Inverse Linear Problems on Hilbert Space and Their Krylov Solvability

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.

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.