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Abstract. We recently proposed an iterative procedure which asymptotically scales the rows and columns of a given matrix to one in a given norm. In this work, we briefly mention some of the properties of that algorithm and discuss its efficient parallelization. We report on a parallel performance study of our implementation on a few computing environments. Key words: sparse matrices; matrix scaling; equilibration; parallel computing 1

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The efficiency of parallel iterative methods for solving linear systems, arising from reallife applications, depends greatly on matrix characteristics and on the amount of parallel overhead. It is often viewed that a major part of this overhead can be caused by parallel matrix-vector multiplications. However, for difficult large linear systems, the preconditioning operations needed to accelerate convergence are to be performed in parallel and may also incur substantial overhead. To obtain an efficient preconditioning, it is desirable to consider certain matrix numerical properties in the matrix partitioning process. In general, graph partitioners consider the nonzero structure of a matrix to balance the number of unknowns and to decrease communication volume among parts. The present work builds upon hypergraph partitioning techniques because of their ability to handle nonsymmetric and irregular structured matrices and because they correctly minimize communication volume. First, several hyperedge weight schemes are proposed to account for the numerical matrix property called diagonal dominance of rows and columns. Then, an algorithm for the independent partitioning of certain submatrices followed by the matching of the obtained parts is presented in detail along with a proof that it correctly minimizes the total communication volume. For the proposed variants of hypergraph partitioning models, numerical experiments compare the iterations to converge, investigate the diagonal dominance of the obtained parts, and show the values of the partitioning cost functions. 1

We present the PaToH MATLAB Matrix Partitioning Interface. The interface provides support for hypergraph-based sparse matrix partitioning methods which are used for efficient parallelization of sparse matrix-vector multiplication operations. The interface also offers tools for visualizing and measuring the quality of a given matrix partition. We propose a novel, multilevel, 2D coarsening-based 2D matrix partitioning method and implement it using the interface. We have performed extensive comparison of the proposed method against our implementation of orthogonal recursive bisection and fine-grain methods on a large set of publicly available test matrices. The conclusion of the experiments is that the new method can compete with the fine-grain method while also suggesting new research directions.

of the inverse of a matrix

Abstract—We discuss efficient shared memory parallelization of sparse matrix computations whose main traits resemble to those of the sparse matrix-vector multiply operation. Such computations are difficult to parallelize because of the relatively small computational granularity characterized by small number of operations per each data access. Our main application is a sparse matrix scaling algorithm which is more memory bound than the sparse matrix vector multiplication operation. We take the application and parallelize it using the standard OpenMP programming principles. Apart from the common race condition avoiding constructs, we do not reorganize the algorithm. Rather, we identify associated performance metrics and describe models to optimize them. By using these models, we implement parallel matrix scaling algorithms for two wellknown sparse matrix storage formats. Experimental results show that simple parallelization attempts which leave data/work partitioning to the runtime scheduler can suffer from the overhead of avoiding race conditions especially when the number of threads increases. The proposed algorithms perform better than these algorithms by optimizing the identified performance metrics and reducing the overhead. Keywords-Shared-memory parallelization, sparse matrices, hypergraphs, matrix scaling I.

On computing inverse entries of a sparse matrix in an out-of-core environment