Fast alignment of a complex tracking detector using advanced track models

The inner silicon detector of the Compact Muon Solenoid experiment (CMS) at CERN's LHC consists of 16\,588 modules. Charged-particle tracks in the detector are used to improve the accuracy to which the position and orientation of the modules is known. This contribution focuses on the Millepede-II algorithm, one of the two alignment algorithms used by CMS. Recently an advanced track model has been introduced into the CMS alignment procedure, which is based on the ``Broken Lines'' model and is able to take multiple Coulomb scattering in the detector material properly into account. We show the unique approach needed for solving the alignment problem in a reasonable amount of time. Emphasis is given to the mathematical treatment of the problem.