Florian Rathgeber

Firedrake: Automating the Finite Element Method by Composing Abstractions

Firedrake is a new tool for automating the numerical solution of partial differential equations. Firedrake adopts the domain-specific language for the finite element method of the FEniCS project, but with a pure Python runtime-only implementation centred on the composition of several existing and new abstractions for particular aspects of scientific computing. The result is a more complete separation of concerns which eases the incorporation of separate contributions from computer scientists, numerical analysts and application specialists. These contributions may add functionality, or improve performance. Firedrake benefits from automatically applying new optimisations. This includes factorising mixed function spaces, transforming and vectorising inner loops, and intrinsically supporting block matrix operations. Importantly, Firedrake presents a simple public API for escaping the UFL abstraction. This allows users to implement common operations that fall outside pure variational formulations, such as flux-limiters.

PyOP2: A High-Level Framework for Performance-Portable Simulations on Unstructured Meshes

Emerging many-core platforms are very difficult to program in a performance portable manner whilst achieving high efficiency on a diverse range of architectures. We present work in progress on PyOP2, a high-level embedded domain-specific language for mesh-based simulation codes that executes numerical kernels in parallel over unstructured meshes. Just-in-time kernel compilation and parallel scheduling are delayed until runtime, when problem-specific parameters are available. Using generative metaprogramming, performance portability is achieved, while details of the parallel implementation are abstracted from the programmer. PyOP2 kernels for finite element computations can be generated automatically from equations given in the domain-specific Unified Form Language. Interfacing to the multi-phase CFD code Fluidity through a very thin layer on top of PyOP2 yields a general purpose finite element solver with an input notation very close to mathematical formulae. Preliminary performance figures show speedups of up to 3.4× compared to Fluiditys built-in solvers when running in parallel.

Cross-Loop Optimization of Arithmetic Intensity for Finite Element Local Assembly

The numerical solution of partial differential equations using the finite element method is one of the key applications of high performance computing. Local assembly is its characteristic operation. This entails the execution of a problem-specific kernel to numerically evaluate an integral for each element in the discretized problem domain. Since the domain size can be huge, executing efficient kernels is fundamental. Their op- timization is, however, a challenging issue. Even though affine loop nests are generally present, the short trip counts and the complexity of mathematical expressions make it hard to determine a single or unique sequence of successful transformations. Therefore, we present the design and systematic evaluation of COF- FEE, a domain-specific compiler for local assembly kernels. COFFEE manipulates abstract syntax trees generated from a high-level domain-specific language for PDEs by introducing domain-aware composable optimizations aimed at improving instruction-level parallelism, especially SIMD vectorization, and register locality. It then generates C code including vector intrinsics. Experiments using a range of finite-element forms of increasing complexity show that significant performance improvement is achieved.

Performance-Portable Finite Element Assembly Using PyOP2 and FEniCS

We describe a toolchain that provides a fully automated compilation pathway from a finite element domain-specific language to low-level code for multicore and GPGPU platforms. We demonstrate that the generated code exceeds the performance of the best available alternatives, without requiring manual tuning or modification of the generated code. The toolchain can easily be integrated with existing finite element solvers, providing a means to add performance portable methods without having to rebuild an entire complex implementation from scratch.