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)是接近包含昂贵科学模型的合成物的关键工具。 其理念是使用指数近似值构建一个比古典蒙特卡洛(Contle Monte Carlo)更精确的测算器。 我们提议通过古典蒙特卡洛(Bayesian ) 的替代模型进一步加强多层次的蒙特卡洛(Monte Carlo ), 重点是高山进程模型和相关的贝耶斯二次测算器。 我们用理论和数字实验都表明,当指数昂贵且光滑,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。 我们用案例研究来结束论文,说明我们的方法在山崩海啸模型中的潜在影响,因为每次指数评估的成本通常对操作环境来说太高。</s>