多保真蒙特卡洛的样本量估计的递归舍入
Recursive rounding of sample size estimation for multi-fidelity Monte Carlo
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中文总结 AI 辅助
本研究针对多保真蒙特卡洛样本分配的标准舍入法无法充分利用预算的问题，通过重新表述问题并结合贝尔曼最优性原理提出递归舍入策略，该策略高效满足方差容差且计算开销更低，还可扩展至多级蒙特卡洛。
中文摘要 AI 辅助
在多保真蒙特卡洛（MFMC）中，最优样本分配通常通过方差最小化问题的连续松弛推导得出，整数解则通过事后舍入获得。这类舍入程序可能无法充分利用可用成本预算，尤其是在预算紧张或模型成本差异显著时。本研究将MFMC分配问题重新表述为方差约束的成本最小化问题，其在连续层面上与标准预算约束公式等价。该重新表述具有递归结构，可基于贝尔曼最优性原理构建整数分配策略。由于所得MFMC分配的数学结构与最优多级蒙特卡洛（MLMC）分配相似，所提策略自然可扩展至MLMC。所得算法构建的整数样本分配更贴合连续的方差-成本权衡，同时更高效地利用规定的方差容差。数值实验表明，与标准舍入策略相比，所提方法在计算开销更低的情况下满足了规定的方差容差。
英文摘要
In multifidelity Monte Carlo (MFMC), optimal sample allocations are typically derived from a continuous relaxation of a variance minimization problem, with integer solutions obtained through post hoc rounding. Such rounding procedures may fail to fully exploit the available cost budget, particularly under tight cost budgets or when model costs vary significantly. In this work, we reformulate the MFMC allocation problem as a variance-constrained cost minimization problem that is equivalent to the standard budget-constrained formulation at the continuous level. This reformulation admits a recursive structure that enables the construction of an integer allocation strategy based on Bellman's principle of optimality. Since the resulting MFMC allocation has a mathematical structure similar to the optimal multilevel Monte Carlo (MLMC) allocation, the proposed strategy naturally extends to MLMC. The resulting algorithm constructs integer-valued sample allocations that more closely follow the continuous variance--cost tradeoff while using the prescribed variance tolerance more efficiently. Numerical experiments demonstrate that the proposed approach satisfies the prescribed variance tolerance with less computational overhead than standard rounding strategies.
