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Towards a solution of the inverse X‐ray diffraction tomography challenge: theory and iterative algorithm for recovering the 3D displacement field function of Coulomb‐type point defects in a crystal
Author(s) -
Chukhovskii Felix N.,
Konarev Petr V.,
Volkov Vladimir V.
Publication year - 2020
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.742
H-Index - 83
ISSN - 2053-2733
DOI - 10.1107/s2053273320000145
Subject(s) - algorithm , diffraction tomography , inverse , diffraction , displacement (psychology) , function (biology) , simulated annealing , inverse problem , vector field , physics , mathematical analysis , mathematics , geometry , quantum mechanics , psychology , evolutionary biology , psychotherapist , biology
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 f Ctpd ( r − r 0 ) = h · u ( r − r 0 ) due to a Coulomb‐type point defect (Ctpd) located at a point r 0 within a crystal [ h is the diffraction vector and u ( r − r 0 ) is the displacement vector]. The joint qNLMSA algorithm operates in a special sequence to optimize the XRDT target function in a χ 2 sense in order to recover the function f Ctpd ( r − r 0 ) [ is the parameter vector that characterizes the 3D function f Ctpd ( r − r 0 ) 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 to tend towards the global minimum even if the vector in the search is initially chosen rather a long way from the true one.