A new strategy for constructing parametric Galerkin reduced order models is presented in this article. This strategy is achieved thanks to the Riemannian manifold, quotient of the set of full-rank matrices by the orthogonal group. Starting from a set of training parametrized subspaces of the same dimension, namely obtained by the proper orthogonal decomposition, the projection subspace for a new untrained parameter value is sought as the Karcher barycenter of the data points. The principal advantage of this strategy is that it leads to parametric reduced order models that are naturally flexible with respect to parameter variations. This property is a result of the simple expressions of the geodesic exponential and logarithmic mappings involved in the calculations of the approximated projection subspace. Confrontation with the usual interpolation in the tangent space to the Grassmann manifold is carried out for the flow problem past a circular cylinder and the flow in a lid-driven cavity. Comparable results in terms of accuracy are obtained, while the proposed approach is shown to be computationally cheaper by allowing real-time update of Galerkin projections.

中文翻译：

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基于黎曼重心插值的参数降阶模型

本文提出了一种构建参数化 Galerkin 降阶模型的新策略。该策略的实现归功于黎曼流形，即正交群的满秩矩阵集的商。从一组相同维度的训练参数化子空间开始，即通过适当的正交分解获得，寻找新的未训练参数值的投影子空间作为数据点的 Karcher 重心。这种策略的主要优点是它导致参数降阶模型在参数变化方面自然灵活。此属性是近似投影子空间计算中涉及的测地线指数和对数映射的简单表达式的结果。与 Grassmann 流形的切线空间中的通常插值的对抗是针对通过圆柱体的流动问题和在盖子驱动的腔中的流动进行的。获得了准确性方面的可比结果，而通过允许实时更新 Galerkin 投影，表明所提出的方法在计算上更便宜。