Krylov Solvability of Unbounded Inverse Linear Problems
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Date
2019
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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, A2g,... , 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.
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Keywords
Inverse linear problems, Conjugate gradient methods, Unbounded operators on Hilbert space, Self-adjoint operators, Krylov subspaces, Krylov solution