  This paper presents a PDE-based planar parameterization framework with
support for Truncated Hierarchical B-Splines (THB-splines). For this, we adopt
the a posteriori refinement strategy of Dual Weighted Residual and present
several adaptive numerical schemes for the purpose of approximating an
inversely harmonic geometry parameterization. Hereby, the combination of
goal-oriented a posteriori strategies and THB-enabled local refinement avoids
over-refinement, in particular in geometries with complex boundaries. To
control the parametric properties of the outcome, we introduce the concept of
domain refinement. Hereby, the properties of the domain into which the mapping
maps inversely harmonically, are optimized in order to fine-tune the parametric
properties of the recomputed geometry parameterization.
