Tridiagonal matrix inversion is an important operation with many applications. It arises frequently in solving discretized one-dimensional elliptic partial differential equations, and forms the basis for many algorithms for block tridiagonal matrix inversion for discretized PDEs in higher-dimensions. In such systems, this operation is often the scaling bottleneck in parallel computation. In this paper, we derive a hybrid multigrid-Thomas algorithm designed to efficiently invert tridiagonal matrix equations in a highly-scalable fashion in the context of time evolving partial differential equation systems. We decompose the domain between processors, using multigrid to solve on a grid consisting of the boundary points of each processor’s local domain. We then reconstruct the solution on each processor using a direct solve with the Thomas algorithm. This algorithm has the same theoretical optimal scaling as cyclic reduction and recursive doubling. We use our algorithm to solve Poisson’s equation as part of the spatial discretization of a time-evolving PDE system. Our algorithm is faster than cyclic reduction per inversion and retains good scaling efficiency to twice as many cores.

三对角矩阵求逆是具有许多应用的重要运算。它经常出现在求解离散化的一维椭圆偏微分方程中，并形成了许多用于高维离散化偏微分方程的块三对角矩阵求逆算法的基础。在此类系统中，此操作通常是并行计算中的扩展瓶颈。在本文中，我们推导出一种混合多重网格-Thomas 算法，旨在在时间演化偏微分方程系统的背景下以高度可扩展的方式有效地反转三对角矩阵方程。我们分解处理器之间的域，使用多重网格在由每个处理器本地域的边界点组成的网格上求解。然后我们使用 Thomas 算法的直接求解在每个处理器上重建解。该算法具有与循环归约和递归加倍相同的理论最优缩放。我们使用我们的算法来求解泊松方程，作为时间演化 PDE 系统空间离散化的一部分。我们的算法比每次反演的循环减少更快，并且保持良好的缩放效率到两倍的内核。