Multilevel sampling methods, such as multilevel and multifidelity Monte Carlo, multilevel stochastic collocation, or delayed acceptance Markov chain Monte Carlo, have become standard uncertainty quantification (UQ) tools for a wide class of forward and inverse problems. The underlying idea is to achieve faster convergence by leveraging a hierarchy of models, such as partial differential equation (PDE) or stochastic differential equation (SDE) discretisations with increasing accuracy. By optimally redistributing work among the levels, multilevel methods can achieve significant performance improvement compared to single level methods working with one high-fidelity model. Intuitively, approximate solutions on coarser levels can tolerate large computational error without affecting the overall accuracy. We show how this can be used in high-performance computing applications to obtain a significant performance gain. As a use case, we analyse the computational error in the standard multilevel Monte Carlo method and formulate an adaptive algorithm which determines a minimum required computational accuracy on each level of discretisation. We show two examples of how the inexactness can be converted into actual gains using an elliptic PDE with lognormal random coefficients. Using a low precision sparse direct solver combined with iterative refinement results in a simulated gain in memory references of up to $3.5\times$ compared to the reference double precision solver; while using a MINRES iterative solver, a practical speedup of up to $1.5\times$ in terms of FLOPs is achieved. These results provide a step in the direction of energy-aware scientific computing, with significant potential for energy savings.

翻译：多层采样方法，如多层与多保真蒙特卡洛、多层随机配置法或延迟接受马尔可夫链蒙特卡洛，已成为解决广泛正向与逆向问题的不确定性量化标准工具。其核心思想是通过利用模型层次结构（例如精度递增的偏微分方程或随机微分方程离散化）来加速收敛。通过在各层级间最优地重新分配计算任务，与仅使用单一高保真模型的单层方法相比，多层方法能实现显著的性能提升。直观上，较粗层级上的近似解可容忍较大的计算误差而不影响整体精度。本文展示了如何在高性能计算应用中利用这一特性以获得显著的性能收益。作为应用案例，我们分析了标准多层蒙特卡洛方法中的计算误差，并构建了一种自适应算法，用于确定各离散化层级所需的最小计算精度。我们通过两个示例展示了如何将非精确性转化为实际收益：使用具有对数正态随机系数的椭圆偏微分方程。采用低精度稀疏直接求解器结合迭代精化，相比参考的双精度求解器，在内存访问方面实现了高达$3.5\times$的模拟增益；而使用MINRES迭代求解器，在浮点运算次数方面实现了高达$1.5\times$的实际加速。这些结果为迈向能源感知的科学计算提供了重要一步，具有显著的节能潜力。