AI 中文总结
本文针对GMRESR收敛慢的问题，提出以HBGMRES为内层循环的HBGMRESR方法，通过真实数据测试验证其性能优于GMRESR与HBGMRES。
AI 中文摘要
重球GMRES（HBGMRES）方法是结合了重启GMRES与优化领域中重球方法的Krylov子空间线性系统求解方法之一。HBGMRES不仅保留了重启GMRES在限制内存使用、控制正交化开销方面的优势，还能解决重启GMRES收敛速度慢的问题。另一类Krylov子空间方法是GMRESR，它以GCR作为外层算法，GMRES的若干步作为内层方法。与HBGMRES相比，GMRESR在GMRES生成的新搜索向量及GCR中所有已保留的搜索向量上给出近似最优解。尽管GMRESR性能优于HBGMRES，但它存在与GMRES及重启GMRES类似的收敛速度慢的问题。受HBGMRES启发，我们提出了重球GMRESR方法（HBGMRESR），该方法采用HBGMRES作为内层循环，替代原有的GMRES，以弥补损失的收敛速度，同时仍保留GMRESR的优势——在Krylov子空间的特定部分完成全局最小化。通过真实数据的数值测试，验证了新方法相比GMRESR和HBGMRES的优越性。
英文摘要
The heavy ball GMRES (HBGMRES) method is one of Krylov subspace methods for linear systems combined with the restarted GMRES and the heavy ball method which is applied in optimization. HBGMRES not only keeps benefit of the restarted GMRES in limiting memory usage and controlling orthogonalization cost, but also is able to cover up the slow convergence problem in the restarted GMRES. Another type of Krylov subspace methods is GMRESR which consists of GCR as the outer algorithm and certain steps of GMRES as an inner method. Compared with HBGMRES, GMRESR gives the approximately optimal solution over the new search vector gained from GMRES and all previously kept search vectors in GCR. Even though GMRESR performs better than HBGMRES, it has slow convergence which is similar to GMRES and the restarted GMRES. Inspired by HBGMRES, we present the heavy ball GMRESR method (HBGMRESR) by using HBGMRES as the inner loop instead of using GMRES to salvage the lost convergence speed while still keeping the benefit of GMRESR that in the sense of a global minimization over some specific part of the Krylov subspace is done. Numerical tests on real data are presented to demonstratee the superiority of the new methods over GMRESR and HBGMRES.
