Geometric Subspace Updates with Applications to Online Adaptive Nonlinear Model Reduction
Geometric Subspace Updates with Applications to Online Adaptive Nonlinear Model Reduction
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DOI：
10.1137/17m1123286
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发表时间：
2018-02
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影响因子：
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通讯作者：
Ralf Zimmermann;B. Peherstorfer;K. Willcox
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文献类型：
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作者：
Ralf Zimmermann;B. Peherstorfer;K. Willcox
在许多科学应用中，包括模型简化和图像处理，子空间被用作用于感兴趣的状态向量的低维近似和重构的近似空间。我们介绍了一个程序，用于适应现有的子空间的基础上的最小二乘问题的近似问题，使相关的最小二乘残差完全消失的信息。该方法建立在低维子空间的格拉斯曼流形上的黎曼优化过程上，即格拉斯曼秩一更新子空间估计（GROUSE）。我们建立了一个封闭形式的表达式，为剩余函数沿着测地线下降方向。在图像处理和非线性偏微分方程系统模型简化的背景下，讨论了子空间自适应的具体应用。
In many scientific applications, including model reduction and image processing, subspaces are used as ansatz spaces for the low-dimensional approximation and reconstruction of the state vectors of interest. We introduce a procedure for adapting an existing subspace based on information from the least-squares problem that underlies the approximation problem of interest such that the associated least-squares residual vanishes exactly. The method builds on a Riemmannian optimization procedure on the Grassmann manifold of low-dimensional subspaces, namely the Grassmannian Rank-One Update Subspace Estimation (GROUSE). We establish for GROUSE a closed-form expression for the residual function along the geodesic descent direction. Specific applications of subspace adaptation are discussed in the context of image processing and model reduction of nonlinear partial differential equation systems.