We present a dimension-incremental algorithm for the nonlinear approximation of high-dimensional functions in an arbitrary bounded orthonormal product basis. Our goal is to detect a suitable truncation of the basis expansion of the function, where the corresponding basis support is assumed to be unknown. Our method is based on point evaluations of the considered function and adaptively builds an index set of a suitable basis support such that the approximately largest basis coefficients are still included. For this purpose, the algorithm only needs a suitable search space that contains the desired index set. Throughout the work, there are various minor modifications of the algorithm discussed as well, which may yield additional benefits in several situations. For the first time, we provide a proof of a detection guarantee for such an index set in the function approximation case under certain assumptions on the sub-methods used within our algorithm, which can be used as a foundation for similar statements in various other situations as well. Some numerical examples in different settings underline the effectiveness and accuracy of our method.

翻译：我们提出了一种用于任意有界正交乘积基中高维函数非线性逼近的维数递增算法。我们的目标是检测函数基展开的合适截断，其中对应的基支撑集假定未知。该方法基于所考虑函数的点评估，自适应地构建合适基支撑集的索引集，使得近似最大的基系数仍被包含在内。为此，该算法仅需要包含目标索引集的适当搜索空间。本文还讨论了该算法的多种细微改进，这些改进可能在多种情况下带来额外收益。我们首次在函数逼近情形下，对所使用子方法满足特定假设时，给出了此类索引集检测保证的证明，该证明可作为其他多种类似情形中相应结论的基础。不同设置下的数值实例进一步验证了我们方法的有效性与精确性。