Model-order reduction methods tackle the following general approximation problem: find an “easily-computable” but accurate approximation \(\hat {\boldsymbol {h}}\) of some target solution h^⋆. In order to achieve this goal, standard methodologies combine two main ingredients: (i) a set of partial observations of h^⋆ and (ii) some “simple” prior model on the set of target solutions. The most common prior models encountered in the literature assume that the target solution h^⋆ is “close” to some low-dimensional subspace. Recently, triggered by the work by Binev et al. (SIAM/ASA Journal on Uncertainty Quantification 5(1), 1–29, 2017), several contributions have shown that refined prior models (based on a set of embedded approximation subspaces) may lead to enhanced approximation performance. In this paper, we focus on a particular decoder exploiting such a “multi-space” information and evaluating \(\hat {\boldsymbol {h}}\) as the solution of a constrained optimization problem. To date, no theoretical results have been derived to support the good empirical performance of this decoder. The goal of the present paper is to fill this gap. More specifically, we provide a mathematical characterization of the approximation performance achievable by this variational “multi-space” decoder and emphasize that, in some specific setups, it has provably better recovery guarantees than its standard “single-space” counterpart. We also discuss the similarities and differences between this decoder and the one proposed in Binev et al. (SIAM/ASA Journal on Uncertainty Quantification 5(1), 1–29, 2017).

中文翻译：

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可变“多空间”解码器的性能保证

模型阶数减少方法解决了以下一般逼近问题：找到某个目标解h ^an的“易于计算”但精确的逼近\（\ hat {\ boldsymbol {h}} \）。为了实现这一目标，标准的方法结合了两种主要成份：（I）一组的部分观测的ħ ^⋆上所设定的目标的解决方案，和（ii）一些“简单”的先验模型。在文献中所遇到的最常见的现有模型假定目标溶液ħ ^⋆是“接近”一些低维子空间。最近，由Binev等人的工作触发。（SIAM / ASA不确定性量化杂志5（1），1-29，2017），一些贡献表明改进的先验模型（基于一组嵌入式近似子空间）可能会导致增强的近似性能。在本文中，我们专注于利用这种“多空间”信息并评估\（\ hat {\ boldsymbol {h}} \）的特定解码器作为约束优化问题的解决方案。迄今为止，尚未获得理论结果来支持该解码器的良好经验性能。本文的目的是填补这一空白。更具体地说，我们提供了这种变体“多空间”解码器可实现的近似性能的数学表征，并强调指出，在某些特定设置中，它具有比标准“单空间”对应物更好的恢复保证。我们还将讨论该解码器与Binev等人提出的解码器之间的异同。（SIAM / ASA不确定性量化期刊5（1），2017年1月29日）。