Yet Another Tensor Toolbox for Discontinuous Galerkin Methods and Other Applications

Abstract

The numerical solution of partial differential equations is at the heart of many grand challenges in supercomputing. Solvers based on high-order discontinuous Galerkin (DG) discretisation have been shown to scale on large supercomputers with excellent performance and efficiency if the implementation exploits all levels of parallelism and is tailored to the specific architecture. However, every year new supercomputers emerge and the list of hardware-specific considerations grows simultaneously with the list of desired features in a DG code. Thus, we believe that a sustainable DG code needs an abstraction layer to implement the numerical scheme in a suitable language. We explore the possibility to abstract the numerical scheme as small tensor operations, describe them in a domain-specific language (DSL) resembling the Einstein notation, and to map them to small General Matrix-Matrix Multiplication routines. The compiler for our DSL implements classic optimisations that are used for large tensor contractions, and we present novel optimisation techniques such as equivalent sparsity patterns and optimal index permutations for temporary tensors. Our application examples, which include the earthquake simulation software SeisSol, show that the generated kernels achieve over 50% peak performance of a recent 48-core Skylake system while the DSL considerably simplifies the implementation.