We introduce a novel random integration algorithm that boasts both high convergence order and polynomial tractability for functions characterized by sparse frequencies or rapidly decaying Fourier coefficients. Specifically, for integration in periodic isotropic Sobolev space and the isotropic Sobolev space with compact support, our approach attains a nearly optimal root mean square error (RMSE) bound. In contrast to previous nearly optimal algorithms, our method exhibits polynomial tractability, ensuring that the number of samples does not scale exponentially with increasing dimensions. Our integration algorithm also enjoys nearly optimal bound for weighted Korobov space. Furthermore, the algorithm can be applied without the need for prior knowledge of weights, distinguishing it from the component-by-component algorithm. For integration in the Wiener algebra, the sample complexity of our algorithm is independent of the decay rate of Fourier coefficients. The effectiveness of the integration is confirmed through numerical experiments.

翻译：本文提出一种新颖的随机积分算法，该算法对于具有稀疏频率或快速衰减傅里叶系数的函数兼具高收敛阶数与多项式可处理性。具体而言，在周期各向同性Sobolev空间及紧支撑各向同性Sobolev空间的积分问题中，本方法达到了近乎最优的均方根误差界。相较于先前近乎最优的算法，本方法展现出多项式可处理性，确保样本数量不随维度增加呈指数级增长。该积分算法在加权Korobov空间中也具备近乎最优的误差界。此外，本算法无需权重先验知识即可实施，此特性使其区别于逐分量构造算法。对于Wiener代数中的积分问题，本算法的样本复杂度与傅里叶系数衰减率无关。数值实验验证了该积分方法的有效性。