Geometry-Oblivious FMM for Compressing Dense SPD Matrices

We present GOFMM (geometry-oblivious FMM), a novel method that creates a hierarchical low-rank approximation, "compression," of an arbitrary dense symmetric positive definite (SPD) matrix. For many applications, GOFMM enables an approximate matrix-vector multiplication in $N \log N$ or even $N$ time, where $N$ is the matrix size. Compression requires $N \log N$ storage and work. In general, our scheme belongs to the family of hierarchical matrix approximation methods. In particular, it generalizes the fast multipole method (FMM) to a purely algebraic setting by only requiring the ability to sample matrix entries. Neither geometric information (i.e., point coordinates) nor knowledge of how the matrix entries have been generated is required, thus the term "geometry-oblivious." Also, we introduce a shared-memory parallel scheme for hierarchical matrix computations that reduces synchronization barriers. We present results on the Intel Knights Landing and Haswell architectures, and on the NVIDIA Pascal architecture for a variety of matrices.Comment: 13 pages, accepted by SC'1

Exploiting Locality and Parallelism with Hierarchically Tiled Arrays

The importance of tiles or blocks in mathematics and thus computer science cannot be overstated. From a high level point of view, they are the natural way to express many algorithms, both in iterative and recursive forms. Tiles or sub-tiles are used as basic units in the algorithm description. From a low level point of view, tiling, either as the unit maintained by the algorithm, or as a class of data layouts, is one of the most effective ways to exploit locality, which is a must to achieve good performance in current computers given the growing gap between memory and processor speed. Finally, tiles and operations on them are also basic to express data distribution and parallelism. Despite the importance of this concept, which makes inevitable its widespread usage, most languages do not support it directly. Programmers have to understand and manage the low-level details along with the introduction of tiling. This gives place to bloated potentially error-prone programs in which opportunities for performance are lost. On the other hand, the disparity between the algorithm and the actual implementation enlarges. This thesis illustrates the power of Hierarchically Tiled Arrays (HTAs), a data type which enables the easy manipulation of tiles in object-oriented languages. The objective is to evolve this data type in order to make the representation of all classes for algorithms with a high degree of parallelism and/or locality as natural as possible. We show in the thesis a set of tile operations which leads to a natural and easy implementation of different algorithms in parallel and in sequential with higher clarity and smaller size. In particular, two new language constructs dynamic partitioning and overlapped tiling are discussed in detail. They are extensions of the HTA data type to improve its capabilities to express algorithms with a high abstraction and free programmers from programming tedious low-level tasks. To prove the claims, two popular languages, C++ and MATLAB are extended with our HTA data type. In addition, several important dense linear algebra kernels, stencil computation kernels, as well as some benchmarks in NAS benchmark suite were implemented. We show that the HTA codes needs less programming effort with a negligible effect on performance

Analyzing and enhancing OSKI for sparse matrix-vector multiplication

Sparse matrix-vector multiplication (SpMxV) is a kernel operation widely used in iterative linear solvers. The same sparse matrix is multiplied by a dense vector repeatedly in these solvers. Matrices with irregular sparsity patterns make it difficult to utilize cache locality effectively in SpMxV computations. In this work, we investigate single- and multiple-SpMxV frameworks for exploiting cache locality in SpMxV computations. For the single-SpMxV framework, we propose two cache-size-aware top-down row/column-reordering methods based on 1D and 2D sparse matrix partitioning by utilizing the column-net and enhancing the row-column-net hypergraph models of sparse matrices. The multiple-SpMxV framework depends on splitting a given matrix into a sum of multiple nonzero-disjoint matrices so that the SpMxV operation is performed as a sequence of multiple input- and output-dependent SpMxV operations. For an effective matrix splitting required in this framework, we propose a cache-size-aware top-down approach based on 2D sparse matrix partitioning by utilizing the row-column-net hypergraph model. The primary objective in all of the three methods is to maximize the exploitation of temporal locality. We evaluate the validity of our models and methods on a wide range of sparse matrices by performing actual runs through using OSKI. Experimental results show that proposed methods and models outperform state-of-the-art schemes.Comment: arXiv admin note: substantial text overlap with arXiv:1202.385

Cache locality exploiting methods and models for sparse matrix-vector multiplication

Ankara : The Department of Computer Engineering and Information Science and the Institute of Engineering and Science of Bilkent University, 2009.Thesis (Master's) -- Bilkent University, 2009.Includes bibliographical references leaves 52-56.The sparse matrix-vector multiplication (SpMxV) is an important kernel operation widely used in linear solvers. The same sparse matrix is multiplied by a dense vector repeatedly in these solvers to solve a system of linear equations. High performance gains can be obtained if we can take the advantage of today’s deep cache hierarchy in SpMxV operations. Matrices with irregular sparsity patterns make it difficult to utilize data locality effectively in SpMxV computations. Different techniques are proposed in the literature to utilize cache hierarchy effectively via exploiting data locality during SpMxV. In this work, we investigate two distinct frameworks for cacheaware/oblivious SpMxV: single matrix-vector multiply and multiple submatrix-vector multiplies. For the single matrix-vector multiply framework, we propose a cache-size aware top-down row/column-reordering approach based on 1D sparse matrix partitioning by utilizing the recently proposed appropriate hypergraph models of sparse matrices, and a cache oblivious bottom-up approach based on hierarchical clustering of rows/columns with similar sparsity patterns. We also propose a column compression scheme as a preprocessing step which makes these two approaches cache-line-size aware. The multiple submatrix-vector multiplies framework depends on the partitioning the matrix into multiple nonzero-disjoint submatrices. For an effective matrixto-submatrix partitioning required in this framework, we propose a cache-size aware top-down approach based on 2D sparse matrix partitioning by utilizing the recently proposed fine-grain hypergraph model. For this framework, we also propose a traveling salesman formulation for an effective ordering of individual submatrix-vector multiply operations. We evaluate the validity of our models and methods on a wide range of sparse matrices. Experimental results show that proposed methods and models outperforms state-of-the-art schemes.Akbudak, KadirM.S

An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling

We present a sparse linear system solver that is based on a multifrontal variant of Gaussian elimination, and exploits low-rank approximation of the resulting dense frontal matrices. We use hierarchically semiseparable (HSS) matrices, which have low-rank off-diagonal blocks, to approximate the frontal matrices. For HSS matrix construction, a randomized sampling algorithm is used together with interpolative decompositions. The combination of the randomized compression with a fast ULV HSS factorization leads to a solver with lower computational complexity than the standard multifrontal method for many applications, resulting in speedups up to 7 fold for problems in our test suite. The implementation targets many-core systems by using task parallelism with dynamic runtime scheduling. Numerical experiments show performance improvements over state-of-the-art sparse direct solvers. The implementation achieves high performance and good scalability on a range of modern shared memory parallel systems, including the Intel Xeon Phi (MIC). The code is part of a software package called STRUMPACK -- STRUctured Matrices PACKage, which also has a distributed memory component for dense rank-structured matrices

Bringing Linear Algebra Objects to Life in a Column-Oriented In-Memory Database

Large numeric matrices and multidimensional data arrays appear in many science domains, as well as in applications of financial and business warehousing. Common applications include eigenvalue determination of large matrices, which decompose into a set of linear algebra operations. With the rise of in-memory databases it is now feasible to execute these complex analytical queries directly in a relational database system without the need of transfering data out of the system and being restricted by hard disc latencies for random accesses. In this paper, we present a way to integrate linear algebra operations and large matrices as first class citizens into an in-memory database following a two-layered architectural model. The architecture consists of a logical component receiving manipulation statements and linear algebra expressions, and of a physical layer, which autonomously administrates multiple matrix storage representations. A cost-based hybrid storage representation is presented and an experimental implementation is evaluated for matrix-vector multiplications