Overview
- Gives a detailed presentation of the theory of abstract inverse linear problems on Hilbert space
- Provides the first comprehensive analysis of Krylov solvability
- Includes a multitude of novel results presented from a broad perspective
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (5 chapters)
Keywords
- inverse linear problems on Hilbert space
- infinite-dimensional Hilbert space
- ill-posed problems
- orthonormal basis discretization
- bounded linear operators
- unbounded operators on Hilbert space
- self-adjoint operators
- cyclic operators
- Krylov subspaces
- Krylov solution
- Krylov solvability
- conjugate gradient methodsspectral measures
- orthogonal polynomials
- cyclic vectors
- spectral theory
- Hausdorff distance
- subspace perturbations
About this book
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.
Reviews
“The material could be used for a single-subject thematic graduate course. Furthermore, it could be used as a reference guide for experts in neighboring fields, such as operator theorists, applied and numerical analysts, etc. … The monograph ends with an appendix with an outlook on general projection methods and weaker convergence. There is an elaborate list of references and a nice index.” (Kees Vuik, Mathematical Reviews, August, 2023)
Authors and Affiliations
About the authors
Noè Angelo Caruso is a postdoctoral researcher at the Gran Sasso Science Institute (GSSI) in L’Aquila, and a recent PhD graduate in Mathematical Analysis, Modelling and Applications from the International School of Advanced Studies (SISSA) in Trieste. His research interests are in operator theory and abstract approximation theory, taking inspiration from topics in theoretical numerical analysis with a particular emphasis on underlying functional analytic aspects.
Alessandro Michelangeli is the Alexander von Humboldt Experienced Researcher at the Institute for Applied Mathematics and the Hausdorff Center for Mathematics, University of Bonn. After obtaining his PhD in mathematical physics from SISSA, Trieste, he was a post-doc and then assistant professor at Ludwig Maximilians University, Munich (2007–2015). He has held numerous other post-doctoral and visiting positions, including at Cambridge University, SISSA, and CIRM (Trento). His research focuses on mathematical methods in physics, and the rigorous operator-theoretic understanding of numerical algorithms. In 2017 he was awarded the Alexander Vasiliev Award for an outstanding paper published in Analysis and Mathematical Physics.
Bibliographic Information
Book Title: Inverse Linear Problems on Hilbert Space and their Krylov Solvability
Authors: Noè Angelo Caruso, Alessandro Michelangeli
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-030-88159-7
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Hardcover ISBN: 978-3-030-88158-0Published: 11 February 2022
Softcover ISBN: 978-3-030-88161-0Published: 11 February 2023
eBook ISBN: 978-3-030-88159-7Published: 10 February 2022
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XI, 140
Number of Illustrations: 8 illustrations in colour
Topics: Analysis, Functional Analysis, Numerical Analysis, Operator Theory