We consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with nonhomogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as well as both distributed and boundary noise. We apply the finite element method for the spatial discretization and the linear implicit Euler method for the temporal discretization. Due to the low regularity induced by the boundary noise, convergence orders above $1/2$ in space and $1/4$ in time cannot be expected. We prove such optimal convergence orders for our full discretization when the distributed noise and the initial condition are sufficiently smooth. Under less smooth conditions the convergence order is further decreased. Our results only assume that the related (deterministic) differential Riccati equation can be approximated with a certain convergence order, which is easy to achieve in practice. We confirm these theoretical results through a numerical experiment in a two-dimensional domain.

中文翻译：

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一种受控随机热方程有限元离散化的线性隐式欧拉法

我们考虑由具有非齐次 Neumann 边界条件的随机热方程约束的线性二次控制问题的数值近似。这涉及分布式和边界控制的组合，以及分布式和边界噪声。我们应用有限元法进行空间离散化和线性隐式欧拉法进行时间离散化。由于边界噪声引起的低规律性，无法预期空间上 1/2 美元以上和时间上 1/4 美元以上的收敛阶数。当分布噪声和初始条件足够平滑时，我们证明了我们完全离散化的最佳收敛顺序。在不太平滑的条件下，收敛阶数会进一步降低。我们的结果只假设相关的（确定的）微分Riccati方程可以用一定的收敛阶来近似，这在实践中很容易实现。我们通过二维域中的数值实验证实了这些理论结果。