Many quantities of interest (qois) arising from differential-equation-centric models can be resolved into functions of scalar fields. Examples of such qois include the lift over an airfoil or the displacement of a loaded structure; examples of corresponding fields are the static pressure field in a computational fluid dynamics solution, and the strain field in the finite element elasticity analysis. These scalar fields are evaluated at each node within a discretised computational domain. In certain scenarios, the field at a certain node is only weakly influenced by far-field perturbations; it is likely to be strongly governed by local perturbations, which in turn can be caused by uncertainties in the geometry. One can interpret this as a strong anisotropy of the field with respect to uncertainties in prescribed inputs. We exploit this notion of localised scalar-field influence for approximating global qois, which often are integrals of certain field quantities. We formalise our ideas by assigning ridge approximations for the field at select nodes. This embedded ridge approximation has favorable theoretical properties for approximating a global qoi in terms of the reduced number of computational evaluations required. Parallels are drawn between our proposed approach, active subspaces and vector-valued dimension reduction. Additionally, we study the ridge directions of adjacent nodes and devise algorithms that can recover field quantities at selected nodes, when storing the ridge profiles at a subset of nodes---paving the way for novel reduced order modeling strategies. Our paper offers analytical and simulation-based examples that expose different facets of embedded ridge approximations.

中文翻译：

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嵌入脊近似

以微分方程为中心的模型产生的许多感兴趣量 (qois) 可以解析为标量场的函数。这种质量的例子包括机翼上方的升力或负载结构的位移；相应场的示例是计算流体动力学解决方案中的静压场和有限元弹性分析中的应变场。这些标量场在离散计算域内的每个节点上进行评估。在某些情况下，某个节点的场仅受远场扰动的影响很小；它很可能受到局部扰动的强烈控制，而局部扰动又可能由几何形状的不确定性引起。人们可以将其解释为与规定输入的不确定性相关的场强各向异性。我们利用局部标量场影响的概念来逼近全局 qois，这通常是某些场量的积分。我们通过为选定节点的场分配脊近似来形式化我们的想法。就减少所需的计算评估数量而言，这种嵌入式脊近似具有用于近似全局 qoi 的有利理论特性。我们提出的方法、活动子空间和向量值降维之间存在相似之处。此外，我们研究了相邻节点的脊方向，并设计了可以在选定节点处恢复场量的算法，当将脊剖面存储在节点子集时——为新的降阶建模策略铺平了道路。