  State estimation aims at approximately reconstructing the solution $u$ to a
parametrized partial differential equation from $m$ linear measurements, when
the parameter vector $y$ is unknown. Fast numerical recovery methods have been
proposed based on reduced models which are linear spaces of moderate dimension
$n$ which are tailored to approximate the solution manifold $\mathcal{M}$ where
the solution sits. These methods can be viewed as deterministic counterparts to
Bayesian estimation approaches, and are proved to be optimal when the prior is
expressed by approximability of the solution with respect to the reduced model.
However, they are inherently limited by their linear nature, which bounds from
below their best possible performance by the Kolmogorov width
$d_m(\mathcal{M})$ of the solution manifold. In this paper we propose to break
this barrier by using simple nonlinear reduced models that consist of a finite
union of linear spaces $V_k$, each having dimension at most $m$ and leading to
different estimators $u_k^*$. A model selection mechanism based on minimizing
the PDE residual over the parameter space is used to select from this
collection the final estimator $u^*$. Our analysis shows that $u^*$ meets
optimal recovery benchmarks that are inherent to the solution manifold and not
tied to its Kolmogorov width. The residual minimization procedure is
computationally simple in the relevant case of affine parameter dependence in
the PDE. In addition, it results in an estimator $y^*$ for the unknown
parameter vector. In this setting, we also discuss an alternating minimization
(coordinate descent) algorithm for joint state and parameter estimation, that
potentially improves the quality of both estimators.
