分野別セミナー

In crystal structure analysis, the discrete structures of atoms in crystalline solids are determined normally from diffraction patterns. The mathematical difficulty is mainly caused by local minimums in the process to fit structure models to experimental patterns by solving nonlinear optimization problems. Most of the problems can be also regarded as subcategories of "phase retrieval", mathematically known as underdetermined systems of quadratic equations (affected by observation errors to some degree).


For the intense demand of global optimization techniques, the semidefinite relaxation (SDR) was applied for the first time, to solve real problems in crystal structure analysis. The solved problems are categorized as quadratically constrained quadratic programs (QCQPs). It is known that under some circumstances, by using SDR, it is possible to obtain all the global minimums in short time (by also using algebraic techniques if there are more than one optimums), and guarantee that they are really global optimums by discussion based on the duality gap and the ranks of the global optimums of the obtained semidefinite programs.


I'll report on our recent finding that magnetic structure analysis and the analysis of atomic occupancies (or atomic species) in crystal structure determination provide a class of QCQPs for which SDR actually shows the above efficient properties. This was based on our numerical test by using more than 3500 crystal structures in a database. SDR also successfully determined and guaranteed global optimums in magnetic structure analysis from a real experimental pattern [1]. Since this was an interdisciplinary research carried out by experimental scientists and an expert of algebra (quadratic forms) and mathematical crystallography, I'll introduce the backgrounds of these fields, in addition to mathematical programming.


[1] K. Tomiyasu, R. Oishi-Tomiyasu, M. Matsuda & K. Matsuhira, Scientific Reports 8: 16228 (2018) https://www.nature.com/articles/s41598-018-34443-2