Reduced-order models are essential tools to deal with parametric problems in the context of optimization, uncertainty quantification, or control and inverse problems. The set of parametric solutions lies in a low-dimensional manifold (with dimension equal to the number of independent parameters) embedded in a large-dimensional space (dimension equal to the number of degrees of freedom of the full-order discrete model). A posteriori model reduction is based on constructing a basis from a family of snapshots (solutions of the full-order model computed offline), and then use this new basis to solve the subsequent instances online. Proper orthogonal decomposition (POD) reduces the problem into a linear subspace of lower dimension, eliminating redundancies in the family of snapshots. The strategy proposed here is to use a nonlinear dimensionality reduction technique, namely, the kernel principal component analysis (kPCA), in order to find a nonlinear manifold, with an expected much lower dimension, and to solve the problem in this low-dimensional manifold. Guided by this paradigm, the methodology devised here introduces different novel ideas, namely, 1) characterizing the nonlinear manifold using local tangent spaces, where the reduced-order problem is linear and based on the neighboring snapshots, 2) the approximation space is enriched with the cross-products of the snapshots, introducing a quadratic description, 3) the kernel for kPCA is defined ad hoc, based on physical considerations, and 4) the iterations in the reduced-dimensional space are performed using an algorithm based on a Delaunay tessellation of the cloud of snapshots in the reduced space. The resulting computational strategy is performing outstandingly in the numerical tests, alleviating many of the problems associated with POD and improving the numerical accuracy.

中文翻译：

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参数化问题的非线性降维：核适当的正交分解

降阶模型是处理优化、不确定性量化或控制和逆问题中的参数问题的重要工具。参数解集位于嵌入大维空间（维数等于全阶离散模型的自由度数）中的低维流形（维数等于独立参数的数量）中。后验模型约简基于从一系列快照（离线计算的全阶模型的解决方案）构建基，然后使用这个新基在线求解后续实例。适当的正交分解 (POD) 将问题减少到较低维度的线性子空间中，从而消除快照系列中的冗余。这里提出的策略是使用非线性降维技术，即核主成分分析（kPCA），以找到一个预期维数低得多的非线性流形，并解决这个低维流形中的问题. 在这种范式的指导下，这里设计的方法引入了不同的新想法，即，1）使用局部切线空间表征非线性流形，其中降阶问题是线性的并且基于相邻快照，2）近似空间丰富快照的交叉产品，引入二次描述，3）kPCA 的内核是基于物理考虑临时定义的，4) 降维空间中的迭代使用基于降维空间中快照云的 Delaunay 细分的算法来执行。由此产生的计算策略在数值测试中表现出色，缓解了与 POD 相关的许多问题并提高了数值精度。