Browsing by Subject "Sparse matrix"

Now showing 1 - 9 of 9

Open Access
Hopping and correlation effects in atomic clusters and networks
(TechConnect, 2001) Kulik, Igor Orestovich
Exact solution for hopping and correlation effects in atomic clusters and mesoscopic/nanoscopic networks is outlined. The program translates the Hamiltonian operator of the cluster written in terms of second-quantized creation and annihilation operators, to sparse matrix of (extremely) large dimension and solves the latter with the help of new compiler termed ABC ("Advanced Basic-C" compiler/converter/programmer). The ABC creates a stand-alone executable or, if proved necessary, source C-code received from the original program written in a simplified Quick Basic dialect. ABC employes mathematical functions including the complex variables, arbitrary precision floating-point numbers, special functions, standard mathematical routines (mulidimensional integrals, eigenvalues of Hermitian marices, in particular a new algorithm for sparce Hermitian matrices, etc.) and is appropriate to practically all software/hardware environments (Windows, OS/2, Linux and UNIX machines).
Open Access
Hypergraph partitioning and reordering for parallel sparse triangular solves and tensor decomposition
(2021-07) Torun, Tuğba
Several scientiﬁc and real-world problems require computations with sparse ma-trices, or more generally, sparse tensors which are multi-dimensional arrays. For sparse matrix computations, parallelization of sparse triangular systems intro-duces signiﬁcant challenges because of the sequential nature of the computations involved. One approach to parallelize sparse triangular systems is to use sparse triangular SPIKE (stSPIKE) algorithm, which was originally proposed for shared memory architectures. stSPIKE decouples the problem into independent smaller systems and requires the solution of a much smaller reduced sparse triangular sys-tem. We extend and implement stSPIKE for distributed-memory architectures. Then we propose distributed-memory parallel Gauss-Seidel (dmpGS) and ILU (dmpILU) algorithms by means of stSPIKE. Furthermore, we propose novel hy-pergraph partitioning models and in-block reordering methods for minimizing the size and nonzero count of the reduced systems that arise in dmpGS and dmpILU. For sparse tensor computations, tensor decomposition is widely used in the anal-ysis of multi-dimensional data. The canonical polyadic decomposition (CPD) is one of the most popular tensor decomposition methods, which is commonly computed by the CPD-ALS algorithm. Due to high computational and mem-ory demands of CPD-ALS, it is inevitable to use a distributed-memory-parallel algorithm for eﬃciency. The medium-grain CPD-ALS algorithm, which adopts multi-dimensional cartesian tensor partitioning, is one of the most successful dis-tributed CPD-ALS algorithms for sparse tensors. We propose a novel hypergraph partitioning model, CartHP, whose partitioning objective correctly encapsulates the minimization of total communication volume of multi-dimensional cartesian tensor partitioning. Extensive experiments on real-world sparse matrices and tensors validate the parallel scalability of the proposed algorithms as well as the eﬀectiveness of the proposed hypergraph partitioning and reordering models.
Open Access
Locality-aware parallel sparse matrix-vector and matrix-transpose-vector multiplication on many-core processors
(Institute of Electrical and Electronics Engineers, 2016) Karsavuran, M. O.; Akbudak K.; Aykanat, Cevdet
Sparse matrix-vector and matrix-transpose-vector multiplication (SpMMTV) repeatedly performed as z ← ATx and y ← A z (or y ← A w) for the same sparse matrix A is a kernel operation widely used in various iterative solvers. One important optimization for serial SpMMTV is reusing A-matrix nonzeros, which halves the memory bandwidth requirement. However, thread-level parallelization of SpMMTV that reuses A-matrix nonzeros necessitates concurrent writes to the same output-vector entries. These concurrent writes can be handled in two ways: via atomic updates or thread-local temporary output vectors that will undergo a reduction operation, both of which are not efficient or scalable on processors with many cores and complicated cache-coherency protocols. In this work, we identify five quality criteria for efficient and scalable thread-level parallelization of SpMMTV that utilizes one-dimensional (1D) matrix partitioning. We also propose two locality-aware 1D partitioning methods, which achieve reusing A-matrix nonzeros and intermediate z-vector entries; exploiting locality in accessing x -, y -, and -vector entries; and reducing the number of concurrent writes to the same output-vector entries. These two methods utilize rowwise and columnwise singly bordered block-diagonal (SB) forms of A. We evaluate the validity of our methods on a wide range of sparse matrices. Experiments on the 60-core cache-coherent Intel Xeon Phi processor show the validity of the identified quality criteria and the validity of the proposed methods in practice. The results also show that the performance improvement from reusing A-matrix nonzeros compensates for the overhead of concurrent writes through the proposed SB-based methods.
Open Access
Minimizing communication through computational redundancy in parallel iterative solvers
(2011) Torun, Fahreddin Şükrü
Sparse matrix vector multiplication (SpMxV) of the form y = Ax is a kernel operation in iterative linear solvers used in scientific applications. In these solvers, the SpMxV operation is performed repeatedly with the same sparse matrix through iterations until convergence. Depending on the matrix and its decomposition, parallel SpMxV operation necessitates communication among processors in the parallel environment. The communication can be reduced by intelligent decomposition. However, we can further decrease the communication through data replication and redundant computation. The communication occurs due to the transfer of x-vector entries in row-parallel SpMxV computation. The input vector x of the next iteration is computed from the output vector of the current iteration through linear vector operations. Hence, a processor may compute a y-vector entry redundantly, which leads to a x-vector entry in the following iteration, instead of receiving that x-vector entry from another processor. Thus, redundant computation of that y-vector entry may lead to reduction in communication. In this thesis, we devise a directed-graph-based model that correctly captures the computation and communication pattern for above-mentioned iterative solvers. Moreover, we formulate the communication minimization by utilizing redundant computation of y-vector entries as a combinatorial problem on this directed graph model. We propose two heuristics to solve this combinatorial problem. Experimental results indicate that the communication reducing strategy by redundantly computing is promising.
Open Access
Optimizing nonzero-based sparse matrix partitioning models via reducing latency
(Academic Press, 2018) Acer, S.; Selvitopi, O.; Aykanat, Cevdet
For the parallelization of sparse matrix-vector multiplication (SpMV) on distributed memory systems, nonzero-based fine-grain and medium-grain partitioning models attain the lowest communication volume and computational imbalance among all partitioning models. This usually comes, however, at the expense of high message count, i.e., high latency overhead. This work addresses this shortcoming by proposing new fine-grain and medium-grain models that are able to minimize communication volume and message count in a single partitioning phase. The new models utilize message nets in order to encapsulate the minimization of total message count. We further fine-tune these models by proposing delayed addition and thresholding for message nets in order to establish a trade-off between the conflicting objectives of minimizing communication volume and message count. The experiments on an extensive dataset of nearly one thousand matrices show that the proposed models improve the total message count of the original nonzero-based models by up to 27% on the average, which is reflected on the parallel runtime of SpMV as an average reduction of 15% on 512 processors.
Open Access
Parallel direct and hybrid methods based on row block partitioning for solving sparse linear systems
(2017-08) Torun, Fahreddin Şükrü
Solving system of linear equations is a kernel operation in many scienti c and industrial applications. These applications usually give rise to linear systems in which the coe cient matrix is very large and sparse. The need for solving these large and sparse systems within a reasonable time necessitates e cient and e ective parallel solution methods. In this thesis, three novel approaches are proposed for reducing the parallel solution time of linear systems. First, a new parallel algorithm, ParBaMiN, is proposed in order to nd the minimum 2-norm solution of underdetermined linear systems, where the coe cient matrix is in the form of column overlapping block diagonal. The conducted experiments demonstrate the scalability of ParBaMiN on both shared and distributed memory architectures. Secondly, a new graph theoretical partitioning method is introduced in order to reduce the number of iterations in block Cimmino algorithm. Experimental results validate the e ectiveness of the proposed partitioning method in terms of reducing the required number of iterations. Finally, we propose a new parallel hybrid method, BCDcols, which further reduces the number of iterations of block Cimmino algorithm for matrices with dense columns. BCDcols combines the block Cimmino iterative algorithm and a dense direct method for solving the system. Experimental results show that BCDcols signi cantly improves the convergence rate of block Cimmino method and hence reduces the parallel solution time.
Open Access
Reduce operations: send volume balancing while minimizing latency
(IEEE, 2020) Karsavuran, M. Ozan; Acer, S.; Aykanat, Cevdet
Communication hypergraph model was proposed in a two-phase setting for encapsulating multiple communication cost metrics (bandwidth and latency), which are proven to be important in parallelizing irregular applications. In the first phase, computational-task-to-processor assignment is performed with the objective of minimizing total volume while maintaining computational load balance. In the second phase, communication-task-to-processor assignment is performed with the objective of minimizing total number of messages while maintaining communication-volume balance. The reduce-communication hypergraph model suffers from failing to correctly encapsulate send-volume balancing. We propose a novel vertex weighting scheme that enables part weights to correctly encode send-volume loads of processors for send-volume balancing. The model also suffers from increasing the total communication volume during partitioning. To decrease this increase, we propose a method that utilizes the recursive bipartitioning framework and refines each bipartition by vertex swaps. For performance evaluation, we consider column-parallel SpMV, which is one of the most widely known applications in which the reduce-task assignment problem arises. Extensive experiments on 313 matrices show that, compared to the existing model, the proposed models achieve considerable improvements in all communication cost metrics. These improvements lead to an average decrease of 30 percent in parallel SpMV time on 512 processors for 70 matrices with high irregularity.
Open Access
Reducing communication overhead in sparse matrix and tensor computations
(2020-08) Karsavuran, Mustafa Ozan
Encapsulating multiple communication cost metrics, i.e., bandwidth and latency, is proven to be important in reducing communication overhead in the parallelization of sparse and irregular applications. Communication hypergraph model was proposed in a two-phase setting for encapsulating multiple communication cost metrics. The reduce-communication hypergraph model suﬀers from failing to correctly encapsulate send-volume balancing. We propose a novel vertex weighting scheme that enables part weights to correctly encode send-volume loads of processors for send-volume balancing. The model also suﬀers from increasing the total communication volume during partitioning. To decrease this increase, we propose a method that utilizes the recursive bipartitioning (RB) paradigm and reﬁnes each bipartition by vertex swaps. For performance evaluation, we consider column-parallel SpMV, which is one of the most widely known applications in which the reduce-task assignment problem arises. Extensive experiments on 313 matrices show that, compared to the existing model, the proposed models achieve considerable improvements in all communication cost metrics. These improvements lead to an average decrease of 30 percent in parallel SpMV time on 512 processors for 70 matrices with high irregularity. We further enhance the reduce-communication hypergraph model so that it also encapsulates the minimization of the maximum number of messages sent by a processor. For this purpose, we propose a novel cutsize metric which we realize using RB paradigm while partitioning the reduce-communication hypergraph. We also introduce a new type of net for the communication hypergraph which models decreasing the increase in the total communication volume directly with the partitioning objective. Experiments on 300 matrices show that the proposed models achieve considerable improvements in communication cost metrics which lead to better column-parallel SpMM time on 1024 processors. We propose a hypergraph model for general medium-grain sparse tensor partitioning which does not enforce any topological constraint on the partitioning. The proposed model is based on splitting the given tensor into nonzero-disjoint component tensors. Then a mode-dependent coarse-grain hypergraph is constructed for each component tensor. A net amalgamation operation is proposed to form a composite medium-grain hypergraph from these mode-dependent coarse-grain hypergraphs to correctly encapsulate the minimization of the communication volume. We propose a heuristic which splits the nonzeros of dense slices to obtain sparse slices in component tensors. We also utilize the well-known RB paradigm to improve the quality of the splitting heuristic. We propose a medium-grain tripartite graph model with the aim of a faster partitioning at the expense of increasing the total communication volume. Parallel experiments conducted on 10 real-world tensors on up to 1024 processors conﬁrm the validity of the proposed hypergraph and graph models.
Open Access
Spatiotemporal graph and hypergraph partitioning models for sparse matrix-vector multiplication on many-core architectures
(IEEE Computer Society, 2019) Abubaker, Nabil; Akbudak, K.; Aykanat, Cevdet
There exist graph/hypergraph partitioning-based row/column reordering methods for encoding either spatial or temporal locality for sparse matrix-vector multiplication (SpMV) operations. Spatial and temporal hypergraph models in these methods are extended to encapsulate both spatial and temporal localities based on cut/uncut net categorization obtained from vertex partitioning. These extensions of spatial and temporal hypergraph models encode the spatial locality primarily and the temporal locality secondarily, and vice-versa, respectively. However, the literature lacks models that simultaneously encode both spatial and temporal localities utilizing only vertex partitioning for further improving the performance of SpMV on shared-memory architectures. In order to fill this gap, we propose a novel spatiotemporal hypergraph model that leads to a one-phase spatiotemporal reordering method which encodes both types of locality simultaneously. We also propose a framework for spatiotemporal methods which encodes both types of locality in two dependent phases and two separate phases. The validity of the proposed spatiotemporal models and methods are tested on a wide range of sparse matrices and the experiments are performed on both a 60-core Intel Xeon Phi processor and a Xeon processor. Results show the validity of the methods via almost doubling the Gflop/s performance through enhancing data locality in parallel SpMV operations.