Least Squares Approach to the Alignment of the Generic High Precision Tracking System

A least squares method to solve a generic alignment problem of high granularity tracking system is presented. The formalism takes advantage of the assumption that the derived corrections are small and consequently uses the first order linear expansion throughout. The algorithm consists of analytical linear expansion allowing for multiple nested fits. E.g. imposing a common vertex for groups of particle tracks is of particular interest. We present a consistent and complete recipe to impose constraints on any set of either implicit or explicit parameters. The baseline solution to the alignment problem is equivalent to the one described in [1]. The latter was derived using purely algebraic methods to reduce the initial large system of linear equations arising from separate fits of tracks and alignment parameters. The method presented here benefits from wider range of applications including problems with implicit vertex fit, physics constraints on track parameters, use of external information to constrain the geometry, etc. The complete formalism is given in [2]. The method has been applied to the full simulation of the ATLAS silicon tracking system. The ultimate goal is to determine ~35,000 degrees of freedom. We present a limited scale exercise exploring various aspects of the solution.