标题： 用于半线性随机偏微分方程的多指标蒙特卡洛方法 显示英文标题

标题： The multi-index Monte Carlo method for semilinear stochastic partial differential equations

摘要： 随机偏微分方程（SPDEs）由于其低正则性和高维性，在数值求解上通常很困难。 这些挑战限制了计算机辅助研究的实际应用，并对SPDE的统计分析构成了重大障碍。 在本工作中，我们引入了一种高效的多指标蒙特卡洛方法（MIMC），旨在近似半线性抛物型SPDE的弱解的统计特性。 我们方法的关键在于证明了由指数积分器数值求解器生成的耦合解的乘法收敛性质，并将其与MIMC结合使用。 我们进一步从理论上描述了当容差趋近于零时，MIMC的渐近计算成本如何以输入精度容差为基准进行界定。 值得注意的是，我们的方法表明，对于低正则性的SPDE，MIMC相比其他可行方法表现出显著的性能提升。 在相关测试问题上，将MIMC与多级蒙特卡洛方法的性能进行比较的数值实验验证了我们的理论结果。 这些结果还表明， MIMC显著优于最先进的多级蒙特卡洛方法，从而突显了其作为解决半线性抛物型SPDE的稳健且可行工具的潜力。 显示英文摘要

摘要： Stochastic partial differential equations (SPDEs) are often difficult to solve numerically due to their low regularity and high dimensionality. These challenges limit the practical use of computer-aided studies and pose significant barriers to statistical analysis of SPDEs. In this work, we introduce a highly efficient multi-index Monte Carlo method (MIMC) designed to approximate statistics of mild solutions to semilinear parabolic SPDEs. Key to our approach is the proof of a multiplicative convergence property for coupled solutions generated by an exponential integrator numerical solver, which we incorporate with MIMC. We further describe theoretically how the asymptotic computational cost of MIMC can be bounded in terms of the input accuracy tolerance, as the tolerance goes to zero. Notably, our methodology illustrates that for an SPDE with low regularity, MIMC offers substantial performance improvements over other viable methods. Numerical experiments comparing the performance of MIMC with the multilevel Monte Carlo method on relevant test problems validate our theoretical findings. These results also demonstrate that MIMC significantly outperforms state-of-the-art multilevel Monte Carlo, thereby underscoring its potential as a robust and tractable tool for solving semilinear parabolic SPDEs.