Nonlinear Reduced Models for State and Parameter Estimation

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 in Maday et al. [Internat. J. Numer. Methods Engrg., 102 (2015), pp. 933--965] based on reduced models which are linear spaces of moderate dimension $n$ that are tailored to approximate the solution manifold ${\cal M}$ 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 [P. Binev et al., SIAM/ASA J. Uncertain. Quantif., 5 (2017), pp. 1--29]. However, they are inherently limited by their linear nature, which bounds from below their best possible performance by the Kolmogorov width $d_m({\cal M})$ 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.