In this short note, we study a concatenation of quasi-Monte Carlo and plain Monte Carlo rules for high-dimensional numerical integration in weighted function spaces. In particular, we consider approximating the integral of periodic functions defined over the $s$-dimensional unit cube by using rank-1 lattice point sets only for the first $d\, (<s)$ coordinates and random points for the remaining $s-d$ coordinates. We prove that, by exploiting a decay of the weights of function spaces, almost the optimal order of the mean squared worst-case error is achieved by such a concatenated quadrature rule as long as $d$ scales at most linearly with the number of points. This result might be useful for numerical integration in extremely high dimensions, such as partial differential equations with random coefficients for which even the standard fast component-by-component algorithm is considered computationally expensive.
翻译:在这个简短的注释中,我们研究了在加权功能空间中高维数字集成的准蒙特卡洛和普通蒙特卡洛规则的组合。特别是,我们考虑通过仅对第一个美元(d),(s)美元坐标和剩余美元(d)坐标随机点使用一级至一级拉蒂点设置,来接近美元单位立方体上界定的定期函数的一体化。我们证明,通过利用功能空间重量的衰减,只要在最大线性值和点数上使用美元尺度,就几乎实现了平均正方形最坏情况错误的最优顺序。这一结果可能有助于极高的数值集成,例如带有随机系数的局部差异方程式,甚至标准快速组件逐部分算法也被认为计算成本昂贵。