Classical a posteriori error analysis for differential equations quantifies the error in a Quantity of Interest (QoI) which is represented as a bounded linear functional of the solution. In this work we consider a posteriori error estimates of a quantity of interest that cannot be represented in this fashion, namely the time at which a threshold is crossed for the first time. We derive two representations for such errors and use an adjoint-based a posteriori approach to estimate unknown terms that appear in our representation. The first representation is based on linearizations using Taylor's Theorem. The second representation is obtained by implementing standard root-finding techniques. We provide several examples which demonstrate the accuracy of the methods. We then embed these error estimates within a framework to provide error bounds on a cumulative distribution function when parameters of the differential equations are uncertain.

中文翻译：

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首次对阈值进行误差估计和不确定性量化

微分方程的经典后验误差分析量化了兴趣量 (QoI) 中的误差，其表示为解的有界线性函数。在这项工作中，我们考虑无法以这种方式表示的感兴趣数量的后验误差估计，即第一次越过阈值的时间。我们为此类错误推导出两种表示，并使用基于伴随的后验方法来估计出现在我们的表示中的未知项。第一种表示基于使用泰勒定理的线性化。第二种表示是通过实施标准的求根技术获得的。我们提供了几个例子来证明这些方法的准确性。