Sparse Tiling for Stationary Iterative Methods

Abstract

In modern computers, a program’s data locality can affect performance significantly. This paper details full sparse tiling, a run-time reordering transformation that improves the data locality for stationary iterative methods such as Gauss–Seidel operating on sparse matrices. In scientific applications such as finite element analysis, these iterative methods dominate the execution time. Full sparse tiling chooses a permutation of the rows and columns of the sparse matrix, and then an order of execution that achieves better data locality. We prove that full sparsetiled Gauss–Seidel generates a solution that is bitwise identical to traditional Gauss–Seidel on the permuted matrix. We also present measurements of the performance improvements and the overheads of full sparse tiling and of cache blocking for irregular grids, a related technique developed by Douglas et al.

Toward Harnessing DOACROSS Parallelism for Multi-GPGPUs

“…In addition, Huang, et al [7] introduced a code tiling technique for improving the cache performance of PDE solvers for uniprocessors. Michelle et al [18] presented a parallel Gauss-Seidel method by applying a full sparse tiling technique to improve the cache locality of a program for uniprocessors and shared-memory machines. Wallin et al [21] considered to temporally tile Gauss-Seidel with pipelining techniques to improve parallelism on shared memory machines.…”

Tiling arbitrarily nested loops by means of the transitive

“…Transformations based on the polyhedral model produce code at compile-time, while the sparse polyhedral framework (Strout et al, 2004) extends the polyhedral model by using uninterpreted function call abstraction for the compile-time specification of run-time reordering transformations. The approach presented in this paper aims at producing code at compile-time, hence we compare it only with techniques producing tiled code at compile time.…”

“…There has been a considerable amount of research into tiling, demonstrating how to aggregate a set of loop nest iterations into tiles with each tile as an atomic macro statement, from pioneer papers (Irigoin and Triolet, 1988;Wolf and Lam, 1991;Ramanujam and Sadayappan, 1992) to those presenting advanced techniques (Bondhugula et al, 2008a;Griebl, 2004;Lim et al, 1999;Wonnacott and Strout, 2013). Several popular frameworks are used to produce tiled code automatically: the classic polyhedral model (Feautrier, 1992a;1992b;Lim and Lam, 1994;Bondhugula et al, 2008a), the sparse polyhedral model (Strout et al, 2004), the non-polyhedral model (Kim and Rajopadhye, 2009), and iteration space slicing (Pugh and Rosser, 1997;.…”

Dynamic Tile Free Scheduling for Code With Acyclic Inter-Tile Dependence Graphs

“…Papers [16,17] introduce tiling methodology for sparse matrix computations. Effective processing of sparse matrices requires the introduction of a memory-efficient data structure, compressed sparse row (CSR), which includes only nonzero values from the corresponding matrix.…”

“…The algorithm is based on the inspector/executor framework for performing run-time iteration reordering, i.e., "For sparse tiling, the inspector examines the non-zero structure of the sparse matrix at run-time, generates a data reordering and a schedule based on a tiling function, and remaps the sparse matrix and vectors based on the data reordering. The executor is a transformed version of the original code that uses the remapped matrix and vectors and the schedule created by the inspector" [17].…”