  The Coronal Multichannel Polarimeter (CoMP) routinely performs coronal
polarimetric measurements using the Fe XIII 10747 $\AA$ and 10798 $\AA$ lines,
which are sensitive to the coronal magnetic field. However, inverting such
polarimetric measurements into magnetic field data is a difficult task because
the corona is optically thin at these wavelengths and the observed signal is
therefore the integrated emission of all the plasma along the line of sight. To
overcome this difficulty, we take on a new approach that combines a
parameterized 3D magnetic field model with forward modeling of the polarization
signal. For that purpose, we develop a new, fast and efficient, optimization
method for model-data fitting: the Radial-basis-functions Optimization
Approximation Method (ROAM). Model-data fitting is achieved by optimizing a
user-specified log-likelihood function that quantifies the differences between
the observed polarization signal and its synthetic/predicted analogue. Speed
and efficiency are obtained by combining sparse evaluation of the magnetic
model with radial-basis-function (RBF) decomposition of the log-likelihood
function. The RBF decomposition provides an analytical expression for the
log-likelihood function that is used to inexpensively estimate the set of
parameter values optimizing it. We test and validate ROAM on a synthetic test
bed of a coronal magnetic flux rope and show that it performs well with a
significantly sparse sample of the parameter space. We conclude that our
optimization method is well-suited for fast and efficient model-data fitting
and can be exploited for converting coronal polarimetric measurements, such as
the ones provided by CoMP, into coronal magnetic field data.
