Robust Krylov Subspace Methods Based on Multiple Lanczos Procedure

Krylov子空间方法在科学和工程应用中广泛应用于大尺度线性系统的迭代求解。标准的Krylov子空间方法是基于单启动Lanczos程序（SSLP）或Arnoldi程序开发的。自多次启动Lanczos程序（MSLP）被发明以来，MSLP对SSLP的鲁棒性已被各种应用证明。从MSLP出发，导出了求解多右手边线性系统的块Krylov子空间方法。然而，对于具有单右手边的系统，从MSLP推导出Krylov子空间方法的研究很少。当然，多重右手边包括单右手边作为特例。然而，PI期望当我们专注于单一的右侧情况时，可以获得更好的方法。Tony Chan和pii最近在这个方向上进行了首次尝试，并分别开发了ML(n)BiCGand ML(n)BiCGSTAB方法。该项目延续了这一趋势，旨在(i)将现有的标准方法从sslp推广到MSLP；（ii）减少ML(n)BiCGSTAB的存储需求；（iii）提高ML(n)BiCG的鲁棒性。这是实现PI开发并行Krylov子空间方法的长期目标的下一步。PI对ML(n)BiCGSTAB和无转置MSLP算法的初步研究证明了所提出目标的可行性。实现目标所采用的方法是基于一定的方法来适应从SSLP推导BiCG和从BiCG推导BiCGSTAB(n)。结果将在工业数据库中进行广泛的测试，例如，Harwell-Boeing稀疏矩阵集合。如果成功，这项研究将具有重大意义，因为它将显著提高现有方法的鲁棒性，从而显著降低计算时间和成本。它还允许开发更健壮的并行Krylov子空间方法，例如，通过将操作重新调度为并行。本研究也将为课堂学习和论文研究提供理想的选题。此外，该PI曾参与怀俄明州大学化学与石油工程的研究，从燃烧废气流中去除氮氧化物。该项目的最终方法将促进未来的合作，因为届时将有更强大的算法可用于解决由合作研究产生的PDE模型。

Krylov subspace methods are widely used for the iterative solution of largesparse linear systems that arise in scientific and engineering applications.The standard Krylov subspace methods have been developed based on either thesingle starting Lanczos procedure (SSLP) or Arnoldi procedure.Since the multiple starting Lanczos procedure (MSLP) was invented,the superior robustness of MSLP to SSLP has been demonstrated through a variety of applications. Block Krylov subspace methods for solving linear systems with multiple right-hand sides have been derived from MSLP. However, there has been rare study of deriving Krylov subspace methods from MSLP for systems with a single right-hand side. Of course, multiple right-hand sides includes the single right-handside as its special case. The PI, however, expects that finer methods can beobtained when we focus on the single right-hand side case. Tony Chan and the PIrecently made the first attempt in this direction and methods named ML(n)BiCGand ML(n)BiCGSTAB, respectively, were developed. This project continues thistrend and aims at (i) generalizing the existing standard methods from SSLPto MSLP; (ii) reducing the storage requirement of ML(n)BiCGSTAB; (iii) improving the robustness of ML(n)BiCG. This is the next step toward attainment of the PI's long-term goal of developing parallel Krylov subspace methods. The feasibility of the proposed aims have been demonstrated by the PI's preliminary studies of ML(n)BiCGSTAB and of a transpose-free MSLP algorithm. The approaches employed to attain the aims are based on certain ways of adapting the derivation of BiCG from SSLP andthe derivation of the BiCGSTAB(n) from BiCG.The results will be tested extensively with industrial databases, for example,the Harwell-Boeing Sparse Matrix Collection. If successful,the research proposed will be significant because it willdramatically improve the robustness of the existing methods and, as a result,significantly drive down the time and cost in computation. It also allows forthe development of more robust parallel Krylov subspace methods, for example,by rescheduling the operations into parallelism. The research will also provide ideal topics for class learning and thesis research. Moreover, the PI once participatedin research in Chemical and Petroleum Engineering, University of Wyoming, onremoving nitrogen oxides from combustion exhaust streams. The resultingmethods of the project will promote the collaboration in future because more robustalgorithms will then be available to use in solving the PDE modelsarising from the collaborative study.