Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands
高阶扰乱数字网对光滑被积函数达到最优均方根误差收敛率AI 解读 ↓
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高阶扰乱数字网对光滑被积函数达到最优均方根误差收敛率
- 研究问题
- 对光滑的多维被积函数，如何设计随机积分算法，使额外光滑性转化为更快的均方根误差收敛，而非停留在既有速率？
- 方法与思路
- 作者提出新的随机化准蒙特卡洛算法，在各变量高阶混合偏导平方可积的条件下证明误差界，并与已知下界比较。
- 材料陈述的贡献
- 摘要称，新算法的误差速率随光滑阶提高，并在允许任意小指数损失的意义下达到所述最优阶；数值例子与理论界一致。
- 阅读关注点
- 应核对数字网及扰乱的构造、混合偏导的精确定义、速率中的任意小指数项，以及维度和光滑性对误差常数的影响。
- 阅读边界
- 仅据版本未注明的索引摘要；最优速率依赖较强光滑性条件，摘要未给出完整算法与常数，不能推断一般积分问题均会受益。
原文摘要
We study a random sampling technique to approximate integrals ∫[0, 1]s f(x) dx by averaging the function at some sampling points. We focus on cases where the integrand is smooth, which is a problem which occurs in statistics. The convergence rate of the approximation error depends on the smoothness of the function f and the sampling technique. For instance, Monte Carlo (MC) sampling yields a convergence of the root mean square error (RMSE) of order N−1/2 (where N is the number of samples) for functions f with finite variance. Randomized QMC (RQMC), a combination of MC and quasi-Monte Carlo (QMC), achieves a RMSE of order N−3/2+ε under the stronger assumption that the integrand has bounded variation. A combination of RQMC with local antithetic sampling achieves a convergence of the RMSE of order N−3/2−1/s+ε (where s ≥ 1 is the dimension) for functions with mixed partial derivatives up to order two. Additional smoothness of the integrand does not improve the rate of convergence of these algorithms in general. On the other hand, it is known that without additional smoothness of the integrand it is not possible to improve the convergence rate. This paper introduces a new RQMC algorithm, for which we prove that it achieves a convergence of the root mean square error (RMSE) of order N−α−1/2+ε provided the integrand satisfies the strong assumption that it has square integrable partial mixed derivatives up to order α > 1 in each variable. Known lower bounds on the RMSE show that this rate of convergence cannot be improved in general for integrands with this smoothness. We provide numerical examples for which the RMSE converges approximately with order N−5/2 and N−7/2, in accordance with the theoretical upper bound.