There is a growing interest in probabilistic numerical solutions to ordinary differential equations. In this paper, the maximum a posteriori estimate is studied under the class of \(\nu \) times differentiable linear time-invariant Gauss–Markov priors, which can be computed with an iterated extended Kalman smoother. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducing kernel Hilbert space associated with the prior, which in the present case is equivalent to a Sobolev space of smoothness \(\nu +1\). Subject to mild conditions on the vector field, convergence rates of the maximum a posteriori estimate are then obtained via methods from nonlinear analysis and scattered data approximation. These results closely resemble classical convergence results in the sense that a \(\nu \) times differentiable prior process obtains a global order of \(\nu \), which is demonstrated in numerical examples.

对常微分方程的概率数值解法的兴趣日益浓厚。在本文中，最大后验估计是在\（\ nu \）乘可微线性时不变高斯-马尔可夫先验的类下研究的，这可以用迭代扩展卡尔曼平滑器来计算。最大后验估计对应于与先验相关联的再现核希尔伯特空间中的最优内插值，在当前情况下，它等于光滑度\（\ nu +1 \）的Sobolev空间。在矢量场的温和条件下，然后通过非线性分析和分散数据逼近的方法获得最大后验估计的收敛速度。这些结果与经典的收敛结果非常相似，在某种意义上，\（\ nu \）倍可微先验过程获得了\（\ nu \）的全局阶数，这在数值示例中得到了证明。