that are variations of block assignment scheme are described. The data traffic generated by these schemes are asymptotically optimal and these schemes allow efficient use of up to <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2</supscrpt>) and &Ogr;(<italic>n</italic>) processors in the dense and the sparse case, respectively. The block based partitioning schemes are shown to provide a better utilization of the data accessed from the shared memory and reduce the total data traffic as compared to the schemes based on the column-wise wrap around assignment.