Efficient mesh management in Firedrake using PETSc-DMPlex

The use of composable abstractions allows the application of new and established algorithms to a wide range of problems while automatically inheriting the benefits of well-known performance optimisations. This work highlights the composition of the PETSc DMPlex domain topology abstraction with the Firedrake automated finite element system to create a PDE solving environment that combines expressiveness, flexibility and high performance. We describe how Firedrake utilises DMPlex to provide the indirection maps required for finite element assembly, while supporting various mesh input formats and runtime domain decomposition. In particular, we describe how DMPlex and its accompanying data structures allow the generic creation of user-defined discretisations, while utilising data layout optimisations that improve cache coherency and ensure overlapped communication during assembly computation.

Unstructured Overlapping Mesh Distribution in Parallel

We present a simple mathematical framework and API for parallel mesh and data distribution, load balancing, and overlap generation. It relies on viewing the mesh as a Hasse diagram, abstracting away information such as cell shape, dimension, and coordinates. The high level of abstraction makes our interface both concise and powerful, as the same algorithm applies to any representable mesh, such as hybrid meshes, meshes embedded in higher dimension, and overlapped meshes in parallel. We present evidence, both theoretical and experimental, that the algorithms are scalable and efficient. A working implementation can be found in the latest release of the PETSc libraries.

Benchmarking mixed-mode PETSc performance on high-performance architectures

The trend towards highly parallel multi-processing is ubiquitous in all modern computer architectures, ranging from handheld devices to large-scale HPC systems; yet many applications are struggling to fully utilise the multiple levels of parallelism exposed in modern high-performance platforms. In order to realise the full potential of recent hardware advances, a mixed-mode between shared-memory programming techniques and inter-node message passing can be adopted which provides high-levels of parallelism with minimal overheads. For scientific applications this entails that not only the simulation code itself, but the whole software stack needs to evolve. In this paper, we evaluate the mixed-mode performance of PETSc, a widely used scientific library for the scalable solution of partial differential equations. We describe the addition of OpenMP threaded functionality to the library, focusing on sparse matrix-vector multiplication. We highlight key challenges in achieving good parallel performance, such as explicit communication overlap using task-based parallelism, and show how to further improve performance by explicitly load balancing threads within MPI processes. Using a set of matrices extracted from Fluidity, a CFD application code which uses the library as its linear solver engine, we then benchmark the parallel performance of mixed-mode PETSc across multiple nodes on several modern HPC architectures. We evaluate the parallel scalability on Uniform Memory Access (UMA) systems, such as the Fujitsu PRIMEHPC FX10 and IBM BlueGene/Q, as well as a Non-Uniform Memory Access (NUMA) Cray XE6 platform. A detailed comparison is performed which highlights the characteristics of each particular architecture, before demonstrating efficient strong scalability of sparse matrix-vector multiplication with significant speedups over the pure-MPI mode.