An Evaluation of Left-Looking, Right-Looking and Multifrontal Approaches to Sparse Cholesky Factorization on Hierarchical-Memory Machines

In this paper we present a comprehensive analysis of the performance of a variety of sparse Cholesky factorization methods on hierarchical-memory machines. We investigate methods that vary along two different axes. Along the first axis, we consider three different high-level approaches to sparse factorization: leftlooking, right-looking, and muhi.frontal. Along the second axis, we consider the implementation of each of these high-level approaches using different sets of primitives. The primitives vary based on the structures they manipulate. One important structure in sparse Cholesky factorization is a single column of the matrix. We first consider primitives that manipulate single columns. These are the most commonly used primitives for expressing the sparse Cholesky computation. Another important structure is the supemode, a set of columns with identical non-zero structures. We consider sets of primitives that exploit the supernodal structure of the matrix to varying degrees. We find that primitives that manipulate larger structures greatly increase the amount of exploitable data reuse, thus leading to dramatically higher performance on hierarchical-memory machines. We observe performance increases of two to three times when comparing methods based on primitives that make extensive use of the supernodal structure to methods based on primitives that manipulate columns. We also find that the overall approach (left-looking, right-looking, or mu.ltifrontal) is less important for performance than the particular set of primitives used to implement the approach.