This work is concerned with inverse problems where a distributed parameter is known a priori to only take on values from a given discrete set. This property can be promoted in Tikhonov regularization with the aid of a suitable convex but nondifferentiable regularization term. Using the specific properties of the regularization term, it can be shown that convergence actually holds pointwise. Furthermore, the resulting Tikhonov functional can be minimized efficiently using a semi-smooth Newton method. Convergence rates of the method is obtained under condition on measure of active sets. Additionally, total variation is added to promote regularity on the boundary of the reconstructions.