Almost every numerical task can be cast as extrapolation with respect to the fidelity or tolerance parameters of a consistent numerical method. This perspective enables probabilistic uncertainty quantification and optimal experimental design functionality to be deployed, and also unlocks the potential for the convergence of numerical methods to be accelerated. Recent work established Probabilistic Richardson Extrapolation as a proof-of-concept, demonstrating how parallel multi-fidelity simulation can be used to accelerate simulation from a whole-heart model. However, the number of simulations was required to increase super-exponentially in $d$, the number of tolerance parameters appearing in the numerical method. This paper develops a refined notion of 'extrapolation dimension', drastically reducing this simulation requirement when multiple tolerance parameters feature in the numerical method. Sparsity-exploiting methodology is developed that is simultaneously simpler and more powerful compared to earlier work, and this is accompanied by sharp theoretical guarantees and substantial empirical support.

翻译：几乎所有数值任务都可以被视作关于一致数值方法的保真度或容差参数的外推。这一视角使概率不确定性量化与最优实验设计功能得以部署，并释放了加速数值方法收敛的潜力。近期工作将概率理查森外推法确立为概念验证，展示了如何利用并行多保真度模拟加速全心脏模型的仿真。然而，所需模拟次数需随数值方法中容差参数数量 $d$ 呈超指数增长。本文发展了“外推维度”的精炼概念，当数值方法包含多个容差参数时大幅降低了模拟需求。我们开发了利用稀疏性的方法论，较之先前工作更简洁且更强大，并辅以严格的理论保证与充分的实证支持。