This work introduces an empirical quadrature-based hyperreduction procedure and greedy training algorithm to effectively reduce the computational cost of solving convection-dominated problems with limited training. The proposed approach circumvents the slowly decaying $n$-width limitation of linear model reduction techniques applied to convection-dominated problems by using a nonlinear approximation manifold systematically defined by composing a low-dimensional affine space with bijections of the underlying domain. The reduced-order model is defined as the solution of a residual minimization problem over the nonlinear manifold. An online-efficient method is obtained by using empirical quadrature to approximate the optimality system such that it can be solved with mesh-independent operations. The proposed reduced-order model is trained using a greedy procedure to systematically sample the parameter domain. The effectiveness of the proposed approach is demonstrated on two shock-dominated computational fluid dynamics benchmarks.

中文翻译：

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使用隐式特征跟踪和经验求积加速求解对流主导的偏微分方程

这项工作引入了基于经验求积的超还原过程和贪婪训练算法，以有效降低通过有限训练解决对流主导问题的计算成本。所提出的方法规避了缓慢衰减的$n$-通过使用非线性逼近流形应用于对流主导问题的线性模型简化技术的宽度限制，该非线性逼近流形通过与基础域的双射组成低维仿射空间而系统地定义。降阶模型被定义为非线性流形上残差最小化问题的解。通过使用经验求积来逼近最优系统，从而可以通过与网格无关的操作来求解，获得了一种在线高效的方法。所提出的降阶模型使用贪婪程序进行训练，以系统地对参数域进行采样。该方法的有效性在两个冲击主导的计算流体动力学基准上得到了证明。