  The method of superposition is proposed in combination with a sparse $\ell_1$
optimisation algorithm with the aim of finding a sparse basis to accurately
reconstruct the structural vibrations of a radiating object from a set of
acoustic pressure values on a conformal surface in the near-field. The nature
of the reconstructions generated by the method differs fundamentally from those
generated via standard Tikhonov regularisation in terms of the level of
sparsity in the distribution of charge strengths specifying the basis. In many
cases, the $\ell_1$ optimisation leads to a solution basis whose size is only a
small fraction of the total number of measured data points. The effects of
changing the wavenumber, the internal source surface and the (noisy) acoustic
pressure data in general will all be studied with reference to a numerical
study on a cuboid of similar dimensions to a typical loudspeaker cabinet. The
development of sparse and accurate reconstructions has a number of advantageous
consequences including improved reconstructions from reduced data sets, the
enhancement of numerical solution methods and wider applications in source
identification problems.
