This book presents a thorough discussion of the theory of abstract inverse linear problems on Hilbert space. Given an unknown vector f in a Hilbert space H, a linear operator A acting on H, and a vector g in H satisfying Af=g, one is interested in approximating f by finite linear combinations of g, Ag, A2g, A3g, … The closed subspace generated by the latter vectors is called the Krylov subspace of H generated by g and A. The possibility of solving this inverse problem by means of projection methods on the Krylov subspace is the main focus of this text.
After giving a broad introduction to the subject, examples and counterexamples of Krylov-solvable and non-solvable inverse problems are provided, together with results on uniqueness of solutions, classes of operators inducing Krylov-solvable inverse problems, and the behaviour of Krylov subspaces under small perturbations. An appendix collects material on weaker convergence phenomena in general projection methods.
This subject of this book lies at the boundary of functional analysis/operator theory and numerical analysis/approximation theory and will be of interest to graduate students and researchers in any of these fields.
Author(s): Noè Angelo Caruso, Alessandro Michelangeli
Series: Springer Monographs in Mathematics
Edition: 1
Publisher: Springer Nature Switzerland AG
Year: 2021
Language: English
Pages: 140
City: Cham, Switzerland
Tags: Inverse Problems, Hilbert Space, Krylov Solvability
Preface
Contents
Acronyms
Chapter 1 Introduction and motivation
1.1 Abstract inverse linear problems on Hilbert space
1.2 General truncation and approximation scheme
1.3 Krylov subspace and Krylov solvability
1.4 Structure of the book
Chapter 2 Krylov solvability of bounded linear inverse problems
2.1 Krylov subspace of a Hilbert space
2.2 Krylov reducibility and Krylov intersection
2.3 Krylov solutions for a bounded linear inverse problem
2.3.1 Krylov solvability. Examples.
2.3.2 General conditions for Krylov solvability: case of injectivity
2.3.3 Krylov reducibility and Krylov solvability
2.3.4 More on Krylov solutions in the lack of injectivity
2.4 Krylov solvability and self-adjointness
2.5 Special classes of Krylov solvable problems
2.6 Some illustrative numerical tests
Chapter 3 An analysis of conjugate-gradient based methods with unbounded operators
3.1 Unbounded posi tive self-adjoint inverse problems and conjugate gradient approach
3.2 Set-up and main results
3.3 Algebraic and measure-theoretic background properties
3.4 Proof of CG-convergence and additional observations
3.5 Unbounded CG-convergence tested numerically
Chapter 4 Krylov solvability of unbounded inverse problems
4.1 Unbounded setting
4.2 The general self-adjoint and skew-adjoint case
4.3 New phenomena in the general unbounded case: ‘Krylov escape’, generalised Krylov reducibility, generalised Krylov intersection
4.4 Krylov solvability in the general unbounded case
4.5 The self-adjoint case revisited: structural properties.
4.6 Remarks on rational Krylov subspaces and solvability of self-adjoint inverse problems
Chapter 5 Krylov solvability in a perturbative framework
5.1 Krylov solvability from a perturbative perspective
5.2 Fundamental perturbative questions
5.3 Gain or loss of Krylov solvability under perturbations
5.3.1 Operator perturbations
5.3.2 Data perturbations
5.3.3 Simultaneous perturbations of operator and data
5.4 Krylov solvability along perturbations of K -class
5.5 Weak gap-metric for weakly closed parts of the unit ball
5.6 Weak gap metric for linear subspaces
5.7 Krylov perturbations in the weak gap-metric
5.7.1 Some technical features of the vicinity of Krylov subspaces
5.7.2 Existence of d_w-limits. Krylov inner approximability.
5.7.3 Krylov solvability along d_w-limits
5.8 Perspectives on the perturbation framework
Appendix A Outlook on general projection methods and weaker convergence
A.1 Standard projection methods and beyond
A.2 Finite-dimensional truncation
A.2.1 Set up and notation
A.2.2 Singularity of the truncated problem
A.2.3 Convergence of the truncated problems
A.3 The compact linear inverse problem
A.4 The bounded linear inverse problem
A.5 Effects of changing the truncation basis: numerical evidence
References
Index
