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 approximate solutions determined by means general projection methods, namely outside standard assumptions Petrov–Galerkin truncation schemes. This includes a discussion mechanisms why error or residual generically fail to vanish norm, and identification practically plausible sufficient conditions for such indicators be small some weaker sense. The presentation is based on theoretical results together with series model examples numerical tests.