  We consider the parametric elliptic PDE $-{\rm div} (a(y)\nabla u)=f$ on a
spatial domain $\Omega$, with $a(y)$ a scalar piecewise constant diffusion
coefficient taking any positive values $y=(y_1, \dots, y_d)\in ]0,\infty[^d$ on
fixed subdomains $\Omega_1,\dots,\Omega_d$. This problem is not uniformly
elliptic as the contrast $\kappa(y)=\frac{\max y_j}{\min y_j}$ can be
arbitrarily high, contrarily to the Uniform Ellipticity Assumption (UEA) that
is commonly made on parametric elliptic PDEs. Based on local polynomial
approximations in the $y$ variable, we construct local and global reduced model
spaces $V_n$ of moderate dimension $n$ that approximate uniformly well all
solutions $u(y)$. Since the solution $u(y)$ blows as $y\to 0$, the solution
manifold is not a compact set and does not have finite $n$-width. Therefore,
our results for approximation by such spaces are formulated in terms of
relative $H^1_0$-projection error, that is, after normalization by
$\|u(y)\|_{H^1_0}$. We prove that this relative error decays exponentially with
$n$, yet exhibiting the curse of dimensionality as the number $d$ of subdomains
grows. We also show similar rates for the Galerkin projection despite the fact
that high contrast is well-known to deteriorate the multiplicative constant
when applying Cea's lemma. We finally establish uniform estimates in relative
error for the state estimation and parameter estimation inverse problems, when
$y$ is unknown and a limited number of linear measurements $\ell_i(u)$ are
observed. A key ingredient in our construction and analysis is the study of the
convergence of $u(y)$ to limit solutions when some of the parameters $y_j$ tend
to infinity.
