论文标题
无界逆线性问题的Krylov溶解度
Krylov solvability of unbounded inverse linear problems
论文作者
论文摘要
对于反问题$ af = g $，“ krylov解决性”的抽象问题进行了广泛的讨论，其中$ a $是无限二维Hilbert Space上的（可能是无界的）线性运算符，而$ g $是$ a $ a $ a的基准。问题包括是否可以通过$ g，ag，a^2 g，\ dots $的有限线性组合在Hilbert Norm中近似于Hilbert Norm，以及此类解决方案是否存在并且是唯一的。在$ a $有限时重新访问已知图片后，我们研究了一个密集定义和关闭$ a $的一般情况。固有的操作者理论机制是确定的，可以保证或防止Krylov的溶解度，并且由于无限制而引起的新功能。在自动化案例中检查了这种机制，基于共轭梯度的技术也证明了Krylov的溶解度。
The abstract issue of 'Krylov solvability' is extensively discussed for the inverse problem $Af = g$ where $A$ is a (possibly unbounded) linear operator on an infinite-dimensional Hilbert space, and $g$ is a datum in the range of $A$. The question consists of whether the solution $f$ can be approximated in the Hilbert norm by finite linear combinations of $g, Ag, A^2 g, \dots$ , and whether solutions of this sort exist and are unique. After revisiting the known picture when $A$ is bounded, we study the general case of a densely defined and closed $A$. Intrinsic operator-theoretic mechanisms are identified that guarantee or prevent Krylov solvability, with new features arising due to the unboundedness. Such mechanisms are checked in the self-adjoint case, where Krylov solvability is also proved by conjugate-gradient-based techniques.
