电磁勘探在矿产勘探、地下水勘探等领域发挥着重要作用，大型多位移多右端向量线性系统的快速求解是其正演过程中亟需解决的重要科学问题之一。由于该类线性系统规模巨大、高度病态且含有特殊的结构特征，现有研究方法普遍存在不高效、不收敛等缺陷。本项目拟研究高效的位移-块Krylov子空间方法和预处理技术，为电磁勘探中大型线性系统设计快速求解算法，并提供理论支持。具体包括：1）利用线性系统的结构特征，研究不精确中断理论以解决多右端向量及各残量近似线性相关性导致的算法中断问题，并结合谱收缩重启技术设计新型位移-块GMRES类方法；2）采用谱信息循环技术，通过收缩小特征值构造加速算法收敛的二层谱预处理子，并进行理论分析；3）针对多频多场源电磁勘探中的线性系统，设计参数选取策略挑选最优种子参数，提出高效的预处理技术。本项目的研究旨在为电磁勘探中的线性系统设计高效稳健的求解方法，丰富Krylov子空间方法及理论。

Electromagnetic exploration plays an important role in mineral exploration, groundwater exploration and other fields. To fast solve the large-scale linear systems with multiple shifts and multiple right-hand sides is one of the important scientific problems that widely exists in the forward process. Because such linear systems are often large, highly ill-conditioned, and have special structural features, most existing methods are inefficient and fail to convergence. This project will study efficient shifted block Krylov subspace methods and preconditioning techniques to provide fast algorithms for large-scale linear systems in electromagnetic exploration, with theoretical support. Specifically, 1) based on the structural features of linear systems, study inexact breakdown theory to handle the issue of breakdown caused by the approximate linear dependence of multiple right-hand sides as well as the shifted block residuals, and design new efficient shifted block GMRES-like methods by combining deflated restarting; 2) by exploiting recycle spectral information technology, develop a spectral two-level preconditioner with deflated small eigenvalues to accelerate the convergence, with theoretical analysis; 3) design a parameter selection strategy to derive the optimal seed parameter and develop the efficient preconditioning technology for linear systems in multi-frequency and multi-source electromagnetic exploration. This project aims to design the efficient and robust methods for the linear systems in electromagnetic exploration, and simultaneously to enrich the Krylov subspace methods and corresponding theories.