  Many modern statistically efficient methods come with tremendous
computational challenges, often leading to large-scale optimisation problems.
In this work, we examine such computational issues for recently developed
estimation methods in nonparametric regression with a specific view on image
denoising. We consider in particular certain variational multiscale estimators
which are statistically optimal in minimax sense, yet computationally
intensive. Such an estimator is computed as the minimiser of a smoothness
functional (e.g., TV norm) over the class of all estimators such that none of
its coefficients with respect to a given multiscale dictionary is statistically
significant. The so obtained multiscale Nemirowski-Dantzig estimator (MIND) can
incorporate any convex smoothness functional and combine it with a proper
dictionary including wavelets, curvelets and shearlets. The computation of MIND
in general requires to solve a high-dimensional constrained convex optimisation
problem with a specific structure of the constraints induced by the statistical
multiscale testing criterion. To solve this explicitly, we discuss three
different algorithmic approaches: the Chambolle-Pock, ADMM and semismooth
Newton algorithms. Algorithmic details and an explicit implementation is
presented and the solutions are then compared numerically in a simulation study
and on various test images. We thereby recommend the Chambolle-Pock algorithm
in most cases for its fast convergence. We stress that our analysis can also be
transferred to signal recovery and other denoising problems to recover more
general objects whenever it is possible to borrow statistical strength from
data patches of similar object structure.
