Locally Optimal Preconditioned Eigenvalue Solvers

用于特征值计算的预条件迭代求解器的领域正在迅速发展。正在编写几个预条件特征解算器的软件实现，特别是由主要研究者(PI)先前开发的局部最优块预条件共轭梯度(LOBPCG)方法。最近的进展为开发高效的内特征值和奇异值计算的预条件迭代求解器提供了新的机会。预条件特征解析器在应用中的使用提出了新的领域特定的重要问题，无论是实践上还是理论上，都需要解决。拟议的研究在国际和平研究所以前工作取得成功的基础上解决了这些问题。PI期望发展一些已知方法的理论，发现新的局部最优算法，并能够提供关于方法选择的具体建议。提出了以下具体和相关的研究项目：降低大块大小的LOBPCG成本；开发高效的内部特征值和奇异值计算的预条件求解器；针对具有高度间断系数的偏微分方程组的特征问题进行有限元误差分析。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，如IBM Bluegene/L超级计算机。拟议研究背后的想法是原创的，并建立在先前工作的基础上。该项目解决了数学上的困难和实际中的重要问题。拟议活动产生的更广泛的影响是双重的：博士教育和软件进步。提案中要求提供资金来支持博士生。目前使用的软件的改进和科学家和工程师新代码的开发带来了潜在的进步，例如，在分析对国家安全至关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.