  For objects belonging to a known model set and observed through a prescribed
linear process, we aim at determining methods to recover linear quantities of
these objects that are optimal from a worst-case perspective. Working in a
Hilbert setting, we show that, if the model set is the intersection of two
hyperellipsoids centered at the origin, then there is an optimal recovery
method which is linear. It is specifically given by a constrained
regularization procedure whose parameters, short of being explicit, can be
precomputed by solving a semidefinite program. This general framework can be
swiftly applied to several scenarios: the two-space problem, the problem of
recovery from $\ell_2$-inaccurate data, and the problem of recovery from a
mixture of accurate and $\ell_2$-inaccurate data. With more effort, it can also
be applied to the problem of recovery from $\ell_1$-inaccurate data. For the
latter, we reach the conclusion of existence of an optimal recovery method
which is linear, again given by constrained regularization, under a
computationally verifiable sufficient condition. Experimentally, this condition
seems to hold whenever the level of $\ell_1$-inaccuracy is small enough. We
also point out that, independently of the inaccuracy level, the minimal
worst-case error of a linear recovery method can be found by semidefinite
programming.
