通过求解矩形上的极小极大问题与消元法改进GMRES的多项式界并加速其收敛
Improved Polynomial Bounds and Acceleration of GMRES by Solving a min-max Problem on Rectangles, and by Deflating
AI总结：
该研究将GMRES的多项式收敛界推广至多种预条件及消元、加权情形，通过复平面矩形极小极大问题求解并结合特定消元空间与预条件子优化界、加速收敛，用数值算例验证了结果。
摘要：
本文研究左、右及分裂预条件GMRES的多项式收敛界，涵盖线性系统Ax=b的加权GMRES与消元型GMRES情形，特别讨论了A为正定矩阵的情况。研究将经典多项式界推广至上述情形，并将界的求解转化为复平面上矩形区域的极小极大问题，对比了多种求解思路。新的界可通过特定消元空间与预条件子进一步优化，进而加速GMRES收敛，数值算例验证了所得结果。
英文摘要：
Polynomial convergence bounds are considered for left, right, and split preconditioned GMRES. They include the cases of Weighted and Deflated GMRES for a linear system Ax = b. In particular, the case of positive definite A is considered. The well-known polynomial bounds are generalized to the cases considered, and then reduced to solving a min-max problem on rectangles on the complex plane. Several approaches are considered and compared. The new bounds can be improved by using specific deflation spaces and preconditioners. This in turn accelerates the convergence of GMRES. Numerical examples illustrate the results obtained.
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