适定的多元样条逼近方法研究

项目名称： 适定的多元样条逼近方法研究

项目编号： No.11471066

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 李崇君

作者单位： 大连理工大学

项目金额： 70万元

中文摘要： 样条方法在数值逼近，计算几何，微分方程数值解等领域有着重要的应用，然而，与相对完善的一元样条理论相比，对一般剖分上的多元样条，许多基本问题的研究都存在本质性的困难。当样条空间的次数与光滑度接近时，空间的维数会出现奇异性，导致样条逼近方法的严重不适定性，这是多元样条理论中一个非常有挑战性的问题，同时也限制了多元样条方法的应用。我们前期的研究发现，适当增加剖分网点的度数，可以消除样条空间的维数奇异性。在本项目中，我们将利用正则化方法研究一般剖分上的适定的多元样条逼近方法。一方面，我们将深入开展样条空间维数稳定性的研究，讨论保持样条空间维数稳定的剖分算法，以根据数据点构造自适应的网格剖分和基函数，形成适合于散乱数据拟合的多元样条逼近方法。另一方面，针对由于样条逼近方法出现的不适定性，我们将结合正则化方法，根据逼近问题选择适当的正则项（罚项）和约束优化算法，构造数值稳定的多元样条逼近算法。

中文关键词： 多元样条；函数逼近；数值逼近

英文摘要： Splines are very important methods in numerical approximation，computational geometry and numerical solutions for differential equations etc. The multivariate splines defined on general partitions are efficient for applications. Compared with the univariate spline theories, there are many essential difficulties in multivariate spline theories. When the degree is close to the smoothness order of the splines, there will be some singularities arising in the dimensions of the spline spaces and cause heavy ill-posedness. This is a big challenge in multivariate splines, and it is also a limitation for applications of splines. In our researches, we find the singularities of the dimensions can be removed by adding some degrees of vertices in the partitions. Then the dimensions are stable and some dimension formulas can be obtained. In this project, we focus on the researches on well-posed approximation methods by multivariate splines. On the one hand, we will study the stabilities of the dimensions of the spline spaces and construct the partition algorithms for keeping the stabilities of the dimensions and spline bases, so as to develop the spline approximation schemes for scatted data. On the other hand, we will adopt the regularization methods to solve the ill-posed problems due to the singularities of the spline spaces, by using some appropriate penalty terms and constrained optimization methods.

英文关键词： Multivariate spline;Function approximation;Numerical approximation