Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al.\ (\textit{Stat.\ Comput.}, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings.
翻译:不确定性量化在一些问题中起着重要作用,这些问题涉及从对解决方案的观察中推断出初始价值问题的一个参数。 Conrad 等人\ (\ textit{Stat.\\comput.},2017年)提议将确定时间整合方法随机化作为因未知的时间分化错误而量化不确定性的战略。我们认为,这一战略适用于由确定性的、可能取决于时间的操作员在Banach空间或Gelfand 3个空间上定义的不同方程式描述的系统。我们的主要结果是在Orlicz规范中测量的随机轨迹上的强烈误差,在对基本确定时间整合方法的局部脱轨错误的较弱假设下证明。我们的分析确定了无限环境中差异方程式随机化时间整合的理论有效性。