Localized model reduction for nonlinear elliptic partial differential equations: localized training, partition of unity, and adaptive enrichment
Localized model reduction for nonlinear elliptic partial differential equations: localized training, partition of unity, and adaptive enrichment
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DOI：
10.1137/22m148402x
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发表时间：
2022-02
期刊：
影响因子：
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通讯作者：
K. Smetana;T. Taddei
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文献类型：
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作者：
K. Smetana;T. Taddei
我们提出了一种用于参数化{非线性}椭圆偏微分方程（PDE）的基于组件（CB）参数模型降阶（pMOR）公式。 CB-pMOR 旨在处理大规模问题，这些问题在合理的时间范围内无法负担全阶求解，或者参数变化会导致拓扑变化，从而阻碍单片 pMOR 技术的应用。我们依靠单位划分方法（PUM）从局部缩减空间设计全局逼近空间，并依靠伽辽金投影来计算全局状态估计。我们提出了一种基于过采样的随机数据压缩算法，用于构建组件的缩减空间：该方法利用过采样边界上受控平滑度的随机边界条件。我们进一步提出了一种基于残差的自适应富集算法，该算法利用代表性系统上的全局降阶求解来更新局部降维空间。我们证明了线性强制问题的富集过程的指数收敛性；我们进一步提出了二维非线性扩散问题的数值结果，以说明我们建议的许多特征并证明其有效性。
We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized {nonlinear} elliptic partial differential equations (PDEs). CB-pMOR is designed to deal with large-scale problems for which full-order solves are not affordable in a reasonable time frame or parameters' variations induce topology changes that prevent the application of monolithic pMOR techniques. We rely on the partition-of-unity method (PUM) to devise global approximation spaces from local reduced spaces, and on Galerkin projection to compute the global state estimate. We propose a randomized data compression algorithm based on oversampling for the construction of the components' reduced spaces: the approach exploits random boundary conditions of controlled smoothness on the oversampling boundary. We further propose an adaptive residual-based enrichment algorithm that exploits global reduced-order solves on representative systems to update the local reduced spaces. We prove exponential convergence of the enrichment procedure for linear coercive problems; we further present numerical results for a two-dimensional nonlinear diffusion problem to illustrate the many features of our proposal and demonstrate its effectiveness.