MQ径向基函数的理论、方法及应用

The Theory、Method and Application of MQ Radial Basis Function

径向基函数方法是近几十年来在计算科学和近似理论研究中热门的研究课题之一,广泛应用于神经网络、图像处理、偏微分方程数值解、机器学习等众多科学领域.该方法作为一个用一元函数描述多元函数的强有力工具,常用于处理大规模散乱数据,并具有较好的逼近能力.本文首先介绍有关径向基函数的发展历程、理论研究和应用背景,其次讨论Multiquadric(MQ)径向基函数方法在检测间断上的具体应用,通过数值实验验证了该方法在一维和二维问题上的有效性和实用性.最后分析MQ径向基函数方法的优缺点并对今后的研究工作提出展望.

In recent decades, Radial Basis Function（RBF） method is one of the hot research topics in the computational science and approximation theory. It has been widely applied in many scientific fields,such as neural network, image processing, numerical approximations of partial differential equations（PDEs）,machine learning and so on. RBF method is known as a powerful tool to deal with large scale scattered data problems, and it has a better approximation ability. In this paper, we first give a brief introduction to the development of the history, theoretical research and application background of the RBF, and we also discuss the application of Multiquadric Radial Basis Function（MQ-RBF） method in the detection of discontinuity.The proposed method verify the validity and practicability by a number of experiments both one and two dimensions. Finally, advantages and disadvantages of this method are summarized and the future work is prospected.