We derive a posteriori error estimates for the (stopped) weak Euler method to discretize SDE systems which emerge from the probabilistic reformulation of elliptic and parabolic (initial) boundary value problems. The a posteriori estimate exploits the use of a scaled random walk to represent noise, and distinguishes between realizations in the interior of the domain and those close to the boundary. We verify an optimal rate of (weak) convergence for the a posteriori error estimate on deterministic meshes. Based on this estimate, we then set up an adaptive method which automatically selects local deterministic mesh sizes, and prove its optimal convergence in terms of given tolerances. Provided with this theoretical backup, and since corresponding Monte-Carlo based realizations are simple to implement, these methods may serve to efficiently approximate solutions of high-dimensional (initial-)boundary value problems.

我们推导出（停止的）弱欧拉方法的后验误差估计，以离散化从椭圆和抛物线（初始）边界值问题的概率重构中出现的 SDE 系统。后验估计利用缩放随机游动来表示噪声，并区分域内部的实现和靠近边界的实现。我们验证后验的最佳（弱）收敛率确定性网格的误差估计。基于这个估计，我们然后建立了一种自动选择局部确定性网格尺寸的自适应方法，并根据给定的公差证明其最佳收敛性。有了这个理论支持，并且由于相应的基于蒙特卡洛的实现很容易实现，这些方法可以有效地近似解决高维（初始）边值问题。