Multilevel Monte Carlo is a key tool for approximating integrals involving expensive scientific models. The idea is to use approximations of the integrand to construct an estimator with improved accuracy over classical Monte Carlo. We propose to further enhance multilevel Monte Carlo through Bayesian surrogate models of the integrand, focusing on Gaussian process models and the associated Bayesian quadrature estimators. We show using both theory and numerical experiments that our approach can lead to significant improvements in accuracy when the integrand is expensive and smooth, and when the dimensionality is small or moderate. We conclude the paper with a case study illustrating the potential impact of our method in landslide-generated tsunami modelling, where the cost of each integrand evaluation is typically too large for operational settings.
翻译:多层次的蒙特卡洛(Monte Carlo)是涉及昂贵科学模型的近似集成体的关键工具。 其想法是使用百万位数的近似值来构建一个比古典蒙特卡洛更精确的估算器。 我们提议通过古典蒙特卡洛(Bayesian supprogate model of the integrand)进一步强化多层次的蒙特卡洛(Monte Carlo),重点是高山进程模型和相关的拜斯海脊估计仪。 我们用理论和数字实验来显示,当指数昂贵和光滑,当维度小或中度小时,我们的方法可以导致精确度的显著提高。 我们最后用案例研究来说明我们的方法在山崩海啸模型中的潜在影响。 在这种模型中,每次巨量评估的成本通常对于操作环境来说太大。