Abstract
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^2g,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.
Highlights
Introduction andSet-up of the ProblemThe question of ‘Krylov solvability’ of an inverse linear problem is an operatortheoretic question, with deep-rooted implications in numerics and scientific computing among others, that in fairly abstract terms is formulated as follows.A linear operator A acting on a real or complex Hilbert space H, and a vector g ∈ H are given such that A is closed and densely or everywhere defined on H, and g is an A-smooth vector in the range of A, i.e., g ∈ ranA ∩ C∞(A) (1.1)Partially supported by the Alexander von Humboldt Foundation.N
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 proved by conjugate-gradient-based techniques
Given the iterative nature of Krylov subspace methods, the largest part of the related literature is mainly concerned with the fundamental issue of convergence of the Krylov approximants to a solution f
Summary
The question of ‘Krylov solvability’ of an inverse linear problem is an operatortheoretic question, with deep-rooted implications in numerics and scientific computing among others, that in fairly abstract terms is formulated as follows.
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