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

数值求解大型稀疏厄米特特征值问题（HEP）和具有少数极端特征值的广义厄米特特征值问题（GHEP）是各种应用中最重要但也是计算密集型的任务之一。例子丰富的光谱图分割，大规模光谱图分析，结构工程，电磁学，晶格量子色动力学，和电子结构的应用，从原子尺度物理学到分子尺度材料科学。与此密切相关的是计算矩阵的部分奇异值分解（SVD）的问题，该问题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佳的方法，使软件包PRIMME击败。PRIMME，或PConditioned Iterative MultiMethod Eigensolver，是2005年开发的用于求解HEP的软件包。 PRIMME将最先进的预处理迭代方法从“前沿”带到生产中，并提供灵活但高度可用的界面。 然而，它是它的重点数值鲁棒性和计算效率，已获得PRIMME的认可，作为最好的特征值包之一。这一成功要求一个新的努力，延长PRIMME与一些期待已久的功能，但也包括新的算法，以解决一些突出的问题，在特征值计算。这项工作对许多研究依赖于晶格QCD和材料科学软件包的团队至关重要，PRIMME将通过合作改进这些软件包。PRIMME已经有一个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，最受欢迎的功能是MATLAB接口。这将向最终用户释放“工业实力”软件的力量。最后，本项目将对两名研究生和几名本科生进行高性能数值软件方面的教育和培训，具体目标包括：GHEP的PRIMME扩展，特别关注病态质量矩阵; SVD的PRIMME扩展，特别注意以高精度获得结果（解决方案不仅必须包括PRIMME的强大组件，还必须包括已知方法和新方法的组合，以及在它们之间进行选择的动态方法）;实施解决高度内部特征值问题和计算大量特征值的新方法和技术;与DOE库和MATLAB的互操作性，以及改进的传播手段。作为一个数值线性代数内核，PRIMME在计算科学界有很大的潜在受众。然而，两个具体的合作将提供现实世界，具有挑战性的问题，并作为一个压力测试评估所产生的方法和软件。一个涉及美国能源部杰斐逊实验室的晶格QCD小组，另一个涉及IBM，苏黎世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.