This monograph is centred at the intersection of three mathematical topics, that are theoretical in nature, yet with motivations and relevance deep rooted in applications: the linear inverse problems on abstract, in general infinite-dimensional Hilbert space; the notion of Krylov subspace associated to an inverse problem, i.e., the cyclic subspace built upon the datum of the inverse problem by repeated application of the linear operator; the possibility to solve the inverse problem by means of Krylov subspace methods, namely projection methods where the finite-dimensional truncation is made with respect to the Krylov subspace and the approximants converge to an exact solution to the inverse problem.

翻译：这份专著的中心是三个数学专题的交叉点,这三个专题是理论性的,但动机和相关性深深扎根于各种应用:抽象的线性反问题,一般是无限的Hilbert空间;与反向问题相关的Krylov子空间概念,即以反向问题基准为基础的环球子空间,通过线性操作员的反复应用;通过Krylov子空间方法解决反向问题的可能性,即对Krylov子空间进行有限维对流的预测方法,以及近身空间与反向问题的确切解决办法相融合。