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Iterative numerical algorithms are typically equipped with a stoppingcriterion, where the iteration process is terminated when some error or misfitmeasure is deemed to be below a given tolerance. This is a useful setting forcomparing algorithm performance, among other purposes. However, in practicalapplications a precise value for such a tolerance is rarely known; rather, onlysome possibly vague idea of the desired quality of the numerical approximationis at hand. We discuss four case studies from different areas of numericalcomputation, where uncertainty in the error tolerance value and in the stoppingcriterion is revealed in different ways. This leads us to think of approachesto relax the notion of exactly satisfying a tolerance value. We thenconcentrate on a {\em probabilistic} relaxation of the given tolerance. Thisallows, for instance, derivation of proven bounds on the sample size of certainMonte Carlo methods. We describe an algorithm that becomes more efficient in acontrolled way as the uncertainty in the tolerance increases, and demonstratethis in the context of some particular applications of inverse problems.

Algorithms that satisfy a stopping criterion, probably