多次元超立方体内の格子点上でのみデータが与えられている場合の最小2乗近似法

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書誌事項

タイトル別名
  • Least Square Approximation with Data Given at Lattice Points in Multidimensional Hypercube
  • An Algorithm Based on Random Sampling with Control Variate Method
  • 制御変量法を伴うランダム・サンプリングによるアルゴリズム

抄録

This paper considers the least square approximation with the data given at the lattice points in the multidimensional hypercube. Since the solution is obtained as the sum of a prohibitively large number of terms, usual termwise calculation of the solution requires a great deal of computational labour. So, we obtained an algorithm which regards these large number of terms as a population and evaluates the solution based on a relatively small number of terms sampled at random from the population. In order to make the scattering due to the random sampling as small as possible, the algorithm is designed to sample the pairs of similar terms in such a way that the terms cancel each other, based on the idea of the control variate method. The control variate, which is a linear apporximation of the data, is constructed by performing a preliminary random sampling. The results of the calculation of a twenty-dimensional example show that the proposed algorithm is very efficient. A considerable reduction of on-line computational labour is expected by using the proposed algorithm together with the algorithm without the control variate method. The proposed algorithm is considered especially effective for the problem with the data with a relatively large linear part.

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