This talk will be about a joint work with M. Dettweiler (Heidelberg). We present a new and elementary approach to Katz' Existence Algorithm for rigid local systems. We find a purely algebraic version of the middle convolution functor used by Katz. This functor commutes with the braiding group action on tuples of matrices. This yields a new approach in inverse Galois theory for realizing subgroups of linear groups regularly as Galois groups over Q.