Locally Optimal Preconditioned Eigenvalue Solvers

特征值计算的预条件迭代求解领域正在迅速发展。几个预条件特征解的软件实现，特别是由首席研究员（PI）早些时候开发的局部最优块预条件共轭梯度（LOBPCG）方法，正在编写中。最近的进展为开发有效的预置迭代求解内特征值和奇异值计算提供了新的机会。在应用中使用预条件特征解提出了新的特定领域的重要问题，包括实践和理论，需要解决。在PI先前工作的成功基础上，提出的研究解决了这些问题。PI期望推进一些已知方法的理论，发现新的局部最优算法，并能够提供有关方法选择的具体建议。建议以下具体和相互关联的研究项目：降低大块大小的LOBPCG成本；内部特征值和奇异值计算的有效预条件解；系数高度不连续偏微分方程特征问题的有限元误差分析。这些项目形成了理论研究和代码开发的平衡组合。数值模拟将在现代并行计算系统上进行，例如IBM BlueGene/L超级计算机。拟议研究背后的想法是原创的，并建立在先前的工作。该项目解决了数学上的困难和实践上的重要问题。拟议的活动产生的更广泛的影响是双重的：博士教育和软件的进步。提案中要求资金支持博士生。目前使用的软件的改进和为科学家和工程师开发的新代码创造了潜在的进步，例如，在分析对国家安全非常重要的超大数据集方面。

The area of preconditioned iterative solvers for eigenvalue computations is rapidly developing. Software implementations of several preconditioned eigensolvers, in particular, the locally optimal block preconditioned conjugate gradient (LOBPCG) method developed by the principal investigator (PI) earlier, are being written. The recent progress opens new opportunities to develop efficient preconditioned iterative solvers for interior eigenvalues and singular value computations. The use of preconditioned eigensolvers in applications raises new area-specific important issues, both practical and theoretical, which need to be resolved. The proposed research addresses these issues based on the success of the previous work of the PI. The PI expects to advance the theory of some known methods, to discover new locally optimal algorithms, and to be able to provide specific recommendations concerning the choice of methods. The following specific and interrelated research projects areproposed: reducing the LOBPCG costs for large block sizes; developing efficient preconditioned solvers for interior eigenvalues and singular value computations; finite element method error analysis for eigenproblems resulting from partial differential equations with highly discontinuous coefficients. The projects form a balanced mix of theoretical research and code development. Numerical simulations are to be performed on modern parallel computing systems, such as the IBM BlueGene/L supercomputer.The ideas behind the proposed research are original, and build on prior work. The project addresses mathematically difficult and practically important problems. The broader impact resulting from the proposed activity is twofold: Ph.D. education and advances in software. Funds are requested in the proposal to support Ph.D. students. Improvement of the software currently used and development of new codes for scientists and engineers creates potential advances, e.g., in analyzing extremely large data sets, which is important for national security.