Locally Optimal Preconditioned Eigenvalue Solvers

特征值计算的预处理迭代求解器领域正在迅速发展。软件实现的几个预处理特征值求解器，特别是局部最优块预处理共轭梯度法（LOBPCG）的主要研究者（PI）早些时候开发的，正在编写。最近的进展开辟了新的机会，开发有效的预处理迭代求解器的内部特征值和奇异值计算。 在应用中使用预处理特征值求解器提出了新的领域特定的重要问题，无论是实际的和理论的，这需要解决。拟议的研究解决了这些问题的基础上，PI以前的工作的成功。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。提出了以下具体的和相互关联的研究项目：降低大块尺寸的LOBPCG成本;开发有效的内部特征值和奇异值计算的预处理求解器;有限元方法误差分析的特征值问题所产生的偏微分方程具有高度不连续系数。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。该项目解决了数学上的困难和实际上的重要问题。拟议活动产生的更广泛影响是双重的：教育和软件的进步。 建议中要求提供资金，以支持博士学位。学生改进目前使用的软件并为科学家和工程师开发新的代码，可以带来潜在的进步，例如，这对国家安全至关重要。

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.