在本文中，我們提出了一個快速的線性系統解，用以解偏微分方程。此偏微分方程主要由Poisson方程的變形以及Dirichlet邊界條件所組成的。我們使用preconditioned generalized minimal residual iterative method(通常簡稱GMRes)作為線性系統的主要算法以及使用fast Poisson solver作預處理。
為了加速運算的性能，我們利用GPU來進行GMRes的平行計算。此計算平台，CPU為I7-4770，記憶體為8G，而GPU則是Nvidia GTX Titian Black。數值實現結果顯示，相較於單核心CPU計算，利用GPU做平行計算可以取得14倍左右的加速效能，而測試矩陣的大小為2的13次方乘以2的13次方。計算結果得出的結論為，GPU的計算具有有效性以及可擴展性。

In this paper, we propose a fast linear system solver for the numerical solutions to partial differential equations mainly consisting of Poisson’s equation with Dirichlet boundary conditions. We pick up the preconditioned generalized minimal residual iterative method (usually abbreviated GMRes) to be linear system solver and the fast
Poisson solver to be the preconditioner.
In order to speed up the computation performance, we implement parallel computing of the GMRes algorithm on the GPU architecture. Our computing platform, CPU is I7 - 4770, RAM is 8G, and GPU is Nvidia GTX TITAN Black. The numerical implementation of the linear system solver for solving the partial differential equation shows that the performance on GPU architecture is about 14 times faster than that on non-parallel architecture for the case of test matrices of sizes 2^13-by-2^13. The numerical results conclude that the GPU computing is efficient and scalable.

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