Application-tailored Linear Algebra Algorithms: A search-based Approach

In this paper, we tackle the problem of automatically generating algorithms for linear algebra operations by taking advantage of problem-specific knowledge. In most situations, users possess much more information about the problem at hand than what current libraries and computing environments accept; evidence shows that if properly exploited, such information leads to uncommon/unexpected speedups. We introduce a knowledge-aware linear algebra compiler that allows users to input matrix equations together with properties about the operands and the problem itself; for instance, they can specify that the equation is part of a sequence, and how successive instances are related to one another. The compiler exploits all this information to guide the generation of algorithms, to limit the size of the search space, and to avoid redundant computations. We applied the compiler to equations arising as part of sensitivity and genome studies; the algorithms produced exhibit, respectively, 100- and 1000-fold speedups

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

Tiramisu: A Polyhedral Compiler for Expressing Fast and Portable Code

This paper introduces Tiramisu, a polyhedral framework designed to generate high performance code for multiple platforms including multicores, GPUs, and distributed machines. Tiramisu introduces a scheduling language with novel extensions to explicitly manage the complexities that arise when targeting these systems. The framework is designed for the areas of image processing, stencils, linear algebra and deep learning. Tiramisu has two main features: it relies on a flexible representation based on the polyhedral model and it has a rich scheduling language allowing fine-grained control of optimizations. Tiramisu uses a four-level intermediate representation that allows full separation between the algorithms, loop transformations, data layouts, and communication. This separation simplifies targeting multiple hardware architectures with the same algorithm. We evaluate Tiramisu by writing a set of image processing, deep learning, and linear algebra benchmarks and compare them with state-of-the-art compilers and hand-tuned libraries. We show that Tiramisu matches or outperforms existing compilers and libraries on different hardware architectures, including multicore CPUs, GPUs, and distributed machines.Comment: arXiv admin note: substantial text overlap with arXiv:1803.0041

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