结合梯度信息的类稀疏网格替代模型
Sparse-grids-like surrogate models enhanced with gradient information
AI总结：
本研究提出结合梯度信息的类稀疏网格替代模型，采用混合方法构建，经数值测试验证，其性能关键取决于QoI导数与QoI评估的相对成本和精度。
AI中文摘要：
本研究针对参数化非线性偏微分方程（PDE）产生的感兴趣量（QoI）开展替代建模工作，更具体地说，我们考虑将类稀疏网格替代建模方法扩展为纳入QoI关于PDE参数的导数。我们探讨了该操作为何并非直接可行，并提出一种混合方法：稀疏网格方案提供参数域内的配置点及合适的多项式空间，而替代模型通过最小二乘方法构建。我们在多个数值测试中展示了该方法，尤其讨论了其性能如何关键取决于QoI导数评估与QoI本身评估相比的相对成本和精度。
英文摘要：
This work concerns surrogate modeling for quantities of interest (QoI) arising from parametric non-linear partial differential equations (PDEs). More specifically, we consider extending the sparse-grids surrogate modeling approach to incorporate derivatives of the QoI with respect to the PDE parameters. We discuss why this operation is not straightforward and propose a hybrid approach in which a sparse-grid scheme provides the collocation points in the parameter domain and a suitable polynomial space, but the surrogate model is built with a least-squares approach. We showcase our approach on several numerical tests, and we discuss in particular how its performance crucially depends on the relative cost and accuracy of evaluating the derivatives of the QoI compared to evaluating the QoI itself.
