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This paper proposes a systematic mathematical analysis of both the direct andinverse acoustic scattering problem given the source in Radon measure space.For the direct problem, we investigate the well-posedness including theexistence, the uniqueness, and the stability by introducing a specialdefinition of the weak solution, i.e. \emph{very} weak solution. For theinverse problem, we choose the Radon measure space instead of the popular $L^1$space to build the sparse reconstruction, which can guarantee the existence ofthe reconstructed solution. The sparse reconstruction problem can be solved bythe semismooth Newton method in the dual space. Numerical examples areincluded.

Sparse Reconstructions of Acoustic Source for Inverse Scattering Problems in Measure Space