Reduced model spaces, such as reduced basis and polynomial chaos, are linear spaces $V_n$ of finite dimension $n$ which are designed for the efficient approximation of families parametrized PDEs in a Hilbert space $V$. The manifold $\mathcal{M}$ that gathers the solutions of the PDE for all admissible parameter values is globally approximated by the space $V_n$ with some controlled accuracy $\epsilon_n$, which is typically much smaller than when using standard approximation spaces of the same dimension such as finite elements. Reduced model spaces have also been proposed in [13] as a vehicle to design a simple linear recovery algorithm of the state $u\in\mathcal{M}$ corresponding to a particular solution when the values of parameters are unknown but a set of data is given by $m$ linear measurements of the state. The measurements are of the form $\ell_j(u)$, $j=1,\dots,m$, where the $\ell_j$ are linear functionals on $V$. The analysis of this approach in [2] shows that the recovery error is bounded by $\mu_n\epsilon_n$, where $\mu_n=\mu(V_n,W)$ is the inverse of an inf-sup constant that describe the angle between $V_n$ and the space $W$ spanned by the Riesz representers of $(\ell_1,\dots,\ell_m)$. A reduced model space which is efficient for approximation might thus be ineffective for recovery if $\mu_n$ is large or infinite. In this paper, we discuss the existence and construction of an optimal reduced model space for this recovery method, and we extend our search to affine spaces. Our basic observation is that this problem is equivalent to the search of an optimal affine algorithm for the recovery of $\mathcal{M}$ in the worst case error sense. This allows us to perform our search by a convex optimization procedure. Numerical tests illustrate that the reduced model spaces constructed from our approach perform better than the classical reduced basis spaces.

中文翻译：

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基于数据状态估计的最优简化模型算法

缩减模型空间，例如缩减基和多项式混沌，是有限维数 $n$ 的线性空间 $V_n$，旨在有效逼近希尔伯特空间 $V$ 中的族参数化偏微分方程。流形 $\mathcal{M}$ 为所有可接受的参数值收集 PDE 的解，由空间 $V_n$ 全局近似，具有一些受控精度 $\epsilon_n$，通常比使用标准近似空间时小得多相同维度的，例如有限元。[13] 中也提出了减少模型空间作为一种工具来设计状态 $u\in\mathcal{M}$ 的简单线性恢复算法，当参数值未知但一组数据由状态的 $m$ 线性测量值给出。测量值的形式为 $\ell_j(u)$, $j=1,\dots,m$，其中 $\ell_j$ 是 $V$ 上的线性泛函。[2]中对这种方法的分析表明，恢复误差以 $\mu_n\epsilon_n$ 为界，其中 $\mu_n=\mu(V_n,W)$ 是描述角度的 inf-sup 常数的倒数$V_n$ 和 $(\ell_1,\dots,\ell_m)$ 的 Riesz 代表所跨越的空间 $W$ 之间。因此，如果 $\mu_n$ 很大或无穷大，则对于近似有效的缩减模型空间可能对恢复无效。在本文中，我们讨论了这种恢复方法的最佳简化模型空间的存在和构造，并将我们的搜索扩展到仿射空间。我们的基本观察是，这个问题等价于在最坏情况错误意义上寻找最佳仿射算法以恢复 $\mathcal{M}$。这允许我们通过凸优化程序执行我们的搜索。数值测试表明，根据我们的方法构建的简化模型空间比经典的简化基空间表现更好。