solution space. In particular [@Owhadi:2014] shows that this process of filtering noise with the inverse operator, when combined with conditioning, produces accurate finite-element basis functions for the solution space whose deterministic worst case errors can be bounded by standard deviation errors using the reproducing kernel structure of the covariance function of the filtered Gaussian field. As already witnessed in [@ChkrebtiiCampbell2015; @schober2014nips; @Owhadi:2014; @Hennig2015; @Hennig2015b; @Briol2015; @Conrad2015], it is natural to expect that the possibilities offered by combining numerical uncertainties/errors with model uncertainties/errors in a unified framework will stimulate a resurgence of the statistical inference approach to numerical analysis.

Linear Algebra with incomplete information {#sec:problem}
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The recovery problem
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The problem of identifying interpolation operators for is equivalent (after discretization or in the algebraic setting) to that of recovering or approximating the solution of a linear system of equations from an incomplete set of measurements (coarse variables) given known norm constraints on the image of the solution.

Let $n\geq 2$ and $A$ be a known real invertible $n\times n$ matrix. Let $b$ be an unknown element of $\R^n$. Our purpose is to approximate the solution $x$ of $$\label{eqn:scalargeneral}
A x=b$$ based on the information that (1) $x$ solves $$\label{eq:ekiudiu3d}
\Phi 