The Tensor Algebra Compiler

Tensor and linear algebra is pervasive in data analytics and the physical sciences. Often the tensors, matrices or even vectors are sparse. Computing expressions involving a mix of sparse and dense tensors, matrices and vectors requires writing kernels for every operation and combination of formats of interest. The number of possibilities is infinite, which makes it impossible to write library code for all. This problem cries out for a compiler approach. This paper presents a new technique that compiles compound tensor algebra expressions combined with descriptions of tensor formats into efficient loops. The technique is evaluated in a prototype compiler called taco, demonstrating competitive performance to best-in-class hand-written codes for tensor and matrix operations

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

The Sparse Abstract Machine

We propose the Sparse Abstract Machine (SAM), an abstract machine model for targeting sparse tensor algebra to reconfigurable and fixed-function spatial dataflow accelerators. SAM defines a streaming dataflow abstraction with sparse primitives that encompass a large space of scheduled tensor algebra expressions. SAM dataflow graphs naturally separate tensor formats from algorithms and are expressive enough to incorporate arbitrary iteration orderings and many hardware-specific optimizations. We also present Custard, a compiler from a high-level language to SAM that demonstrates SAM's usefulness as an intermediate representation. We automatically bind from SAM to a streaming dataflow simulator. We evaluate the generality and extensibility of SAM, explore the performance space of sparse tensor algebra optimizations using SAM, and show SAM's ability to represent dataflow hardware.Comment: 18 pages, 17 figures, 3 table

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

Unified Framework for Finite Element Assembly

At the heart of any finite element simulation is the assembly of matrices and vectors from discrete variational forms. We propose a general interface between problem-specific and general-purpose components of finite element programs. This interface is called Unified Form-assembly Code (UFC). A wide range of finite element problems is covered, including mixed finite elements and discontinuous Galerkin methods. We discuss how the UFC interface enables implementations of variational form evaluation to be independent of mesh and linear algebra components. UFC does not depend on any external libraries, and is released into the public domain

Automated code generation for discontinuous Galerkin methods

A compiler approach for generating low-level computer code from high-level input for discontinuous Galerkin finite element forms is presented. The input language mirrors conventional mathematical notation, and the compiler generates efficient code in a standard programming language. This facilitates the rapid generation of efficient code for general equations in varying spatial dimensions. Key concepts underlying the compiler approach and the automated generation of computer code are elaborated. The approach is demonstrated for a range of common problems, including the Poisson, biharmonic, advection--diffusion and Stokes equations