• A Random Integration Algorithm for High-dimensional Function Spaces

    分类: 数学 >> 数值分析 提交时间: 2024-06-16

    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.

  • Approximation-Degree-Based Interpolation: A New Interpolation Method

    分类: 数学 >> 数值分析 提交时间: 2019-12-29

    摘要: This paper introduces the measure of approximate-degree and the concept of approximate-degree function between numerical values, thus developing a new interpolation method —— approximation-degree-based interpolation, i.e., AD interpolation. One-dimensional AD interpolation is done directly by using correlative interpolation formulas; n(n>1)-dimensional AD interpolation is firstly separated into n parallel one-dimensional AD interpolation computations to do respectively, and then got results are synthesized by Sum-Times-Difference formula into a value as the result value of the n-dimensional interpolation. If the parallel processing is used, the efficiency of n-dimensional AD interpolation is almost the same as that of the one-dimensional AD interpolation. Thus it starts a feasible and convenient approach and provides an effective method for high-dimensional interpolations. Furthermore, if AD interpolation is introduced into machine learning, a new instance-based learning method is expected to be realized.