  MR-STAT is a recently proposed framework that allows the reconstruction of
multiple quantitative parameter maps from a single short scan by performing
spatial localisation and parameter estimation on the time domain data
simultaneously, without relying on the FFT. To do this at high-resolution,
specialized algorithms are required to solve the underlying large-scale
non-linear optimisation problem. We propose a matrix-free and parallelized
inexact Gauss-Newton based reconstruction algorithm for this purpose. The
proposed algorithm is implemented on a high performance computing cluster and
is demonstrated to be able to generate high-resolution ($1mm \times 1mm$
in-plane resolution) quantitative parameter maps in simulation, phantom and
in-vivo brain experiments. Reconstructed $T_1$ and $T_2$ values for the gel
phantoms are in agreement with results from gold standard measurements and for
the in-vivo experiments the quantitative values show good agreement with
literature values. In all experiments short pulse sequences with robust
Cartesian sampling are used for which conventional MR Fingerprinting
reconstructions are shown to fail.
