The solution of linear systems of equations is a central task in a number of scientific and engineering applications. In many cases the solution of linear systems may take most of the simulation time thus representing a major bottleneck in the further development of scientific and technical software. For large scale simulations, nowadays accounting for several millions or even billions of unknowns, it is quite common to resort to preconditioned iterative solvers for exploiting their low memory requirements and, at least potential, parallelism. Approximate inverses have been shown to be robust and effective preconditioners in various contexts. In this work, we show how adaptive Factored Sparse Approximate Inverse (aFSAI), characterized by a very high degree of parallelism, can be successfully implemented on a distributed memory computer equipped with GPU accelerators. Taking advantage of GPUs in adaptive FSAI set-up is not a trivial task, nevertheless we show through an extensive numerical experimentation how the proposed approach outperforms more traditional preconditioners and results in a close-to-ideal behavior in challenging linear algebra problems.

线性方程组的解决方案是许多科学和工程应用程序中的中心任务。在许多情况下，线性系统的解决方案可能会花费大部分仿真时间，因此代表了科学和技术软件进一步开发中的主要瓶颈。对于大规模仿真而言，如今已解决了数百万甚至数十亿个未知数，很常见的是使用预处理的迭代求解器来利用其低内存需求以及至少潜在的并行性。在各种情况下，近似逆已被证明是强大且有效的预处理器。在这项工作中，我们展示了以非常高的并行度为特征的自适应因子稀疏近似逆（aFSAI），可以在配备GPU加速器的分布式存储计算机上成功实现。在自适应FSAI设置中利用GPU并不是一件容易的事，尽管如此，我们还是通过广泛的数值实验证明了所提出的方法在性能上比传统的前置条件更好，并且在挑战线性代数问题时表现出接近理想的行为。