标题： 基于参数域分解与重构的降阶基方法自适应贪心算法 显示英文标题

标题： Adaptive greedy algorithms based on parameter-domain decomposition and reconstruction for the reduced basis method

摘要： 通过离线-在线分解，减少基方法（RBM）能够对参数化偏微分方程进行重复和快速评估，也称为学习-执行过程。该方法的一个关键特征是贪婪算法反复扫描训练集，即参数域的精细离散化，以确定沿参数诱导解流形的下一个维度，从而扩展代理解空间。尽管该方法已被成功应用于具有相当高参数维度的问题，但挑战在于这种扫描成本主导了离线成本，因为它与训练集的基数成正比，而训练集的基数随着参数维度呈指数增长。在本工作中，我们回顾了三种有效延缓维度灾难的最新尝试，并通过逐步细化和在训练集上对误差估计进行多级最大化提出了两种新的混合策略。所有五种离线增强方法和原始贪婪算法都在两种类型的问题上进行了测试和比较：热块问题和几何参数化的赫姆霍兹问题。 显示英文摘要

摘要： The reduced basis method (RBM) empowers repeated and rapid evaluation of parametrized partial differential equations through an offline-online decomposition, a.k.a. a learning-execution process. A key feature of the method is a greedy algorithm repeatedly scanning the training set, a fine discretization of the parameter domain, to identify the next dimension of the parameter-induced solution manifold along which we expand the surrogate solution space. Although successfully applied to problems with fairly high parametric dimensions, the challenge is that this scanning cost dominates the offline cost due to it being proportional to the cardinality of the training set which is exponential with respect to the parameter dimension. In this work, we review three recent attempts in effectively delaying this curse of dimensionality, and propose two new hybrid strategies through successive refinement and multilevel maximization of the error estimate over the training set. All five offline-enhanced methods and the original greedy algorithm are tested and compared on {two types of problems: the thermal block problem and the geometrically parameterized Helmholtz problem.