Optimal Reduced Model Algorithms for Data-Based State Estimation
Optimal Reduced Model Algorithms for Data-Based State Estimation
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DOI：
10.1137/19m1255185
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发表时间：
2019-03
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通讯作者：
A. Cohen;W. Dahmen;R. DeVore;M. Fadili;Olga Mula;James Nichols
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文献类型：
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作者：
A. Cohen;W. Dahmen;R. DeVore;M. Fadili;Olga Mula;James Nichols
降维模型空间，如简化基和多项式混沌，是有限维线性空间V_n的有效逼近，是为了有效逼近Hilbert空间中的族参数偏微分方程族而设计的。集合了所有允许参数值的偏微分方程解的流形$\数学{M}$被具有一定控制精度的空间$V_n$全局逼近，这通常比使用相同维度的标准逼近空间(如有限元)时要小得多。在文献[13]中还提出了简化的模型空间，作为一种工具来设计一种简单的线性恢复算法，当参数值未知，但由状态的$m$线性测量给出一组数据时，对应于特定解的状态的简单的线性恢复算法。度量的形式为$\ell_j(U)$，$j=1，\dots，m$，其中$\ell_j$是$V$上的线性泛函。文[2]中对这种方法的分析表明，恢复误差由$Mu_n\epsilon_n$有界，其中$\Mu_n=\Mu(V_n，W)$是描述$V_n$与$(\ell_1，\dots，\ell_m)$的Riesz表示所跨越的空间$W之间的夹角的inf-sup常数的逆。因此，如果$MU_n$很大或无穷大，则简化的模型空间对于近似是有效的，对于恢复可能是无效的。在这篇文章中，我们讨论了这种恢复方法的最优约简模型空间的存在和构造，并将我们的搜索扩展到仿射空间。我们的基本观察是，这个问题等价于在最坏情况误差意义下寻找一个最优仿射算法来恢复$\mathcal{M}$。这允许我们通过凸优化过程来执行我们的搜索。数值试验表明，用该方法构造的约简模型空间比经典的约简基空间具有更好的性能。
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.