Elimination forest guided 2D sparse LU factorization

Abstract

Sparse LU factorization with partial pivoting is important for many scienti c applications and delivering high performance for this problem is di cult on distributed memory machines. Our previous work has developed an approach called S that incorporates static symbolic factorization, supernode partitioning and graph scheduling. This paper studies the properties of elimination forests and uses them to guide supernode partitioning/amalgamation and execution scheduling. The new design with 2D mapping e ectively identi es dense structures without introducing too many zeros in the BLAS computation and exploits asynchronous parallelism with low bu er space cost. The implementation of this code, called S + , uses supernodal matrix multiplication which retains the BLAS-3 level e ciency and avoids unnecessary arithmetic operations. The experiments show that S + improves our previous code substantially and can achieve up to 11.04GFLOPS on 128 Cray T3E 450MHz nodes, which is the highest performance reported in the literature.

Parallel sparse LU factorization on different message passing platforms

“…As a result, S + with static symbolic factorization has relatively slow uni-processor performance. On the other hand, it exhibits competitive parallel performance [8,9] due to the exploitation of dense data structures and the absence of runtime data structure variation.…”

“…We use the S + solver [8,9] to demonstrate the performance of parallel sparse LU factorization on platforms with different message passing performance. S + uses static symbolic factorization, L/U supernode partitioning, and 2D data mapping described in Section 2.…”

“…ORI: the original S + [8,9] using 2D data mapping. This version does not contain any techniques described in Section 4.…”

“…Parallel solvers such as SuperLU MT [3,4], WSMP [5], and PAR-DISO [6] were developed specifically for shared memory machines and interprocessor communications in them take place through access to the shared memory. Other solvers like van der Stappen et al [7], S + [8,9], SPOOLES [10], MUMPS [11], and SuperLU DIST [12] employ explicit message passing which allows them to run on non-cache-coherent distributed memory computing platforms.…”

“…The key to improve the parallel application performance on platforms with slow message passing is to reduce the application inter-processor communications. This paper studies communication-reduction techniques in the context of the S + solver [8,9], which uses static symbolic factorization, supernodal matrix partitioning, and two-dimensional data mapping. In such a solver, the columnby-column pivot selection and accompanied row exchanges result in significant message passing overhead when selected pivots do not reside on the same processors as corresponding diagonals.…”