基于 PDE 特性的特征值计算新型计算模式研究

本项目进行PDE 特征值计算的高精度格式构造及基于特征信息的预条件子新型算法研究：基于先验估计同时逼近特征值的上下界，构造了一系列高精度格式及相应的多级精度提升格式，测试表明对于提高奇异特征函数及高频特征值的精度均有显著效果；针对 CFD 扩散项为背景的数值离散矩阵，利用谱图理论，构造了基于 Low Stretch生成树的LST类预条件子新型算法,谱分布分析与 数值实验均表明，与现有的MILU、ILU等传统预条件子相比，其可扩展性显著增强，其预处理加速效果明显提升。培育项目延续期间，将进一步发展前期提出的特征值计算多级精度提升算法，并进行高效程序实现，从模型测试发展到应用测试；将针对LST类预条件子的Sparsification算法进行深入研究与分析，针对Partitioning进行程序实现，力图得到具有高可扩展性的并行预条件子，解决传统预条件子的应用缺陷。

During the past work of the project, we obtained the high accuracy schemes of the PDE eigen-computation and studied the new-type preconditioned algorithm based on the feature information: we approximated the upper and lower bound of the eigenvalues simultaneously based on a priori estimate, and then constructed a series of high-accuracy schemes and multi-stage upgrading format, numerical tests show that they have a significant effect on improving the accuracy of the singular eigen functions and high-frequency eigenvalues; according to the numerical discrete background of CFD, we constructed a new matrix preconditioned algorithm based on LST by using spectral graph theory, numerical experimental results show that it has significant pre-accelerated effects compared with IC, MIC, Vaidya maximum weight spanning tree preconditioner. Our aims are the further development of the early precision Upgrading algorithm, the efficient program implementation, and the model test also the applied test; another work is the theoretical analysis and algorithm implementation for Sparsification sector of LST-like preconditioner, in order to get a much higher scalable parallel preconditioner than that of traditional preconditioners especially for application-oriented PDE solver.