Sparse grids based on Lagrange polynomials have become one of the staple methods for approximating functions that are high-dimensional and expensive to evaluate, in the context e.g. of PDE-based parametric design exploration. They are however known to be inefficient for problems requiring local refinement, such as when the target function exhibits localized features or sharp gradients. While locally-refined sparse grids based e.g. on piecewise linear polynomials are a well-established alternative to circumvent this problem, in this work we present a strategy for improving the local efficiency of Lagrangian sparse grids. We do so by building the sparse grid approximation incrementally and evaluating the function only at collocation points at which a suitable (and crucially, zero-cost) error indicator suggest that incorporating the function evaluation would significantly change the landscape of the approximation. The remaining collocation points are instead assigned values predicted by the already available sparse grid, i.e., following a bifidelity approach that reduces costs while preserving accuracy. The effectiveness of this methodology is demonstrated on several benchmark analytical functions and an engineering application concerning flashback phenomena in hydrogen-fueled perforated burners.

翻译：基于拉格朗日多项式的稀疏网格已成为高维昂贵函数逼近的核心方法之一，尤其在基于偏微分方程的参数化设计探索等场景中应用广泛。然而，该方法在处理需要局部细化的问题时效率较低，例如当目标函数呈现局部特征或陡峭梯度时。虽然基于分段线性多项式等方法的局部细化稀疏网格已成为解决此问题的成熟替代方案，但本研究提出了一种提升拉格朗日型稀疏网格局部效率的策略。该策略通过增量式构建稀疏网格逼近，并仅在特定配置点处进行函数求值——这些配置点需满足以下条件：合适的（且关键的是零计算成本的）误差指标表明，纳入该点的函数值将显著改变逼近的整体形态。其余配置点的函数值则由已构建的稀疏网格进行预测赋值，即采用双保真度方法，在保持精度的同时降低计算成本。该方法的有效性在多个基准解析函数及涉及氢燃料多孔燃烧器回火现象的工程应用案例中得到了验证。