论文标题
首次估计和不确定性定量到阈值
Error estimation and uncertainty quantification for first time to a threshold value
论文作者
论文摘要
经典A后验误差分析用于微分方程，量化了利益量（QOI）中的误差，该误差表示为溶液的有界线性函数。在这项工作中，我们考虑了无法以这种方式表示的一定兴趣的后验错误估计，即首次跨越阈值的时间。我们为此类错误得出了两个表示形式，并使用基于伴随的后验方法来估计我们表示中出现的未知术语。第一个表示基于使用泰勒定理的线性化。第二个表示是通过实施标准的扎根技术获得的。我们提供了几个示例，这些示例证明了方法的准确性。然后，当不确定微分方程的参数时，我们将这些误差估计嵌入累积分布函数上，以在累积分布函数上提供错误。
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.
