Rational approximation is a powerful tool to obtain accurate surrogates for nonlinear functions that are easy to evaluate and linearize. The interpolatory adaptive Antoulas--Anderson (AAA) method is one approach to construct such approximants numerically. For large-scale vector- and matrix-valued functions, however, the direct application of the set-valued variant of AAA becomes inefficient. We propose and analyze a new sketching approach for such functions called sketchAAA that, with high probability, leads to much better approximants than previously suggested approaches while retaining efficiency. The sketching approach works in a black-box fashion where only evaluations of the nonlinear function at sampling points are needed. Numerical tests with nonlinear eigenvalue problems illustrate the efficacy of our approach, with speedups above 200 for sampling large-scale black-box functions without sacrificing on accuracy.

翻译：有理逼近是获取非线性函数精确代理模型的有力工具，此类代理模型易于求值与线性化。插值自适应Antoulas--Anderson（AAA）方法是数值构造此类逼近算子的途径之一。然而对于大规模向量值与矩阵值函数，直接应用AAA算法的集值变体会导致效率低下。我们针对此类函数提出并分析了一种名为sketchAAA的新型素描方法，该方法以高概率生成比以往方案更优的逼近算子，同时保持计算效率。该素描方法以黑箱方式运行，仅需在采样点处对非线性函数进行求值。针对非线性特征值问题的数值实验验证了本方法的有效性，在对大规模黑箱函数采样时实现了200倍以上的加速，且未牺牲计算精度。