Preamble Verified numerical methods empower scientific computing to solve challenging mathematical problems whilst providing error guarantees. Differently from other methods, in verified methods, the error is a constituent component, it is contextually quantified and it can be reduced with more computational effort. These methods have been used to find rigorous—thus proven to be correct—solutions to notoriously difficult problems. An example of a challenging mathematical problem in engineering is the computation of the failure probability. Computing the failure probability of a complex system involves the solution of high dimensional integrals over a domain characterised by the preimage of a nonlinear function. Because of the complexity of this task, analytical rigour is sacrificed to find suitable numerical approximations. With verified methods the failure probability can be bounded numerically, while controlling the magnitude of the approximation. Motivations Bounding the failure probability of a complex system with the desired precision means being able to provide an interval where the reference solution resides with a given level of confidence. So far, only the Monte Carlo method has been used to provide reference values to failure probability problems whose analytical solution is not possible. Monte Carlo however, has a huge limitation: the desired precision cannot be reached on problems with very small target failure probability. Moreover, the efficiency of Monte Carlo deprecates when the problem is formulated to allow imprecise probability. Several adaptations of Monte Carlo have been proposed to allow for imprecise probability but none with error guarantees [1, 3]. Monte Carlo variations that allow for imprecision, e.g. second-order Monte Carlo,
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