Efficient Code Generation and Performance Optimization for Matrix-Free Finite Element Methods on Hybrid Tetrahedral Grids

Core Concepts

This paper presents a code generator that automatically generates highly optimized, extreme-scalable, matrix-free finite element operators on hybrid tetrahedral grids. The generated operators achieve high node-level performance through various techniques including tabulation, relocation of loop invariants, and inter-element vectorization.

Abstract

The paper introduces the HyTeG Operator Generator (HOG), a unified pipeline that automatically generates matrix-free finite element compute kernels from a symbolic description of a differential operator and respective finite element spaces. The key contributions are:

Development and analysis of efficient loop patterns that exploit the local structure of the underlying tetrahedral grid to enhance cache locality and arithmetic intensity.
Comprehensive performance analysis guiding the generation of optimized compute kernels through resource-based performance models. Optimizations include tabulation, relocation of loop invariants, and inter-element vectorization.
Detailed node-level performance analysis demonstrating the generated operators achieve a throughput of 1.3 to 2.1 GDoF/s, reaching up to 62% of the machine's peak performance on a 36-core Intel Ice Lake socket.
Demonstration of the generated operators' extreme scalability by solving a curl-curl problem with more than a trillion degrees of freedom on 21,504 processes in less than 50 seconds as part of a full multigrid solver.

Stats

The simulation of convection in the Earth's mantle with a global resolution of 1 km requires about a trillion (10^12) elements. Assuming one degree of freedom per element and a 7-point stencil, the memory requirement for the system matrix would be 56 TB, which is extremely optimistic.

Quotes

"Matrix-free finite element methods [10, 28, 30, 36, 40] address two main limitations faced by conventional approaches that follow the assemble-solve cycle. Matrix-vector operations using standard sparse storage formats are typically bandwidth-limited on state-of-the-art architectures [32]. The characteristic machine balance, i.e., the ratio of memory bandwidth (B/s) to performance (FLOP/s) of current hardware favors on-the-fly evaluation that reduces bandwidth-pressure at the cost of additional arithmetic operations [25, 31, 32, 35]."
"Expecting a memory requirement one order of magnitude higher than (1.1), storing the system matrix becomes infeasible, even on the majority of the largest available supercomputers. Since most iterative linear solvers only require the results of matrix-vector operations but no explicit access to the matrix entries, matrix-free methods enable the solution of linear systems with trillions (10^12) of unknowns [16, 28] that could not be realized with standard sparse assembly."

Key Insights Distilled From

Code Generation and Performance Engineering for Matrix-Free Finite Element Methods on Hybrid Tetrahedral Grids

by Fabi... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08371.pdf

Deeper Inquiries

How can the code generator be extended to support a wider range of discretizations and PDEs beyond the current focus on extreme scalability and high node-level performance

To extend the code generator to support a wider range of discretizations and PDEs beyond extreme scalability and high node-level performance, several enhancements can be considered:

Additional Finite Element Spaces: Introduce support for a broader range of finite element spaces, such as higher-order Lagrange elements, discontinuous Galerkin methods, or mixed finite element formulations. This would involve incorporating the necessary basis functions, integration rules, and interpolation techniques into the code generation process.

Generalized Weak Forms: Enable the code generator to handle a variety of weak forms beyond the specific set considered in the current implementation. This could involve defining a more flexible input format for specifying the weak form of the PDE, allowing users to input custom equations and boundary conditions.

Domain-Specific Optimizations: Implement domain-specific optimizations tailored to different types of PDEs. For example, for problems with specific symmetries or structures, custom optimizations can be applied to exploit these characteristics and improve the efficiency of the generated operators.

Integration with External Libraries: Enhance the code generator to interface with external libraries or tools that provide specialized functionality for certain types of discretizations or PDEs. This could involve integrating with libraries for specific types of finite element methods or solvers to leverage existing capabilities.

Parameterization and Customization: Introduce parameters and customization options in the code generator to allow users to specify additional details about the discretization, solver settings, or optimization preferences. This would provide flexibility for users to tailor the generated code to their specific requirements.

What are the potential limitations or drawbacks of the tight coupling between discretization and solver in the matrix-free approach compared to the discretization-solver split implemented in frameworks like FEniCS

The tight coupling between discretization and solver in the matrix-free approach offers several advantages but also presents potential limitations compared to the discretization-solver split implemented in frameworks like FEniCS:
Limitations:

Flexibility: The tight coupling may limit the flexibility to interchange different discretizations or solvers easily. In contrast, a more modular approach allows for greater flexibility in swapping out components without affecting the entire workflow.

Complexity: The interdependence between the discretization and solver components can lead to increased complexity in the codebase. Changes in one part may have cascading effects on other components, making maintenance and extension more challenging.

Portability: The tight coupling may restrict the portability of the code to different architectures or environments. A more modular design can facilitate porting the code to diverse platforms with minimal modifications.

Drawbacks:

Performance Overhead: The tight integration may introduce additional overhead in terms of memory usage or computational cost. Decoupling the discretization and solver components can sometimes lead to more optimized implementations for each part.

Scalability: Depending on the specific implementation, the tight coupling may impact the scalability of the code. In some cases, a more modular design can offer better scalability by allowing for parallel development and optimization of individual components.

Maintenance: Changes or updates to either the discretization or solver may require modifications in multiple places within a tightly coupled system, potentially increasing the maintenance burden.

What other techniques or optimizations could be explored to further improve the performance and scalability of the generated matrix-free operators, especially for more compute-intensive operators beyond the low-order examples presented in the paper

To further improve the performance and scalability of the generated matrix-free operators, especially for more compute-intensive operators, the following techniques and optimizations could be explored:

Advanced Parallelization: Implement more sophisticated parallelization strategies, such as task-based parallelism or hybrid parallel models combining shared and distributed memory paradigms. This can enhance scalability on modern multi- and many-core architectures.

Dynamic Load Balancing: Introduce dynamic load balancing techniques to distribute computational workloads efficiently across processing units, ensuring optimal resource utilization and minimizing idle time.

Adaptive Mesh Refinement: Incorporate adaptive mesh refinement algorithms to dynamically adjust the mesh resolution based on solution characteristics, focusing computational resources where they are most needed and improving accuracy and efficiency.

Hardware-Specific Optimizations: Explore hardware-specific optimizations, such as utilizing specialized instructions sets (e.g., AVX-512) or memory hierarchies to maximize performance on specific architectures.

Machine Learning Techniques: Investigate the use of machine learning algorithms to optimize code generation, performance tuning, or adaptive mesh refinement, leveraging data-driven approaches to enhance efficiency and scalability.

By incorporating these advanced techniques and optimizations, the generated matrix-free operators can achieve higher performance and scalability for a broader range of compute-intensive applications.