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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(rr0) = h · u(rr0) due to a Coulomb-type point defect (Ctpd) located at a point r0 within a crystal [h is the diffraction vector and u(rr0) 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(rr0) [{\cal P} is the parameter vector that characterizes the 3D function fCtpd(rr0) 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.

Supporting information

mpg

Moving Picture Experts Group (MPG) video file https://doi.org/10.1107/S2053273320000145/lk5053sup1.mpg
An animation of the true 3D displacement field function


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