标题： 参数Hermitian特征值问题的认证模型阶约简 显示英文标题

标题： Certified Model Order Reduction for parametric Hermitian eigenproblems

摘要： 本文研究了大规模参数化埃尔米特矩阵最小特征值及其相关特征空间的有效且经过认证的数值逼近方法。 为此，我们依赖于基于投影的模型降阶（MOR），即通过将其投影到一个合适的子空间上，将大规模问题约化为维数小得多的问题。 这种子空间通过弱贪婪型策略构造。 在详细阐述与源问题的约化基方法的联系之后，我们引入了一种新的误差估计，用于与最小特征值相关的特征空间的逼近误差。 由于第二小特征值与最小特征值之间的差值，即所谓的谱隙，对于误差估计的可靠性至关重要，我们提出了计算高效且可验证的高阶特征值及其谱隙的上下界，从而可以构建用于MOR近似谱隙的子空间。 在此基础上，生成第二个子空间以用于最小特征值相关特征空间的MOR近似。 我们还提供了计算高效的条件，以确保最小特征值的重数在降维空间中被完全捕获。 这项工作受到特定应用的推动：即重复识别参数化量子自旋系统模型中的最小能量状态，即所谓的基态。 显示英文摘要

摘要： This article deals with the efficient and certified numerical approximation of the smallest eigenvalue and the associated eigenspace of a large-scale parametric Hermitian matrix. For this aim, we rely on projection-based model order reduction (MOR), i.e., we approximate the large-scale problem by projecting it onto a suitable subspace and reducing it to one of a much smaller dimension. Such a subspace is constructed by means of weak greedy-type strategies. After detailing the connections with the reduced basis method for source problems, we introduce a novel error estimate for the approximation error related to the eigenspace associated with the smallest eigenvalue. Since the difference between the second smallest and the smallest eigenvalue, the so-called spectral gap, is crucial for the reliability of the error estimate, we propose efficiently computable upper and lower bounds for higher eigenvalues and for the spectral gap, which enable the assembly of a subspace for the MOR approximation of the spectral gap. Based on that, a second subspace is then generated for the MOR approximation of the eigenspace associated with the smallest eigenvalue. We also provide efficiently computable conditions to ensure that the multiplicity of the smallest eigenvalue is fully captured in the reduced space. This work is motivated by a specific application: the repeated identifications of the states with minimal energy, the so-called ground states, of parametric quantum spin system models.