We present a new method for performing differential emission measure (DEM) inversions on narrowband EUV images from the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO). The method yields positive definite DEM solutions by solving a linear program. This method has been validated against a diverse set of thermal models of varying complexity and realism. These include (1) idealized gaussian DEM distributions, (2) 3D models of NOAA Active Region 11158 comprising quasi-steady loop atmospheres in a non-linear force-free field, and (3) thermodynamic models from a fully-compressible, 3D MHD simulation of AR corona formation following magnetic flux emergence. We then present results from the application of the method to AIA observations of Active Region 11158, comparing the region's thermal structure on two successive solar rotations. Additionally, we show how the DEM inversion method can be adapted to simultaneously invert AIA and XRT data, and how supplementing AIA data with the latter improves the inversion result. The speed of the method allows for routine production of DEM maps, thus facilitating science studies that require tracking of the thermal structure of the solar corona in time and space.

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