Communication Optimal Parallel and Sequential Cholesky Factorization
|Title||Communication Optimal Parallel and Sequential Cholesky Factorization|
|Publication Type||Conference Paper|
|Year of Publication||2009|
|Authors||Ballard, G., Demmel J., Holtz O., & Schwartz O.|
|Conference Name||Symposium on Parallelism in Algorithms and Architectures|
|Conference Location||Johns Hopkins University|
Numerical algorithms have two kinds of costs: arithmetic and communication, by which we mean either moving data between levels of a memory hierarchy (in the sequential case) or over a network connecting processors (in the parallel case). Communication costs often dominate arithmetic costs, so it is of interest to design algorithms minimizing communication. In this paper we ﬁrst extend known lower bounds on the communication cost (both for bandwidth and for latency) of conventional (O(n3 )) matrix multiplication to Cholesky factorization, which is used for solving dense symmetric positive deﬁnite linear systems. Second, we compare the cost of various Cholesky decomposition implementations to this lower bound, and draw the following conclusions:
(1) “Na¨ive” sequential algorithms for Cholesky attain neither the bandwidth nor latency lower bounds.
(2) The sequential blocked algorithm in LAPACK (with the right block size), as well as various recursive algorithms [AP00, GJ01, AGW01, ST04], and one based on work of Toledo [Tol97], can attain the bandwidth lower bound.
(3) The LAPACK algorithm can also attain the latency bound if used with blocked data structures rather than column-wise or row-wise matrix data structures, though the Toledo algorithm cannot.
(4) The recursive sequential algorithm due to [AP00] attains the bandwidth and latency lower bounds at every level of a multi-level memory hierarchy, in a “cache-oblivious” way.
(5) The parallel implementation of Cholesky in the ScaLAPACK library (again with the right block-size) attains both the bandwidth and latency lower bounds to within a poly- logarithmic factor.
Combined with prior results in [DGHL08a, DGHL08b, DGX08] this gives a complete set of communication-optimal algorithms for O(n3 ) implementations of three basic factorizations of dense linear algebra: LU with pivoting, QR and Cholesky. But it goes beyond this prior work on sequential LU and QR by optimizing communication for any number of levels of memory hierarchy.