We propose a fast algorithm for the probabilistic solution of boundary value problems (BVPs), which are ordinary differential equations subject to boundary conditions. In contrast to previous work, we introduce a Gauss--Markov prior and tailor it specifically to BVPs, which allows computing a posterior distribution over the solution in linear time, at a quality and cost comparable to that of well-established, non-probabilistic methods. Our model further delivers uncertainty quantification, mesh refinement, and hyperparameter adaptation. We demonstrate how these practical considerations positively impact the efficiency of the scheme. Altogether, this results in a practically usable probabilistic BVP solver that is (in contrast to non-probabilistic algorithms) natively compatible with other parts of the statistical modelling tool-chain.

中文翻译：

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边值问题的线性时间概率解

我们为边界值问题 (BVP) 的概率解提出了一种快速算法，边界值问题是受边界条件约束的常微分方程。与之前的工作相比，我们引入了高斯-马尔可夫先验，并专门针对 BVP 进行了定制，这允许在线性时间内计算解决方案的后验分布，其质量和成本可与成熟的、非概率的方法。我们的模型进一步提供了不确定性量化、网格细化和超参数适应。我们展示了这些实际考虑如何对计划的效率产生积极影响。总而言之，这导致了一个实际可用的概率 BVP 求解器（与非概率算法相反）与统计建模工具链的其他部分本机兼容。