  We propose an automated nonlinear model reduction and mesh adaptation
framework for rapid and reliable solution of parameterized advection-dominated
problems, with emphasis on compressible flows. The key features of our approach
are threefold: (i) a metric-based mesh adaptation technique to generate an
accurate mesh for a range of parameters, (ii) a general (i.e., independent of
the underlying equations) registration procedure for the computation of a
mapping $\Phi$ that tracks moving features of the solution field, and (iii) an
hyper-reduced least-square Petrov-Galerkin reduced-order model for the rapid
and reliable estimation of the mapped solution. We discuss a general paradigm
-- which mimics the refinement loop considered in mesh adaptation -- to
simultaneously construct the high-fidelity and the reduced-order
approximations, and we discuss actionable strategies to accelerate the offline
phase. We present extensive numerical investigations for a quasi-1D nozzle
problem and for a two-dimensional inviscid flow past a Gaussian bump to display
the many features of the methodology and to assess the performance for problems
with discontinuous solutions.
