Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the first and second moments, as well as the covariance. In the first part, we focus on stochastic ordinary differential equations. For the canonical examples with additive noise (Ornstein-Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic equations in variational form and discuss their well-posedness in detail. Notably, the second moment equation in the multiplicative case is naturally posed on projective-injective tensor product spaces as trial-test spaces. We construct Petrov-Galerkin discretizations based on tensor product piecewise polynomials and analyze their stability and convergence in these natural norms. In the second part, we proceed with parabolic stochastic partial differential equations with affine multiplicative noise. We prove well-posedness of the deterministic variational problem for the second moment, improving an earlier result. We then propose conforming space-time Petrov-Galerkin discretizations, which we show to be stable and quasi-optimal. In both parts, the outcomes are illustrated by numerical examples.

中文翻译：

---

抛物线随机偏微分方程确定性二阶矩方程的数值方法

随机偏微分方程的数值方法通常从采样路径估计解的矩。相反，我们将直接针对由一阶和二阶矩以及协方差满足的确定性方程。在第一部分，我们关注随机常微分方程。对于具有加性噪声（Ornstein-Uhlenbeck 过程）或乘性噪声（几何布朗运动）的典型示例，我们以变分形式导出这些确定性方程并详细讨论它们的适定性。值得注意的是，乘法情况下的二阶矩方程自然是在射影-射张量积空间上作为试测空间提出的。我们基于张量积分段多项式构建 Petrov-Galerkin 离散化，并分析它们在这些自然范数中的稳定性和收敛性。在第二部分，我们继续研究具有仿射乘法噪声的抛物线随机偏微分方程。我们在第二时刻证明了确定性变分问题的适定性，改进了早期的结果。然后，我们提出了一致的时空 Petrov-Galerkin 离散化，我们证明它是稳定和准最优的。在这两部分中，结果都通过数值例子来说明。然后，我们提出了一致的时空 Petrov-Galerkin 离散化，我们证明它是稳定和准最优的。在这两部分中，结果都通过数值例子来说明。然后，我们提出了一致的时空 Petrov-Galerkin 离散化，我们证明它是稳定和准最优的。在这两部分中，结果都通过数值例子来说明。