  We introduce the sparse direct sampling method (DSM) to estimate properties
of a region from signals that probe the region. We demonstrate the sparse-DSM
on two separate problems: estimating both the angle-of-arrival of a radio wave
impinging on an array and the location and shape of an inhomogeneity from
scattered acoustic waves. The sparse-DSM is qualitative in nature, so it does
not require the simulation of a forward problem to solve the inverse problem.
The method generalizes of two older qualitative methods, one which has
low-resolution reconstructions but uses few measurements and one which is
high-resolution but has higher measurement cost. The sparse-DSM inherits
positive qualities from both. We demonstrate the technique on measured and
simulated examples.
