When a solution to an abstract inverse linear problem on Hilbert space is approximable by finite linear combinations of vectors from the cyclic subspace associated with the datum and with the linear operator of the problem, the solution is said to be a Krylov solution, i.e., it belongs to the Krylov subspace of the problem. Krylov solvability of the inverse problem allows for solution approximations that, in applications, correspond to the very efficient and popular Krylov subspace methods. We study here the possible behaviours of persistence, gain, or loss of Krylov solvability under suitable small perturbations of the inverse problem -- the underlying motivations being the stability or instability of Krylov methods under small noise or uncertainties, as well as the possibility to decide a priori whether an inverse problem is Krylov solvable by investigating a potentially easier, perturbed problem. We present a whole scenario of occurrences in the first part of the work. In the second, we exploit the weak gap metric induced, in the sense of Hausdorff distance, by the Hilbert weak topology, in order to conveniently monitor the distance between perturbed and unperturbed Krylov subspaces.

中文翻译：

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抽象逆线性问题摄动下的Krylov可解性

当希尔伯特空间上的抽象逆线性问题的解决方案可以通过与该问题相关联的循环子空间和该问题的线性算子的向量的有限线性组合来逼近时，该解决方案被称为克雷洛夫解决方案，即属于问题的Krylov子空间。反问题的Krylov可解性允许求解近似值，在应用中，该近似值与非常有效且流行的Krylov子空间方法相对应。我们在这里研究逆问题的适当小扰动下Krylov可解性的持久性，获得或损失的可能行为-潜在动机是在小噪声或不确定性下Krylov方法的稳定性或不稳定性，还可以通过研究潜在的更容易解决的扰动问题来事先确定逆问题是否可以解决Krylov问题。我们在工作的第一部分中介绍了发生的整个情况。在第二部分中，我们利用希尔伯特弱拓扑在Hausdorff距离的意义上诱导的弱间隙度量，以便方便地监视摄动和非摄动Krylov子空间之间的距离。