Improved efficiency of multilevel Monte Carlo for stochastic PDE through strong pairwise coupling

Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in the multilevel hierarchy is needed to obtain efficiency gains. In this work, we show that strong pairwise coupling indeed is also important when (MLMC) is applied to stochastic partial differential equations (SPDE) of reaction-diffusion type, as it can improve the rate of convergence and thus improve tractability. For the (MLMC) method with strong pairwise coupling that was developed and studied numerically on filtering problems in [{\it Chernov et al., Numer. Math., 147 (2021), 71-125}], we prove that the rate of computational efficiency is higher than for existing methods. We also provide numerical comparisons with alternative coupling ideas illustrate the importance of this feature. The comparisons are conducted on a range of SPDE, which include a linear SPDE, a stochastic reaction-diffusion equation, and stochastic Allen--Cahn equation.