We present a novel multilevel Monte Carlo approach for estimating quantities of interest for stochastic partial differential equations (SPDEs). Drawing inspiration from [Giles and Szpruch: Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L\'evy area simulation, Annals of Appl. Prob., 2014], we extend the antithetic Milstein scheme for finite-dimensional stochastic differential equations to Hilbert space-valued SPDEs. Our method has the advantages of both Euler and Milstein discretizations, as it is easy to implement and does not involve intractable L\'evy area terms. Moreover, the antithetic correction in our method leads to the same variance decay in a MLMC algorithm as the standard Milstein method, resulting in significantly lower computational complexity than a corresponding MLMC Euler scheme. Our approach is applicable to a broader range of non-linear diffusion coefficients and does not require any commutative properties. The key component of our MLMC algorithm is a truncated Milstein-type time stepping scheme for SPDEs, which accelerates the rate of variance decay in the MLMC method when combined with an antithetic coupling on the fine scales. We combine the truncated Milstein scheme with appropriate spatial discretizations and noise approximations on all scales to obtain a fully discrete scheme and show that the antithetic coupling does not introduce an additional bias.

翻译：我们提出了一种新颖的多层级蒙特卡罗方法，用于估计随机偏微分方程（SPDE）的感兴趣量。受[Giles and Szpruch: Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation, Annals of Appl. Prob., 2014]启发，我们将有限维随机微分方程的反变量米尔斯坦格式推广至希尔伯特空间取值的SPDE。该方法兼具欧拉与米尔斯坦离散化的优点：易于实现且不涉及棘手的莱维区域项。此外，本文方法中的反变量校正可使多层级蒙特卡罗（MLMC）算法达到与标准米尔斯坦方法相同的方差衰减率，从而显著降低计算复杂度（相较于对应的MLMC欧拉格式）。该方法适用于更广泛的非线性扩散系数，且无需任何交换性条件。本MLMC算法的核心组件是一种用于SPDE的截断米尔斯坦型时间推进格式，结合细尺度上的反变量耦合可加速MLMC方法中的方差衰减速率。我们将截断米尔斯坦格式与各尺度上恰当的空间离散化及噪声近似相结合，得到全离散格式，并证明反变量耦合不会引入额外偏差。