Discrete Least Square Approximation of Multivariable Functions by Random Sampling Method

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  • ランダム・サンプリングを用いた多変数関数の最小2乗近似法

Abstract

This paper deals with the discrete least square approximation of multivariable functions whose values are given only at the lattice points.<br>First, the solution is obtained as a linear combination of the values of the function at the lattice points, and it is shown that the deterministic method requires evaluation of the sum of an extremely large number of terms.<br>Next, to overcome this difficulty, the problem is converted into that of evaluating the expected values of two probabilistic models. An algorithm based on simulating these models by means of choosing the lattice points at random is proposed for the evaluation of these expected values. The method of sampling the lattice points is also discussed.<br>Moreover, the following points are shown.<br>(i) The values of the parameters which are necessary to execute the algorithm can be obtained easily regardless of the number of independent variables.<br>(ii) In two typical cases the proposed algorithm requires less computational labour than the deterministic method.<br>(iii) The algorithm is effective in the following situations:<br>(a) The situation when the values (data) of the function to be approximated can be measured only at the lattice points<br>(b) The situation when the function is a composite function<br>A numerical example is also presented.

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