On general convergence behaviours of finite-dimensional approximants for abstract linear inverse problems

Article type: Research Article

Authors: Caruso, Noè Angelo^{a; b; *} | Michelangeli, Alessandro^c | Novati, Paolo^d

Affiliations: [a] International School for Advanced Studies, via Bonomea 265, I-34136 Trieste, Italy | [b] Gran Sasso Science Institute, Viale F. Crispi 7, I-67100 L’Aquila, Italy. E-mail: noe.caruso@gssi.it | [c] Institute for Applied Mathematics and Hausdorff Center for Mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany. E-mail: michelangeli@iam.uni-bonn.de | [d] Università degli Studi di Trieste, Piazzale Europa 1, I-34127 Trieste, Italy. E-mail: novati@units.it

Correspondence: [*] Corresponding author. E-mail: noe.caruso@gssi.it.

Abstract: In the framework of abstract linear inverse problems in infinite-dimensional Hilbert space we discuss generic convergence behaviours of approximate solutions determined by means of general projection methods, namely outside the standard assumptions of Petrov–Galerkin truncation schemes. This includes a discussion of the mechanisms why the error or the residual generically fail to vanish in norm, and the identification of practically plausible sufficient conditions for such indicators to be small in some weaker sense. The presentation is based on theoretical results together with a series of model examples and numerical tests.

Keywords: Linear inverse problems, infinite-dimensional Hilbert space, ill-posed problems, orthonormal basis discretisation, bounded linear operators, Krylov subspaces, Krylov solution, GMRES, conjugate gradient, LSQR

DOI: 10.3233/ASY-211678

Journal: Asymptotic Analysis, vol. 127, no. 1-2, pp. 167-189, 2022