Inverse linear problems on a Hilbert space and their Krylov solvability / / Noè Angelo Caruso, Alessandro Michelangeli

• Record Nr.: UNISA996466545403316
• Autore: Noè Angelo Caruso
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2021]; ©2021
• ISBN: 3-030-88159-8
• Descrizione fisica: 1 online resource (150 pages)
• Collana: Springer Monographs in Mathematics
• Disciplina: 515.357
• Soggetti: Hilbert space; Problemes inversos (Equacions diferencials); Espais de Hilbert; Operadors lineals; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.