论文标题
非线性椭圆形偏微分方程的局部模型减少：局部训练，统一分配和自适应富集
Localized model reduction for nonlinear elliptic partial differential equations: localized training, partition of unity, and adaptive enrichment
论文作者
论文摘要
我们为参数化{nonlinear}椭圆形偏微分方程（PDES）提出了一个基于组件的（CB）参数模型订购（PMOR）公式。 CB-PMOR旨在处理大规模问题，在合理的时间范围或参数的变化中，全阶求解无法负担得起拓扑变化，从而阻止了单层PMOR技术的应用。我们依靠 - 划分方法（PUM）来从局部减小的空间和Galerkin投影中设计全球近似空间来计算全球状态估计。我们提出了一种基于对组件减少空间的构建的过采样的随机数据压缩算法：该方法利用了在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自适应残差的富集算法，该算法利用了代表性系统上的全局减少订单解决方案，以更新局部减少的空间。我们证明了线性强制性问题的富集程序的指数融合；我们进一步提出了二维非线性扩散问题的数值结果，以说明我们提案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.
