A multiscale method is proposed for a parabolic stochastic partial differential equation with additive noise and highly oscillatory diffusion. The framework is based on the localized orthogonal decomposition (LOD) method and computes a coarse-scale representation of the elliptic operator, enriched by fine-scale information on the diffusion. Optimal order strong convergence is derived. The LOD technique is combined with a (multilevel) Monte-Carlo estimator and the weak error is analyzed. Numerical examples that confirm the theoretical findings are provided, and the computational efficiency of the method is highlighted.

翻译：针对具有加性噪声和高振荡扩散系数的抛物型随机偏微分方程，提出了一种多尺度方法。该框架基于局部化正交分解（LOD）方法，通过融入扩散系数的细尺度信息，计算椭圆算子的粗尺度表示。推导了最优阶强收敛性。将LOD技术与（多层）蒙特卡罗估计器相结合，并分析了弱误差。文中提供了验证理论结果的数值算例，并突出了该方法的计算效率。