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A Random Integration Algorithm for High-dimensional Function Spaces

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摘要: 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 near-optimal root mean square error. 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 near-optimal bound for weighted Korobov space. Furthermore, the algorithm can be applied without the need for prior knowledge of weights, distinguishing it from component-by-component algorithms. 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.

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[V2] 2024-06-16 16:28:45 ChinaXiv:202406.00231V2 下载全文
[V1] 2024-06-14 03:05:48 ChinaXiv:202406.00231v1 查看此版本 下载全文
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