Bayesian Probabilistic Numerical Methods
Bayesian Probabilistic Numerical Methods
复制标题
DOI：
10.1137/17m1139357
复制
发表时间：
2019-12-01
期刊：
影响因子：
10.2
通讯作者：
Girolami, Mark
中科院分区：
文献类型：
--
作者：
Cockayne, Jon;Oates, Chris J.;Girolami, Mark
四十多年前，应用数学文献中提出了平均情况误差作为评估数值方法的替代标准。与最坏情况下的错误相反，该标准依赖于对候选数值任务的概率度量的构建，并且数值方法基于其相对于这些任务的平均性能进行评估。本文更进一步，建立贝叶斯概率数值方法作为解决某些反问题的基础上，贝叶斯框架内的数值任务。这使我们能够建立贝叶斯概率数值方法定义良好的一般条件，包括非线性和非高斯背景。对于一般的计算，提出了一个数值逼近方案，并建立了其渐近收敛性。理论发展扩展到计算管道，其中概率数值方法组成，以解决更具挑战性的数值任务。的贡献突出了一个重要的研究前沿的接口的数值分析和不确定性量化，并提出了一个具有挑战性的工业应用。
Over forty years ago average-case error was proposed in the applied mathematics literature as an alternative criterion with which to assess numerical methods. In contrast to worst-case error, this criterion relies on the construction of a probability measure over candidate numerical tasks, and numerical methods are assessed based on their average performance over those tasks with respect to the measure. This paper goes further and establishes Bayesian probabilistic numerical methods as solutions to certain inverse problems based upon the numerical task within the Bayesian framework. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well defined, encompassing both the nonlinear and non-Gaussian contexts. For general computation, a numerical approximation scheme is proposed and its asymptotic convergence established. The theoretical development is extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, and a challenging industrial application is presented.