Graphulo Implementation of Server-Side Sparse Matrix Multiply in the Accumulo Database

The Apache Accumulo database excels at distributed storage and indexing and is ideally suited for storing graph data. Many big data analytics compute on graph data and persist their results back to the database. These graph calculations are often best performed inside the database server. The GraphBLAS standard provides a compact and efficient basis for a wide range of graph applications through a small number of sparse matrix operations. In this article, we implement GraphBLAS sparse matrix multiplication server-side by leveraging Accumulo's native, high-performance iterators. We compare the mathematics and performance of inner and outer product implementations, and show how an outer product implementation achieves optimal performance near Accumulo's peak write rate. We offer our work as a core component to the Graphulo library that will deliver matrix math primitives for graph analytics within Accumulo.Comment: To be presented at IEEE HPEC 2015: http://www.ieee-hpec.org

Format Abstraction for Sparse Tensor Algebra Compilers

This paper shows how to build a sparse tensor algebra compiler that is agnostic to tensor formats (data layouts). We develop an interface that describes formats in terms of their capabilities and properties, and show how to build a modular code generator where new formats can be added as plugins. We then describe six implementations of the interface that compose to form the dense, CSR/CSF, COO, DIA, ELL, and HASH tensor formats and countless variants thereof. With these implementations at hand, our code generator can generate code to compute any tensor algebra expression on any combination of the aforementioned formats. To demonstrate our technique, we have implemented it in the taco tensor algebra compiler. Our modular code generator design makes it simple to add support for new tensor formats, and the performance of the generated code is competitive with hand-optimized implementations. Furthermore, by extending taco to support a wider range of formats specialized for different application and data characteristics, we can improve end-user application performance. For example, if input data is provided in the COO format, our technique allows computing a single matrix-vector multiplication directly with the data in COO, which is up to 3.6$\times$ faster than by first converting the data to CSR.Comment: Presented at OOPSLA 201

Distributed Triangle Counting in the Graphulo Matrix Math Library

Triangle counting is a key algorithm for large graph analysis. The Graphulo library provides a framework for implementing graph algorithms on the Apache Accumulo distributed database. In this work we adapt two algorithms for counting triangles, one that uses the adjacency matrix and another that also uses the incidence matrix, to the Graphulo library for server-side processing inside Accumulo. Cloud-based experiments show a similar performance profile for these different approaches on the family of power law Graph500 graphs, for which data skew increasingly bottlenecks. These results motivate the design of skew-aware hybrid algorithms that we propose for future work.Comment: Honorable mention in the 2017 IEEE HPEC's Graph Challeng

Algorithms and data structures for matrix-free finite element operators with MPI-parallel sparse multi-vectors

Traditional solution approaches for problems in quantum mechanics scale as $\mathcal O(M^3)$, where $M$ is the number of electrons. Various methods have been proposed to address this issue and obtain linear scaling $\mathcal O(M)$. One promising formulation is the direct minimization of energy. Such methods take advantage of physical localization of the solution, namely that the solution can be sought in terms of non-orthogonal orbitals with local support. In this work a numerically efficient implementation of sparse parallel vectors within the open-source finite element library deal.II is proposed. The main algorithmic ingredient is the matrix-free evaluation of the Hamiltonian operator by cell-wise quadrature. Based on an a-priori chosen support for each vector we develop algorithms and data structures to perform (i) matrix-free sparse matrix multivector products (SpMM), (ii) the projection of an operator onto a sparse sub-space (inner products), and (iii) post-multiplication of a sparse multivector with a square matrix. The node-level performance is analyzed using a roofline model. Our matrix-free implementation of finite element operators with sparse multivectors achieves the performance of 157 GFlop/s on Intel Cascade Lake architecture. Strong and weak scaling results are reported for a typical benchmark problem using quadratic and quartic finite element bases.Comment: 29 pages, 12 figure

Doctor of Philosophy

dissertationSparse matrix codes are found in numerous applications ranging from iterative numerical solvers to graph analytics. Achieving high performance on these codes has however been a significant challenge, mainly due to array access indirection, for example, of the form A[B[i]]. Indirect accesses make precise dependence analysis impossible at compile-time, and hence prevent many parallelizing and locality optimizing transformations from being applied. The expert user relies on manually written libraries to tailor the sparse code and data representations best suited to the target architecture from a general sparse matrix representation. However libraries have limited composability, address very specific optimization strategies, and have to be rewritten as new architectures emerge. In this dissertation, we explore the use of the inspector/executor methodology to accomplish the code and data transformations to tailor high performance sparse matrix representations. We devise and embed abstractions for such inspector/executor transformations within a compiler framework so that they can be composed with a rich set of existing polyhedral compiler transformations to derive complex transformation sequences for high performance. We demonstrate the automatic generation of inspector/executor code, which orchestrates code and data transformations to derive high performance representations for the Sparse Matrix Vector Multiply kernel in particular. We also show how the same transformations may be integrated into sparse matrix and graph applications such as Sparse Matrix Matrix Multiply and Stochastic Gradient Descent, respectively. The specific constraints of these applications, such as problem size and dependence structure, necessitate unique sparse matrix representations that can be realized using our transformations. Computations such as Gauss Seidel, with loop carried dependences at the outer most loop necessitate different strategies for high performance. Specifically, we organize the computation into level sets or wavefronts of irregular size, such that iterations of a wavefront may be scheduled in parallel but different wavefronts have to be synchronized. We demonstrate automatic code generation of high performance inspectors that do explicit dependence testing and level set construction at runtime, as well as high performance executors, which are the actual parallelized computations. For the above sparse matrix applications, we automatically generate inspector/executor code comparable in performance to manually tuned libraries

Polyhedral+Dataflow Graphs

This research presents an intermediate compiler representation that is designed for optimization, and emphasizes the temporary storage requirements and execution schedule of a given computation to guide optimization decisions. The representation is expressed as a dataflow graph that describes computational statements and data mappings within the polyhedral compilation model. The targeted applications include both the regular and irregular scientific domains. The intermediate representation can be integrated into existing compiler infrastructures. A specification language implemented as a domain specific language in C++ describes the graph components and the transformations that can be applied. The visual representation allows users to reason about optimizations. Graph variants can be translated into source code or other representation. The language, intermediate representation, and associated transformations have been applied to improve the performance of differential equation solvers, or sparse matrix operations, tensor decomposition, and structured multigrid methods

DOLFIN: Automated Finite Element Computing

We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness with efficient computation. Finite element variational forms may be expressed in near mathematical notation, from which low-level code is automatically generated, compiled and seamlessly integrated with efficient implementations of computational meshes and high-performance linear algebra. Easy-to-use object-oriented interfaces to the library are provided in the form of a C++ library and a Python module. This paper discusses the mathematical abstractions and methods used in the design of the library and its implementation. A number of examples are presented to demonstrate the use of the library in application code