Rasterization-based discontinuous integration on the GPU

Abstract

In this paper, we propose a method based on rasterization in computer graphics for the integration of discontinuous functions accurately and efficiently on the GPU. The proposed method takes advantage of the massively parallel architecture of GPUs via a modified graphics pipeline, where discontinuities are encoded through geometric primitives and processed in the rasterization stage, and multisample anti-aliasing (MSAA) is used to improve integration accuracy across subdomain interfaces. We present both hardware- and software-based pipelines and compare their performance in terms of accuracy as well as computational effort on modern GPU hardware with adaptive quadrature on the CPU. For the tested hardware, at approximately comparable median relative error, the hardware pipeline is up to around $100\text{x}$ faster than adaptive quadrature, while the single- and double-precision software pipelines achieve up to $840\text{x}$ and $1200\text{x}$ speedups compared to their adaptive quadrature counterparts. We use the finite cell method as model problem, where cutcells represent discontinuous integrands at the interface of the physical and fictitious domains and demonstrate the performance of the proposed method using numerical benchmarks. The end-to-end finite cell solution runtime is reduced by a factor of $8.3$ to $9.5$ in the single-precision pipeline and by a factor of $62$ to $72$ in the double-precision pipeline at the prescribed integration settings. Therefore, results indicate that the rasterization-based integration method is a promising alternative for the accurate and efficient integration of discontinuous functions in demanding applications on the GPU.