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 non-standard 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.

翻译：富化逼近空间是由常规基函数与若干额外函数（例如用于捕获计算问题解已知特征的函数）张成的空间。向基中添加函数会导致其过完备，进而使对应的离散逼近问题可能需要求解病态系统。近期研究表明，在合理条件下，这些系统仍能提供高精度的数值逼近。本文提出一种计算此类逼近的高效算法。该算法基于过完备集合与框架的AZ算法，并在富化基的情况下得到简化。此外，对原始AZ算法及其改进变体的分析，为如何利用富化基实现最优且稳定的离散化提供了建设性见解。我们将该算法应用于文献中的富化逼近空间实例，包括若干非标准逼近问题及一个二维边值问题的富化谱方法，结果表明简化后的AZ算法确实具有稳定性、精确性和高效性。