论文标题
线性隐式Euler方法，用于受控随机热方程的有限元离散化
A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation
论文作者
论文摘要
我们考虑通过非均匀的诺伊曼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以及分布式和边界噪声的组合。我们将有限元方法应用于空间离散化和线性隐式Euler方法进行时间离散化。由于边界噪声引起的规律较低，因此无法预期的是，空间中的收敛顺序超过1/2。当分布式噪声和初始条件足够光滑时，我们证明了我们的全部离散化的最佳收敛顺序。在较不光滑的条件下，收敛顺序进一步降低。我们的结果仅假定相关（确定性的）差异riccati方程可以通过一定的收敛顺序近似，这在实践中很容易实现。我们通过在二维结构域中的数值实验来确认这些理论结果。
We consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with non-homogeneous 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.
