Multigrid methods are fast and scalable numerical solvers for partial differential equations (PDEs) that possess a large design space for implementing their algorithmic components. Code generation approaches allow formulating multigrid methods on a higher level of abstraction that can then be used to derive a problem- and hardware-specific solutions. Since these problems have a considerable implementation variability, it is crucial to investigate a general mapping of core components in multigrid methods to the target software. With SYCL there exists a high-level C++ abstraction layer that is capable of targeting a multitude of architectures. We contribute a general way to map multigrid components to SYCL functionality and provide a performance evaluation for specific algorithmic components.

In scientific computing, unstructured meshes are a crucial foundation for the simulation of real-world physical phenomena. Compared to regular grids, they allow resembling the computational domain with a much higher accuracy, which in turn leads to more efficient computations. There exists a wealth of supporting libraries and frameworks that aid programmers with the implementation of applications working on such grids, each built on top of existing parallelization technologies. However, many approaches require the programmer to introduce a different programming paradigm into their application or provide different variants of the code. SYCL is a new programming standard providing a remedy to this dilemma by building on standard C++ 17 with its so-called single-source approach: Programmers write standard C++ code and expose parallelism using C++ 17 keywords. The application is then transformed into a concrete implementation by the SYCL implementation. By encapsulating the OpenCL ecosystem, different SYCL implementations enable not only the programming of CPUs but also of heterogeneous platforms such as GPUs or other devices. For the first time, this paper showcases a SY CL-based solver for the nodal Discontinuous Galerkin method for Maxwell's equations on unstructured meshes. We compare our solution to a previous C-based implementation with respect to programmability and performance on heterogeneous platforms.

The exploration of FPGAs as accelerators for scientific simulations has so far mostly been focused on small kernels of methods working on regular data structures, for example in the form of stencil computations for finite difference methods. In computational sciences, often more advanced methods are employed that promise better stability, convergence, locality and scaling. Unstructured meshes are shown to be more effective and more accurate, compared to regular grids, in representing computation domains of various shapes. Using unstructured meshes, the discontinuous Galerkin method preserves the ability to perform explicit local update operations for simulations in the time domain. In this work, we investigate FPGAs as target platform for an implementation of the nodal discontinuous Galerkin method to find time-domain solutions of Maxwell's equations in an unstructured mesh. When maximizing data reuse and fitting constant coefficients into suitably partitioned on-chip memory, high computational intensity allows us to implement and feed wide data paths with hundreds of floating point operators. By decoupling off-chip memory accesses from the computations, high memory bandwidth can be sustained, even for the irregular access pattern required by parts of the application. Using the Intel/Altera OpenCL SDK for FPGAs, we present different implementation variants for different polynomial orders of the method. In different phases of the algorithm, either computational or bandwidth limits of the Arria 10 platform are almost reached, thus outperforming a highly multithreaded CPU implementation by around 2x.

The purpose of a domain-specific language (DSL) is to enable the application programmer to specify a problem, or an abstract algorithm description, in his/her domain of expertise without being burdened by implementation details. The ideal scenario is that the implementation detail is added in an automatic process of program translation and code generation. The approach of domain-specific program generation has lately received increasing attention in the area of computational science and engineering. In this paper, we introduce the new code generation framework Athariac. Its goal is to support the quick implementation of a language processing and program optimization platform for a given DSL based on stepwise term rewriting. We demonstrate the framework's use on our DSL ExaSlang for the specification and optimization of multigrid solvers. On this example, we provide evidence of Athariac's potential for making domain-specific software engineering more productive.