A recently introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution x and its first q derivatives a priori as a Gauss–Markov process \({\varvec{X}}\), which is then iteratively conditioned on information about \({\dot{x}}\). This article establishes worst-case local convergence rates of order \(q+1\) for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order q in the case of \(q=1\) and an integrated Brownian motion prior, and analyses how inaccurate information on \({\dot{x}}\) coming from approximate evaluations of f affects these rates. Moreover, we show that, in the globally convergent case, the posterior credible intervals are well calibrated in the sense that they globally contract at the same rate as the truncation error. We illustrate these theoretical results by numerical experiments which might indicate their generalizability to \(q \in \{2,3,\ldots \}\).

最近引入的一类用于常微分方程（ODE）的概率（不确定性）求解器将高斯（Kalman）滤波应用于初始值问题。这些方法将真实解x及其第一个q导数作为高斯-马尔可夫过程\（{\ varvec {X}} \）进行先验建模，然后以有关\（{\ dot {x}} \ ）。本文为该高斯ODE滤波器的各种版本建立了阶（\ q + 1 \）的最坏情况的局部收敛速度，并在\（q = 1 \）的情况下建立了阶q的全局收敛速度。以及集成的布朗运动先验，然后分析来自f的近似评估的\（{\ dot {x}} \）上不正确的信息如何影响这些比率。此外，我们表明，在全局收敛的情况下，从可信度上讲，后可信区间在全局范围内以与截断误差相同的速率收缩，因此得到了很好的校准。我们通过数值实验说明了这些理论结果，这些实验可能表明它们对\（q \ in \ {2,3，\ ldots \} \）的推广。