可穿透非均匀介质正反散射问题的非多项式有限元算法研究

The direct and inverse inhomogeneous medium scattering problem is a typical kind of all direct and inverse scattering problems, and has very important applications in practice. Although there are many reach results in this field, some problems still need to be considered. In this project, we will design the non-polynomial finite element methods for both direct and inverse inhomogeneous medium scattering problems. Compared to the traditional polynomial finite element methods, constructing bases for the finite element methods by using plane wave functions, Fourier-Bessel functions, etc. can better capture the behaviors of waves. So the non-polynomial finite element methods are of high precision, and they can deal well with large wave-number scattering problems. We will first study the non-polynomial finite element methods for the direct inhomogeneous scattering problems, involving the design, theoretical analysis and numerical simulation of the methods. Then we will use the methods to compute the Fréchet derivatives of the measured data with respect to the parameter of the inhomogeneous medium numerically. At last, we will combine the numerical Fréchet derivatives with some optimization methods or Bayesian inference methods to give numerical methods for solving the inverse inhomogeneous medium scattering problems, also we will analysis the methods and give numerical simulation for the methods.

非均匀介质的正反散射问题是诸多正反散射问题中的一类, 在实际中有着非常重要的应用. 虽然目前国际上针对正反散射问题的研究很多, 但仍有很多问题有待解决. 本项目拟分别对非均匀介质正反散射问题设计非多项式有限元法. 相对于传统以多项式为基底的有限元法, 使用平面波、Fourier-Bessel函数等非多项式函数构造基底设计有限元算法能更好的描述波动问题, 因此精度较高, 并且在处理大波数问题上能收到非常好的效果. 本项目将首先研究正散射问题的非多项式有限元算法, 包括算法设计, 理论分析和数值实现. 其次将正散射问题的数值算法应用于反散射问题中, 即利用非多项式有限元法数值求解已知数据关于介质参数的Fréchet导数. 最后将该Fréchet导数与优化算法或贝叶斯推断法相结合, 给出反散射问题的数值方法, 并进行理论分析和数值模拟.

本项目主要研究了非多项式有限元法在可穿透非均匀介质散射和反散射问题中的应用. 首先利用 Fourier-Bessel 函数构造有限元空间, 提出了正散射问题的数值算法，给出了相应的误差估计和数值实验. 其次利用非多项式有限元算法近似计算介质参数反演所需的 Fréchet 导数, 并将其与 Newton 法相结合得到反问的数值算法. 本项目所提出的非均匀介质正反散射问题的数值算法可以为医学成像、地质和石油勘探等领域的研究提供理论基础及计算机模拟技术.