An enriched approximation space is the span of a conventional basis with a few extra functions included, for example to capture known features of the solution to a computational problem. Adding functions to a basis makes it overcomplete and, consequently, the corresponding discretized approximation problem may require solving an ill-conditioned system. Recent research indicates that these systems can still provide highly accurate numerical approximations under reasonable conditions. In this paper we propose an efficient algorithm to compute such approximations. It is based on the AZ algorithm for overcomplete sets and frames, which simplifies in the case of an enriched basis. In addition, analysis of the original AZ algorithm and of the proposed variant gives constructive insights on how to achieve optimal and stable discretizations using enriched bases. We apply the algorithm to examples of enriched approximation spaces in literature, including a few nonstandard approximation problems and an enriched spectral method for a 2D boundary value problem, and show that the simplified AZ algorithm is indeed stable, accurate and efficient.

中文翻译：

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丰富逼近空间中的高效函数逼近

丰富的近似空间是包含一些额外函数的传统基础的范围，例如捕获计算问题解决方案的已知特征。向基础添加函数会使其变得过完备，因此，相应的离散逼近问题可能需要求解病态系统。最近的研究表明，这些系统在合理的条件下仍然可以提供高精度的数值近似。在本文中，我们提出了一种有效的算法来计算此类近似值。它基于针对超完备集和框架的 AZ 算法，在丰富基础的情况下进行了简化。此外，对原始 AZ 算法和所提出的变体的分析为如何使用丰富的基数实现最佳和稳定的离散化提供了建设性的见解。我们将该算法应用于文献中丰富的近似空间的示例，包括一些非标准近似问题和二维边值问题的丰富谱方法，并表明简化的 AZ 算法确实稳定、准确和高效。