基于紧支径向基函数的支持向量机多尺度反演算法及其应用

The content of this project is mainly focusd on the compactly supported radial basis functions (RBFs) based multiscale algorithm with support vector machine regularization and its application. In the process of the establishment of the ill-posed problem, when we are given a large amount of scattered discrete data distributed on a bounded domain or a unit sphere, this algorithm will divide the data into several groups, define the scaled RBFs via given scale parameters sequence, combine with support vector machine regularization method, and reconstruct the solution step by step, scale by scale. On one hand, the compactly supported RBFs based multiscale algorithm will reduce condition number of the system, bring more stable and accurate result. On the other hand, the utilization of the support vector machine regularization will bring more robustness, with the a priori selection of the cut-off parameter, the reconstruction process will be stable, thus the regularization parameter can be chosen a priorily as well, independent of noise level and residual, therefore more applicable and efficient. In addition, the parameter strategy gives the same order or error estimate as the a posteriori strategy. In this project, both theoretical analysis, computational cost and the application scope of this multiscale method will be investigated, including convergence analysis, error estimation and numerical implementation etc.. We wish the scientific exploration of this algorithm is not only limited in theoretical research, but also can be utilized in real industrial and engineering applications, and expand the research filed of the inverse problem.

本项目将研究基于紧支径向基函数的支持向量机多尺度反演算法及其应用。在不适定的反演模型建立过程中，当定解条件是分布于有界区域或单位球面的大规模带噪离散数据时，该算法将数据分层，通过预先给定的尺度序列定义放缩紧支径向基函数，结合支持向量机正则化方法进行逐层逐尺度反演。一方面，基于紧支径向基函数的多尺度反演算法能有效降低反演系统的条件数，相较于单层反演，结果更加稳定和精确，另一方面，结合支持向量机正则化方法，先验选取的截断参数保证反演解的稳定性，使得正则化参数能够独立于噪声水平及残差先验选取，实用性强，提升效率，并且算法能获得与后验选取相同的误差阶。本项目从理论分析，算法效率，适用范围等方面深入研究基于紧支径向基函数的支持向量机多尺度反演算法的收敛性，稳定性，数值实现等关键问题，希望对该算法的科学探索不仅局限在理论研究方面，并且能用于解决工业工程等实际应用领域的具体问题，拓展反问题新的研究领域。

本项目研究基于紧支径向基函数的支持向量机多尺度反演算法及其应用。在不适定的反演模型建立过程中，当定解条件是分布于有界区域或单位球面的大规模带噪离散数据时，该算法将数据分层，通过预先给定的尺度序列定义放缩紧支径向基函数，结合支持向量机正则化方法进行逐层逐尺度反演。本项目从理论分析，算法效率，适用范围等方面深入研究基于紧支径向基函数的支持向量机多尺度反演算法的收敛性，稳定性，数值实现等关键问题。项目负责人结合项目的预定研究内容和目标，给出了几类支持向量机正则化方法的多尺度算法，进行了研究的理论分析，并结合数据压缩大大提升了算法的效率，达到了预期的目标。