  Stellar spectra are often modeled and fit by interpolating within a
rectilinear grid of synthetic spectra to derive the stars' labels: stellar
parameters and elemental abundances. However, the number of synthetic spectra
needed for a rectilinear grid grows exponentially with the label space
dimensions, precluding the simultaneous and self-consistent fitting of more
than a few elemental abundances. Shortcuts such as fitting subsets of labels
separately can introduce unknown systematics and do not produce correct error
covariances in the derived labels. In this paper we present a new approach --
CHAT (Convex Hull Adaptive Tessellation) -- which includes several new ideas
for inexpensively generating a sufficient stellar synthetic library, using
linear algebra and the concept of an adaptive, data-driven grid. A convex hull
approximates the region where the data lie in the label space. A variety of
tests with mock datasets demonstrate that CHAT can reduce the number of
required synthetic model calculations by three orders of magnitude in an 8D
label space. The reduction will be even larger for higher-dimensional label
spaces. In CHAT the computational effort increases only linearly with the
number of labels that are fit simultaneously. Around each of these grid points
in label space an approximate synthetic spectrum can be generated through
linear expansion using a set of "gradient spectra" that represent flux
derivatives at every wavelength point with respect to all labels. These
techniques provide new opportunities to fit the full stellar spectra from large
surveys with 15-30 labels simultaneously.
