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We consider the parametric elliptic PDE $-{\rm div} (a(y)\nabla u)=f$ on aspatial domain $\Omega$, with $a(y)$ a scalar piecewise constant diffusioncoefficient taking any positive values $y=(y_1, \dots, y_d)\in ]0,\infty[^d$ onfixed subdomains $\Omega_1,\dots,\Omega_d$. This problem is not uniformlyelliptic as the contrast $\kappa(y)=\frac{\max y_j}{\min y_j}$ can bearbitrarily high, contrarily to the Uniform Ellipticity Assumption (UEA) thatis commonly made on parametric elliptic PDEs. Based on local polynomialapproximations in the $y$ variable, we construct local and global reduced modelspaces $V_n$ of moderate dimension $n$ that approximate uniformly well allsolutions $u(y)$. Since the solution $u(y)$ blows as $y\to 0$, the solutionmanifold is not a compact set and does not have finite $n$-width. Therefore,our results for approximation by such spaces are formulated in terms ofrelative $H^1_0$-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 subdomainsgrows. We also show similar rates for the Galerkin projection despite the factthat high contrast is well-known to deteriorate the multiplicative constantwhen applying Cea's lemma. We finally establish uniform estimates in relativeerror for the state estimation and parameter estimation inverse problems, when$y$ is unknown and a limited number of linear measurements $\ell_i(u)$ areobserved. A key ingredient in our construction and analysis is the study of theconvergence of $u(y)$ to limit solutions when some of the parameters $y_j$ tendto infinity.

Reduced order modeling for elliptic problems with high contrast diffusion coefficients