The theoretical framework and a joint quasi-Newton-Levenberg-Marquardt-simulated annealing (qNLMSA) algorithm are established to treat an inverse X-ray diffraction tomography (XRDT) problem for recovering the 3D displacement field function fCtpd(r - r0) = h · u(r - r0) due to a Coulomb-type point defect (Ctpd) located at a point r0 within a crystal [h is the diffraction vector and u(r - r0) is the displacement vector]. The joint qNLMSA algorithm operates in a special sequence to optimize the XRDT target function {\cal F}\{ {\cal P} \} in a χ2 sense in order to recover the function fCtpd(r - r0) [{\cal P} is the parameter vector that characterizes the 3D function fCtpd(r - r0) in the algorithm search]. A theoretical framework based on the analytical solution of the Takagi-Taupin equations in the semi-kinematical approach is elaborated. In the case of true 2D imaging patterns (2D-IPs) with low counting statistics (noise-free), the joint qNLMSA algorithm enforces the target function {\cal F} \{ {\cal P} \} to tend towards the global minimum even if the vector {\cal P} in the search is initially chosen rather a long way from the true one.

中文翻译：

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寻求反X射线衍射层析成像挑战的解决方案：用于恢复晶体中库仑型点缺陷的3D位移场函数的理论和迭代算法。

建立了理论框架和联合准牛顿-莱文贝格-马夸特模拟退火（qNLMSA）算法来处理逆X射线衍射层析成像（XRDT）问题，以恢复3D位移场函数fCtpd（r-r0）= h ·由于位于晶体内点r0的库仑型点缺陷（Ctpd）而导致的u（r-r0）[h是衍射矢量，u（r-r0）是位移矢量]。联合qNLMSA算法以特殊顺序运行，以在χ2的意义上优化XRDT目标函数{\ cal F} \ {{\ cal P} \}，以恢复函数fCtpd（r-r0）[{\ cal P }是表征算法搜索中3D函数fCtpd（r-r0）的参数向量。阐述了基于半运动学方法的高木-陶平方程解析解的理论框架。