本篇論文開發了預處理 BiCGstab 算法的 GPU 平行化，該算法能夠減 少離子通道模擬運算中矩陣的條件數。 兩個使用的預處理是不完整 的 LU 分解(ILU)和對稱的連續過度鬆弛(SSOR)。 我們使用 CUSPARSE 中的子程序來精確求解 ILU 和 SSOR 預處理的下三角和上 三角線性系統。 此外，我們還開發了 GPU 內核來執行不精確的 SSOR 預處理。結果表明，使用預處理可以減少迭代次數，不精確的 SSOR 預處理在這些方法中具有最佳性能。

This thesis develops the GPU parallelization for the preconditioned BiCGstab algorithm which is able to reduce the condition numbers of matrices in ion channel simulations. Two used preconditioners are the incomplete LU factorizations(ILU) and symmetric successive over-relaxation(SSOR). We use the subroutines in CUS- PARSE to exactly solve the lower and upper triangular linear systems for ILU and SSOR preconditioner. Moreover,we also develop a GPU kernel to perform the inexact SSOR preconditioner.The results show that the use of preconditioner can reduce the number of iteration, and the inexact SSOR preconditioner has the best performance among these methods.

Abstract-------------------------i
Acknowledgement-------------------------ii
Content-------------------------iii
Introduction-------------------------1
Ion channel-------------------------2
Preconditioner techniques---------------------5
GPU implementation --------------------------11
Conclusion------------------------------------18
Reference-------------------------------------19

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