分类： 数学 >> 数值分析 提交时间： 2024-06-14
摘要： We introduce a novel random integration algorithm that boasts both high con#2;vergence order and polynomial tractability for functions characterized by sparsefrequencies or rapidly decaying Fourier coefficients. Specifically, for integration inperiodic isotropic Sobolev space and the isotropic Sobolev space with compact sup#2;port, our approach attains a near-optimal root mean square error. In contrast toprevious nearly optimal algorithms, our method exhibits polynomial tractability,ensuring that the number of samples does not scale exponentially with increasingdimensions. Our integration algorithm also enjoys near-optimal bound for weightedKorobov space. Furthermore, the algorithm can be applied without the need forprior knowledge of weights, distinguishing it from component-by-component algo#2;rithms. For integration in the Wiener algebra, the sample complexity of our algo#2;rithm is independent of the decay rate of Fourier coefficients. The effectiveness ofthe integration is confirmed through numerical experiments.
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.
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