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.

中文翻译：

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抽象线性反问题的有限维近似的一般收敛性

在无穷维希尔伯特空间中的抽象线性逆问题的框架中，我们讨论了通过一般投影方法确定的近似解的一般收敛性，即在彼得罗夫-加勒金截断方案的标准假设之外。这包括对错误或残差通常无法正常消失的机制的讨论，以及确定实际上合理的充分条件，以使此类指标在某种程度上较弱。演示基于理论结果以及一系列模型示例和数值测试。