In this paper we present a methodology for data accesses when solving batches of Tridiagonal and Pentadiagonal matrices that all share the same left-hand-side (LHS) matrix. The intended application is to the numerical solution of Partial Differential Equations via the finite-difference method, although the methodology is applicable more broadly. By only storing one copy of this matrix, a significant reduction in storage overheads is obtained, together with a corresponding decrease in compute time. Taken together, these two performance enhancements lead to an overall more efficient implementation over the current state of the art algorithms cuThomasBatch and cuPentBatch, allowing for a greater number of systems to be solved on a single GPU. We demonstrate the methodology in the case of the Diffusion Equation, Hyperdiffusion Equation, and the Cahn--Hilliard Equation, all in one spatial dimension. In this last example, we demonstrate how the method can be used to perform $2^{20}$ independent simulations of phase separation in one dimension. In this way, we build up a robust statistical description of the coarsening phenomenon which is the defining behavior of phase separation. We anticipate that the method will be of further use in other similar contexts requiring statistical simulation of physical systems.

翻译：在本文中,我们提出了一个数据存取方法,用于解决分批的Tridiagonal 和 Pentadiagonal 矩阵,这些矩阵都具有相同的左侧矩阵。 打算应用的方法是通过有限差异法解决部分差异方程式的数字解决方案, 尽管该方法适用范围更广。 我们通过存储该矩阵的复制件, 存储管理器的显著减少, 并相应减少计算时间 。 合并起来, 这两种性能增强导致在整体上更有效地实施对当前水平的艺术算法 cuThomasBatch 和 cuPentBatch 进行独立模拟, 使更多的系统能够在单一的GPU上解决。 我们展示了在分离、 超异化和 Cahn- Hillard Equation 的情况下采用的方法, 都集中在一个空间层面。 在最后一个例子中, 我们展示了如何使用这种方法在一个层面对阶段的阶段分离进行独立模拟。 通过这种方式, 我们构建了一个可靠的统计学描述, 将进一步使用类似的系统。