嵌入岭近似
Embedded Ridge Approximations
摘要：
由微分方程中心模型产生的许多感兴趣的量（qois）可以被分解为标量场的函数。此类qois的例子包括翼型上的升力或受载结构的位移;相应场的例子是计算流体动力学解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处被评估。在某些情况下，某个节点的场只受到远场扰动的微弱影响;它很可能受到局部扰动的强烈影响，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相对于规定输入中的不确定性的场的强各向异性。我们利用这一概念的局部标量场的影响近似全球qois，这往往是某些领域的数量的积分。我们正式我们的想法通过指定脊近似的字段在选择节点。这种嵌入式脊近似具有良好的理论性质，用于近似全局qoi，减少了所需的计算评估的数量。平行画我们提出的方法，积极的子空间和向量值降维。此外，我们研究了相邻节点的脊方向，并设计了算法，可以恢复在选定的节点的字段数量，当存储在节点的一个子集的脊配置文件-铺平了道路，为新的降阶建模策略。我们的论文提供了分析和模拟为基础的例子，暴露不同方面的嵌入式脊近似。
英文摘要：
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.
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