  We present a unifying view on various statistical estimation techniques
including penalization, variational and thresholding methods. These estimators
will be analyzed in the context of statistical linear inverse problems
including nonparametric and change point regression, and high dimensional
linear models as examples. Our approach reveals many seemingly unrelated
estimation schemes as special instances of a general class of variational
multiscale estimators, named MIND (MultIscale Nemirovskii--Dantzig). These
estimators result from minimizing certain regularization functionals under
convex constraints that can be seen as multiple statistical tests for local
hypotheses.
  For computational purposes, we recast MIND in terms of simpler unconstraint
optimization problems via Lagrangian penalization as well as Fenchel duality.
Performance of several MINDs is demonstrated on numerical examples.
