有理滤波器非线性最小二乘优化用于求解内部本征值问题
Non-Linear Least-Squares Optimization of Rational Filters for the Solution of Interior Eigenvalue Problems
AI总结：
本文提出基于非凸加权最小二乘方案的有理滤波器优化框架，结合FEAST库在多个基准问题中表现优异，通过利用对称性优化过程、梯度下降算法、起始位置选择及约束优化方案提升本征值求解性能。
摘要：
有理滤波函数可用于提高基于轮廓的本征求解器的收敛性，这是求解内部本征值问题的流行算法家族。我们提出了一种基于非凸加权最小二乘方案的有理滤波器优化框架。当与FEAST库结合使用时，我们的滤波器在大量代表性基准问题中优于现有方法。本工作详细描述了：(1) 利用滤波函数对称性优化过程的设置，用于Hermitian本征问题；(2) 利用对称性的梯度下降和Levenberg-Marquardt算法公式；(3) 选择优化算法起始位置的方法，以可靠产生有效的滤波器；(4) 产生具有特定属性的滤波函数的约束优化方案，这些属性可能对使用它们的本征求解器性能有益。
英文摘要：
Rational filter functions can be used to improve convergence of contour-based eigensolvers, a popular family of algorithms for the solution of the interior eigenvalue problem. We present a framework for the optimization of rational filters based on a non-convex weighted Least-Squares scheme. When used in combination with the FEAST library, our filters out-perform existing ones on a large and representative set of benchmark problems. This work provides a detailed description of: (1) a set up of the optimization process that exploits symmetries of the filter function for Hermitian eigenproblems, (2) a formulation of the gradient descent and Levenberg-Marquardt algorithms that exploits the symmetries, (3) a method to select the starting position for the optimization algorithms that reliably produces effective filters, (4) a constrained optimization scheme that produces filter functions with specific properties that may be beneficial to the performance of the eigensolver that employs them.
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