Firedrake: automating the finite element method by composing abstractions (1501.01809v3)

Abstract: 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.

• The paper’s key contribution is automating the finite element method by composing high-level abstractions, simplifying complex PDE problem formulation.
• It integrates established tools like UFL, FIAT, and PETSc with PyOP2 to optimize kernel execution and enable efficient parallel computations.
• Experimental results demonstrate superior scalability and performance in assembling and solving mixed function PDE systems compared to traditional methods.

Overview of "Firedrake: Automating the Finite Element Method by Composing Abstractions"

The paper "Firedrake: Automating the Finite Element Method by Composing Abstractions" introduces Firedrake, a tool designed to facilitate the numerical solution of partial differential equations (PDEs) using the finite element method (FEM). Firedrake leverages a runtime-only Python implementation, centered around various abstractions to simplify collaboration among computer scientists, numerical analysts, and domain experts. This enhances functionality and performance without requiring deep cross-discipline expertise.

Firedrake builds upon the FEniCS project, utilizing its domain-specific language while presenting a higher level of abstraction through PyOP2, an embedded DSL for mesh data operations. This abstraction supports parallel execution, enabling efficient mapping between mathematical formulations and computational implementations.

Key Contributions

1. Abstraction and Automation: Firedrake exemplifies the decomposition of the finite element method into distinct automated abstractions using PyOP2. This approach allows separation between local discretization and parallel execution, which is crucial for handling complex PDE solves efficiently. By incorporating PyOP2, Firedrake facilitates high-performance computations and compact code bases through refined data management.
2. Integration of Existing Tools: The system integrates several established tools, including UFL, FIAT, and PETSc. The use of UFL allows expression of PDEs in a high-level mathematical syntax, while FIAT supports a broad range of finite elements, facilitating versatility in element-specific computations. PETSc delivers robust solver capabilities, ensuring Firedrake's adaptability to various computational challenges.
3. Kernel Optimization and Execution: COFFEE, an integral component, provides kernel optimizations like vectorization and loop transformations. These optimizations accommodate the execution of high-performance, platform-specific code, ensuring computational efficiency.
4. Handling Complex Systems: Firedrake's robust handling of mixed function spaces and assembly of systems involving nested matrices highlights its capability in solving complex PDE systems effectively. PyOP2's support for efficiently executing these mixed operations enhances performance considerably, as seen in comparisons with DOLFIN.
5. Scalability: The paper presents comprehensive experiments demonstrating Firedrake's efficacy across various orders of polynomial basis functions and mesh sizes. The results indicate superior performance in terms of assembly times and solver efficiency compared to comparable tools, especially in mixed function scenarios. Firedrake exhibits excellent strong and weak scalability, benefiting from efficient parallel execution facilitated by PyOP2.

Implications and Future Directions

The abstraction model leveraged by Firedrake provides significant implications for future developments in scientific computing:

• Efficiency in Research: By lowering barriers to entry, Firedrake enables researchers to focus more on their domain-specific analyses rather than on the underlying computational complexities. This promises potential advances in fields that extensively use FEM, such as fluid dynamics and structural analysis.
• Extensibility: The modular abstraction approach adopted by Firedrake supports extensibility, allowing new functionalities or optimizations to be integrated without overhauling the existing framework.
• Potential for Hybrid Architectures: Although current results focus on MPI-driven CPU backends, the design paves the way for extensions to hybrid parallel architectures involving accelerators, aligning with evolving hardware trends.
• Adjoint and Gradient Computations: Integration with tools like Firedrake-adjoint showcases how symbolic abstractions can enable automated derivation of adjoint models, assisting in optimization and control problems associated with PDEs.

In summary, Firedrake successfully demonstrates a high level of abstraction in automating the finite element method, providing both theoretical insights and practical solutions for scalable and maintainable scientific computations. The approach represents a significant stride in computational PDE solving, merging state-of-the-art techniques with accessible high-level programming operations.