Parameter space reduction has been proved to be a crucial tool to speed-up the execution of many numerical tasks such as optimization, inverse problems, sensitivity analysis, and surrogate models' design, especially when in presence of high-dimensional parametrized systems. In this work we propose a new method called local active subspaces (LAS), which explores the synergies of active subspaces with supervised clustering techniques in order to carry out a more efficient dimension reduction in the parameter space. The clustering is performed without losing the input-output relations by introducing a distance metric induced by the global active subspace. We present two possible clustering algorithms: K-medoids and a hierarchical top-down approach, which is able to impose a variety of subdivision criteria specifically tailored for parameter space reduction tasks. This method is particularly useful for the community working on surrogate modelling. Frequently, the parameter space presents subdomains where the objective function of interest varies less on average along different directions. So, it could be approximated more accurately if restricted to those subdomains and studied separately. We tested the new method over several numerical experiments of increasing complexity, we show how to deal with vectorial outputs, and how to classify the different regions with respect to the local active subspace dimension. Employing this classification technique as a preprocessing step in the parameter space, or output space in case of vectorial outputs, brings remarkable results for the purpose of surrogate modelling.

翻译：参数空间缩减已被证明是加速优化、反问题、敏感性分析及代理模型设计等数值任务的关键工具，尤其适用于高维参数化系统。本文提出一种名为局部活性子空间（LAS）的新方法，通过探索活性子空间与监督聚类技术的协同作用，实现参数空间更高效的降维。该方法引入由全局活性子空间诱导的距离度量，在不损失输入-输出关联的前提下完成聚类。我们提出了两种聚类算法：K-medoids算法与层次化自上而下方法，后者能够针对参数空间缩减任务定制多种细分准则。该方法对从事代理模型研究的工作者尤为实用。由于参数空间常存在子区域，其中目标函数沿不同方向的平均变化幅度较小，若局限于这些子区域进行独立研究，则可实现更精确的近似。我们通过一系列复杂度递增的数值实验验证了新方法，探讨了向量型输出的处理方式，以及如何根据局部活性子空间维度对不同区域进行分类。将这种分类技术作为参数空间（或向量型输出场景下的输出空间）预处理步骤，可为代理建模带来显著成效。