论文标题
有界正规产品碱基中的非线性近似
Nonlinear approximation in bounded orthonormal product bases
论文作者
论文摘要
我们提出了一种在任意界限的正统产物基础上的高维函数的非线性近似值的维度收入算法。我们的目标是检测合适的截断功能的基础扩展，其中假定相应的基础支持是未知的。我们的方法基于对考虑函数的点评估，并自适应地构建了合适基础支持的索引集，以便仍然包括大约最大的基础系数。为此，该算法仅需要一个包含所需索引集的合适搜索空间。在整个工作中，也讨论了算法的各种次要修改，这在几种情况下可能会产生其他好处。我们第一次提供了在函数近似情况下在我们算法中使用的子方法的某些假设下设置的索引的检测保证的证明，该假设也可以用作其他各种情况下类似陈述的基础。不同设置中的一些数值示例突显了我们方法的有效性和准确性。
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.
