Browsing by Subject "Sparse matrices."

Now showing 1 - 3 of 3

Open Access
Locality aware reordering for sparse triangular solve
(2014) Torun, Tuğba
Sparse Triangular Solve (SpTS) is a commonly used kernel in a wide variety of scientific and engineering applications. Efficient implementation of this kernel on current architectures that involve deep cache hierarchy is crucial for attaining high performance. In this work, we propose an effective framework for cache-aware SpTS. Solution of sparse linear symmetric systems utilizing the direct methods require the triangular solve of the form LUz = b, where L is lower triangular factor and U is upper triangular factor. For cache utilization, we reorder the rows and columns of the L factor regarding the data dependencies of the triangular solve. We represent the data dependencies of the triangular solve as a directed hypergraph and construct an ordered partitioning model on this structure. For this purpose, we developed a variant of Fiduccia-Mattheyses (FM) algorithm which respects the dependency constraints. We also adopt the idea of splitting L factors into dense and sparse components and solving them seperately with different autotuned kernels for achieving more flexibility in this process. We investigate the performance variation of different storage schemes of L factors and the corresponding sparse and dense components. We utilize autotuning provided by Optimized Sparse Kernel Interface (OSKI) to reduce performance degradation that incurs due to the gap between processors and memory speeds. Experiments performed on real-world datasets verify the effectiveness of the proposed framework.
Open Access
Parallel sparse matrix vector multiplication techniques for shared memory architectures
(2014) Başaran, Mehmet
SpMxV (Sparse matrix vector multiplication) is a kernel operation in linear solvers in which a sparse matrix is multiplied with a dense vector repeatedly. Due to random memory access patterns exhibited by SpMxV operation, hardware components such as prefetchers, CPU caches, and built in SIMD units are under-utilized. Consequently, limiting parallelization efficieny. In this study we developed; • an adaptive runtime scheduling and load balancing algorithms for shared memory systems, • a hybrid storage format to help effectively vectorize sub-matrices, • an algorithm to extract proposed hybrid sub-matrix storage format. Implemented techniques are designed to be used by both hypergraph partitioning powered and spontaneous SpMxV operations. Tests are carried out on Knights Corner (KNC) coprocessor which is an x86 based many-core architecture employing NoC (network on chip) communication subsystem. However, proposed techniques can also be implemented for GPUs (graphical processing units).
Open Access
A recursive graph bipartitioning algorithm by vertex separators with fixed vertices for permuting sparse matrices into block diagonal form with overlap
(2011) Acer, Seher
Solving sparse system of linear equations Ax=b using preconditioners can be effi- ciently parallelized using graph partitioning tools. In this thesis, we investigate the problem of permuting a sparse matrix into a block diagonal form with overlap which is to be used in the parallelization of the multiplicative schwarz preconditioner. A matrix is said to be in block diagonal form with overlap if the diagonal blocks may overlap. In order to formulate this permutation problem as a graph-theoretical problem, we introduce a restricted version of the graph partitioning by vertex separator problem (GPVS), where the objective is to find a vertex partition whose parts are only connected by a vertex separator. The modified problem, we refer as ordered GPVS problem (oGPVS), is restricted such that the parts should exhibit an ordered form where the consecutive parts can only be connected by a separator. The existing graph partitioning tools are unable to solve the oGPVS problem. Thus, we present a recursive graph bipartitioning algorithm by vertex separators together with a novel vertex fixation scheme so that a GPVS tool supporting fixed vertices can effectively and efficiently be utilized. We also theoretically verified the correctness of the proposed approach devising a necessary and sufficient condition to the feasibility of a oGPVS solution. Experimental results on a wide range of matrices confirm the validity of the proposed approach.