  Extreme-ultraviolet (EUV) observations provide considerable insight into
evolving physical conditions in the active solar atmosphere. For a prescribed
density and temperature structure, it is straightforward to construct the
corresponding differential emission measure profile $\xi(T)$, such that $\xi(T)
\, dT$ is proportional to the emissivity from plasma in the temperature range
$[T, T + dT]$. Here we study the inverse problem of obtaining a valid $\xi(T)$
profile from a set of EUV spectral line intensities observed at a pixel within
a solar image. Our goal is to introduce and develop a regularized maximum
likelihood (RML) algorithm designed to address the mathematically ill-posed
problem of constructing differential emission measure profiles from a discrete
set of EUV intensities in specified wavelength bands, specifically those
observed by the Atmospheric Imaging Assembly (AIA) on the NASA Solar Dynamics
Observatory. The RML method combines features of maximum likelihood and
regularized approaches used by other authors. It is also guaranteed to produce
a positive definite differential emission measure profile. We evaluate and
compare the effectiveness of the method against other published algorithms,
using both simulated data generated from parametric differential emission
profile forms, and AIA data from a solar eruptive event on 2010 November 3.
Similarities and differences between the differential emission measure profiles
and maps reconstructed by the various algorithms are discussed. The RML
inversion method is mathematically rigorous, computationally efficient, and
produces acceptable measures of performance in the following three key areas:
fidelity to the data, accuracy in the reconstruction, and robustness in the
presence of data noise. As such, it shows considerable promise for computing
differential emission measure profiles from datasets of discrete spectral
lines.
