  We present a newly developed approach to solar coronal magnetic field
extrapolation from vector magnetograms, based on the Principle of Minimum
Dissipation Rate (MDR). The MDR system was derived from a variational problem
that is more suitable for an open and externally driven system, like the solar
corona. The resulting magnetic field equation is more general than force-free.
Its solution can be expressed as the superposition of two linear
(constant-$\alpha$) force-free fields (LFFFs) with distinct $\alpha$
parameters, and one potential field. Thus the original extrapolation problem is
decomposed into three LFFF extrapolations, utilizing boundary data. The full
MDR-based approach requires two layers of vector magnetograph measurements on
solar surface, while a slightly modified practical approach only requires one.
We test both approaches against 3D MHD simulation data in a finite volume. Both
yield quantitatively good results. The errors in the magnetic energy estimate
are within a few percents. In particular, the main features of relatively
strong perpendicular current density structures, representative of the
non-force freeness of the solution, are well recovered.
