SI2-SSE: Enhancing the PReconditioned Iterative MultiMethod Eigensolver Software with New Methods and Functionality for Eigenvalue and Singular Value Decomposition (SVD) Problems

大型稀疏厄米特征值问题（HEP）和少数极值特征值的广义厄米特征值问题（HEP）的数值解是各种应用中最重要但也是计算量最大的任务之一。在谱图划分、大规模谱图分析、结构工程、电磁学、晶格量子色动力学以及从原子尺度物理到分子尺度材料科学的电子结构应用中，例子比比皆是。密切相关的是计算矩阵的部分奇异值分解（SVD）的问题，它在包括数据挖掘和机器学习在内的许多应用中都有日常使用。在过去的十年中，在开发高质量的特征值软件包方面投入了大量的资源，这证明了问题的重要性。然而，这些软件包仍然不包括接近最优的方法，这些方法使软件包prime成为软件的首选。PRIMME，即预条件迭代多方法特征求解器，是2005年开发的用于求解HEP的软件包。PRIMME带来了最先进的预置迭代方法，从“前沿”到生产，具有灵活但高度可用的界面。然而，它对数值鲁棒性和计算效率的关注使prime成为最好的特征值包之一。这一成功需要新的努力来扩展prime的一些期待已久的功能，同时也包括新的算法来解决特征值计算中的一些突出问题。这项工作对许多依赖晶格QCD和材料科学软件包的研究小组至关重要，prime将通过合作改进这些软件包。prime已经有了一个PETSc接口，随着Hypre和Trilinos接口的拟议开发，它将被更广泛的用户社区访问。然而，最需要的功能是MATLAB接口。这将向终端用户释放“工业实力”软件的力量。最后但并非最不重要的是，该项目将在高性能数值软件艺术方面教育和培训两名研究生和几名本科生。这些项目的具体目标包括：将质数模型扩展到全球环境模型，特别注意病态质量矩阵；PRIMME扩展到SVD，特别注意获得高精度的结果（解决方案不仅必须包括PRIMME的鲁棒组件，还必须包括已知和新方法的组合，以及在它们之间进行动态选择的方法）；实现求解高度内蕴特征值问题和计算大量特征值的新方法和新技术；与DOE库和MATLAB的互操作性，以及改进的传播手段。作为一种数值线性代数内核，prime在计算科学界拥有大量潜在的受众。然而，两个特定的协作将提供现实世界中具有挑战性的问题，并作为结果方法和软件的压力测试评估器。一个项目涉及美国能源部杰弗逊实验室的晶格QCD小组，另一个项目涉及苏黎世IBM的高性能计算和材料科学小组。

The numerical solution of large, sparse Hermitian Eigenvalue Problems (HEP) and Generalized HEP (GHEP) for a few extreme eigenvalues is one of the most important but also computationally intensive tasks in a variety of applications. Examples abound in spectral graph partitioning, large scale spectral graph analysis, structural engineering, electromagnetics, lattice Quantum Chromodynamics, and electronic structure applications from atomic scale physics to molecular scale materials science. Closely related is the problem of computing a partial Singular Value Decomposition (SVD) of a matrix, which finds everyday use in numerous applications including data mining and machine learning. The importance of the problem is evidenced by the significant resources that have been invested over the last decade in developing high quality eigenvalue software packages. However, these packages still do not include the near-optimal methods that have made the package PRIMME the software to beat. PRIMME, or PReconditioned Iterative MultiMethod Eigensolver, is a software package developed in 2005 for the solution of HEP. PRIMME brings state-of-the-art preconditioned iterative methods from "bleeding edge" to production, with a flexible, yet highly usable interface. Yet, it is its focus on numerical robustness and computational efficiency that has gained PRIMME the recognition as one of the best eigenvalue packages. This success calls for a new effort to extend PRIMME with some long awaited functionality but also to include new algorithms to address some outstanding problems in eigenvalue computations. This work is critical to many groups whose research depends on the lattice QCD and materials science software packages that PRIMME will improve through collaborations. PRIMME already has a PETSc interface, and with the proposed development of Hypre and Trilinos interfaces, it will be accessible by a far wider community of users. The most requested feature, however, has been a MATLAB interface. This will unleash the power of an "industrial strength" software to end users. Last but not least, this project will educate and train two graduate and several undergraduate students in the art of high performance numerical software.Specific goals for this projects include: PRIMME extension to GHEP, with special attention to ill conditioned mass matrices; PRIMME extension to SVD, with special attention to obtaining results at high accuracy (the solution must include not only PRIMME's robust components but a combination of known and new methods, as well as a dynamic way to choose between them); implementation of new methods and techniques for the solution of highly interior eigenvalue problems and for the computation of a large number of eigenvalues; interoperability with DOE libraries and MATLAB, and improved means of dissemination. As a numerical linear algebra kernel, PRIMME has a large potential audience in the computational sciences community. However, two specific collaborations will provide real-world, challenging problems and serve as a stress-test evaluator of the resulting methods and software. One involves the lattice QCD group at the DOE's Jefferson Lab, and the other involves the high performance computing and materials science group at IBM, Zurich.