  Finding a sparse representation of a possibly noisy signal can be modeled as
a variational minimization with l_q-sparsity constraints for q less than one.
Especially for real-time, on-line, or iterative applications, in which problems
of this type have to be solved multiple times, one needs fast algorithms to
compute these minimizers. Identifying the exact minimizers is computationally
expensive. We consider minimization up to a constant factor to circumvent this
limitation. We verify that q-dependent modifications of shrinkage rules provide
closed formulas for such minimizers. Therefore, their computation is extremely
fast. We also introduce a new shrinkage rule which is adapted to q. To support
the theoretical results, the proposed method is applied to Landweber iteration
with shrinkage used at each iteration step. This approach is utilized to solve
the ill-posed problem of analytic ultracentrifugation, a method to determine
the size distribution of macromolecules. For relatively pure solutes, our
proposed scheme leads to sparser solutions with sharper peaks, higher
resolution, and smaller residuals than standard regularization for this
problem.
