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

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.