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.
翻译:国家估算的目的是在参数矢量为美元以内线性测量时,将解决方案的美元大约重建成一个美化的部分差异方程,如果参数矢量值为美元,则该值为美元; 提议了快速数字回收方法,其依据的模型是较低的中度线性空间,这些模型是专门为接近解决方案所在的方块数 $\ mathcal{M}$。 这些方法可以被视为是巴伊西亚估算方法的确定性对应方,当先前的公式以对降低的模型的准度表示时,这些方法被证明是最佳的。然而,这些方法本质上受线性特性的限制,而线性能与科诺洛洛夫(Kolmogorov)宽度为美元(m) (m) (m) (mathcalcalcal{M}) 。在本文件中,我们建议通过简单的非线性减少模型来打破这一屏障屏障,这些模型由线性空间的最小性结合值组成,每个尺寸最多为美元,并导致不同的估测算 $+$ 。 最终选择机制, 用于在最精确的平面的平面平面平面分析。