近场光学反问题的同伦方法

Near-field optics is a new subject since the 1980s, breaking the traditional limit of optical resolution. We take the photon scanning tunneling microscope (PSTM), which is the most advanced model at present, as a model to study the numerical methods. The near-field optical imaging is to obtain the nature of the scatterer by detecting the scattering field in the near-field region, whose mathematical model is solving a nonlinear ill-posed (inverse) problem. We have generalized the finite-dimensional homotopy method to the Hilbert space, and obtained some preliminary theoretical results and numerical methods, so we aim to apply it to inverse near-field optical problem in this project. And secondly, we aim to construct more optimal homotopy curve and more efficient numerical tracking algorithm, so that it does not contain the higher order derivatives of the operator, even without the first derivative in the numerical algorithm, in order that the calculation efficiency can be improved. Finally, the calculation of the direct problem has to be considered in the tracking algorithm, so an efficient algorithm for the direct problem is an indispensable part of the project.

近场光学是上世纪80年代以来出现的一个新兴学科，突破了传统光学的分辨率极限. 我们以目前最先进的光子扫描隧道显微镜(PSTM)为模型，研究近场光学成像的数值方法. 近场光学显微镜通过探测近场区域的散射场，来反演散射体的性质，其数学模型的本质是求解一个非线性不适定(反)问题. 申请人在攻读博士学位期间将有限维空间的同伦方法推广到Hilbert空间，并做了初步的理论分析和数值实验，本项目中我们拟将Hilbert空间的同伦方法应用于近场光学反问题的计算当中，在同伦曲线的跟踪过程中同时实现正则化. 其次，本项目要进一步构造更优化的同伦曲线以及曲线的数值跟踪算法，使其离散化的格式中不含算子的高阶导数，甚至不含一阶导数，以提高曲线跟踪的计算效率. 最后，在同伦曲线的跟踪算法中，会涉及到正问题或与正问题相关的Fréchet导数的计算，所以正问题的快速算法也是本项目所考虑的问题之一.

近场光学突破了传统光学的分辨率极限，但其计算模型涉及严重不适定与非线性两大困难。本项目使用同伦方法，旨在同伦曲线的追踪过程中同时实现正则化，即同时解决问题的不适定性和非线性。本项目首先对同伦方法做了进一步探讨，构建了Hilbert空间的同伦方法的基本理论框架，给出了Hilbert空间同伦曲线存在的条件，构造了两类用于求解不适定问题的同伦——带Tikhonov正则化的同伦和无导数同伦，给出了同伦曲线的若干种数值跟踪算法——Stefenssen方法、牛顿法、Euler法等，并从计算精度和计算效率上通过数值算例进行了对比。最后，本项目将Hilbert空间带Tikhonov正则化的同伦应用于近场光学反问题，得到了理论算法。本项目的研究结果对于一般的非线性不适定问题的求解具有借鉴意义。