Krilov solvability of unbounded inverse linear problems
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Date
2020-01-23
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Publisher
SISSA
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
Description
21 p.
Keywords
inverse linear problems, conjugate gradient methods, unbounded operators on Hilbert space, self-adjoint operators, Krylov subspaces, Krylov solution