标题： 加权内积用于GMRES和GMRES-DR 显示英文标题

标题： Weighted Inner Products for GMRES and GMRES-DR

摘要： 通过在每次重启时使用变化的加权内积，可以显著改善某些问题的重启GMRES方法的收敛性。 这种加权如何影响收敛性，何时是有用的？ 我们表明，加权内积可以通过两种不同的方式有所帮助：当系数矩阵具有局部化特征向量时，加权可以使重启的GMRES专注于那些 otherwise 会减缓收敛的特征值；对于一般问题，加权可以打破重启GMRES通常陷入的循环收敛模式。 从微分方程导出的矩阵的特征向量通常不是局部化的，因此限制了加权的影响。 对于这些问题，将离散余弦变换纳入内积中可以显著改善GMRES的收敛性，我们称这种方法为W-GMRES-DCT。 将加权与通过GMRES-DR进行的特征值消去相结合也可以给出有效的解决方案。 显示英文摘要

摘要： The convergence of the restarted GMRES method can be significantly improved, for some problems, by using a weighted inner product that changes at each restart. How does this weighting affect convergence, and when is it useful? We show that weighted inner products can help in two distinct ways: when the coefficient matrix has localized eigenvectors, weighting can allow restarted GMRES to focus on eigenvalues that otherwise slow convergence; for general problems, weighting can break the cyclic convergence pattern into which restarted GMRES often settles. The eigenvectors of matrices derived from differential equations are often not localized, thus limiting the impact of weighting. For such problems, incorporating the discrete cosine transform into the inner product can significantly improve GMRES convergence, giving a method we call W-GMRES-DCT. Integrating weighting with eigenvalue deflation via GMRES-DR also can give effective solutions.