  Signal reconstruction in compressive sensing involves finding a sparse
solution that satisfies a set of linear constraints. Several approaches to this
problem have been considered in existing reconstruction algorithms. They each
provide a trade-off between reconstruction capabilities and required
computation time. In an attempt to push the limits for this trade-off, we
consider a smoothed l0 norm (SL0) algorithm in a noiseless setup. We argue that
using a set of carefully chosen parameters in our proposed adaptive SL0
algorithm may result in significantly better reconstruction capabilities in
terms of phase transition while retaining the same required computation time as
existing SL0 algorithms. A large set of simulations further support this claim.
Simulations even reveal that the theoretical l1 curve may be surpassed in major
parts of the phase space.
