  We develop and apply an enhanced regularization algorithm, used in RHESSI
X-ray spectral analysis, to constrain the ill-posed inverse problem that is
determining the DEM from solar observations. We demonstrate this
computationally fast technique applied to a range of DEM models simulating
broadband imaging data from SDO/AIA and high resolution line spectra from
Hinode/EIS, as well as actual active region observations with Hinode/EIS and
XRT. As this regularization method naturally provides both vertical and
horizontal (temperature resolution) error bars we are able to test the role of
uncertainties in the data and response functions. The regularization method is
able to successfully recover the DEM from simulated data of a variety of model
DEMs (single Gaussian, multiple Gaussians and CHIANTI DEM models). It is able
to do this, at best, to over four orders of magnitude in DEM space but
typically over two orders of magnitude from peak emission. The combination of
horizontal and vertical error bars and the regularized solution matrix allows
us to easily determine the accuracy and robustness of the regularized DEM. We
find that the typical range for the horizontal errors is $\Delta$log$T\approx
0.1 -0.5$ and this is dependent on the observed signal to noise, uncertainty in
the response functions as well as the source model and temperature. With
Hinode/EIS an uncertainty of 20% greatly broadens the regularized DEMs for both
Gaussian and CHIANTI models although information about the underlying DEMs is
still recoverable. When applied to real active region observations with
Hinode/EIS and XRT the regularization method is able to recover a DEM similar
to that found via a MCMC method but in considerably less computational time.
