基于赫尔米特函数展开的磨光方法研究

The main difficulty of inverse problems is related to their ill posed nature in the sense of Hadamard. Mollification methods are of great importance for solving ill posed problems. The main drawback of existing mollification methods is that the principles for choosing mollification parameter are a priori in most of these methods. This hinders the application of these methods. Moreover, our previous research found that different solution processes have to be used for different a priori condition in order to obtain the corresponding convergence results in many methods for solving ill posed problems. In this research, we consider the mollification methods based on Hermite functions expansion to solve some ill posed problems related to unbounded domain. We will construct the mollification operator from various angles, analyse the basic features of the Hermite expansion and research the applicability of various principles for choosing mollification parameter. Moreover, theoretical analysis frames and stable algorithms for several typical problems will be obtained. Our research attempts to present new mollification methods based on Hermite functions expansion, in which, the principles of choosing parameter are a posteriori, the solution processes are uniform for different a priori conditions and the convergence rates are self adaptive. In general, the research will provide new effective tools for solving some ill posed problems and enrich theories and application range of Hermite functions expansion.

数学物理反问题一般具有不适定性的特点，这给反问题的求解带来实质困难。磨光方法是求解不适定问题的一类重要方法。现有磨光方法中，一般参数选取的方式都是先验的，这在一定程度上限制了这些方法的应用。并且，申请人前期工作中发现，很多求解不适定问题的方法，对不同的先验条件，需要调节求解过程才能获得相应的收敛结果。本课题研究将针对一些涉及无界区域的不适定问题，研究基于赫尔米特展开的磨光方法。课题将从几个不同的角度研究应用赫尔米特展开构造磨光算子的方法，分析赫尔米特展开求解不适定问题的基本性质，并研究不同的参数选取准则对方法的适用性。针对几类典型问题建立理论分析，进一步在数值上建立稳定的算法。研究旨在建立能够结合后验的参数选取方式、对不同先验条件求解过程具有一致性、并且能够自适应获得相应收敛结果的不适定问题求解方法。课题研究将丰富赫尔米特展开的理论与应用范围，进一步为不适定问题的求解提供新的有效工具。

磨光方法是不适定问题求解中的一类重要方法。以往磨光方法的参数选取大部分是先验的，本课题中，我们以赫尔米特展开为基础构造磨光方法，该方法能够通过后验停止准则来获得正则化参数，并且能够自适应的获得相应收敛阶。课题组首先针对几类典型的不适定问题获得了赫尔米特展开逼近性质及误差传播性质，拓展了前期成果中所提出方法的适用范围,为进一步研究其它磨光方式奠定了基础。进而，通过对希尔伯特尺度下的吉洪诺夫正则化方法的研究，指出了现有一些正则化方法的缺点，并考虑以希尔伯特尺度下的吉洪诺夫方法作为正则化手段来稳定赫尔米特展开的逼近过程，从而形成对数据的磨光处理。再者，利用赫尔米特函数傅里叶变换的特点，结合共轭算子，将问题解的展开系数通过扰动数据进行计算，通过截断等正则化方式实现对解的直接磨光处理。课题的研究丰富了赫尔米特展开方法的理论及应用范围，为不适定问题的求解提供了新的有效工具。