Numerical Methods for the Nonlinear Eigenvalue Problems in High Performance Materials Design

该项目的目标是开发自适应数值方法，用于预测具有重要技术价值的材料的第一原理。这些材料的特征可以是具有复杂的结构和组成、大量的颗粒(100)和很少或没有对称性。正在研究的方法包括一套迭代、加速和最小化方案，用于求解材料性质第一原理预测中出现的非线性特征值问题和伴随的椭圆型问题。该套方法包括Rayleigh-Ritz、Longine和McCormick的特征值迭代法；最小化法，如共轭梯度法、最陡下降法和Sameh和Wisniewski的迹最小化方法；以及多层方法，如多重网格法和快速多极子方法。这些方法将与自适应网格一起使用，以开发电子波函数所固有的潜在的动态、局域结构。现代并行体系结构将被用来减少计算时间和内存成本。这个项目的各个组成部分将通过来自化学、物理、计算机科学和数值分析的研究人员的密切合作来开发。

The objective of this project is to develop adaptive numerical methods for the first principles prediction of the properties of technologically important materials. These materials may be characterized as having complex structures and compositions, large numbers of particles ( 100) and little or no symmetry. The methods being investigated include a suite of iterative, acceleration, and minimization schemes for the nonlinear eigenvalue problem and accompanying elliptic problem arising in first principles prediction of material properties. This suite includes the eigenvalue iterative methods of Rayleigh-Ritz and Longsine and McCormick; minimization methods such us conjugate gradient, steepest descent, and the trace minimization method by Sameh and Wisniewski; and multilevel methods such as multigrid and fast multipole method. These methods will be used with adaptive grid to exploit the underlying dynamical, localized structure inherent in electronic wave functions. Modern parallel architectures will be used to reduce the computational time and memory costs. The various components of this project will be developed through the close collaboration of researchers from chemistry, physics, computer science, and numerical analysis.