Linear Response Eigenvalue Problem: New Minimization Principles and Efficient Algorithms

线性响应本征值问题，又称随机相位近似本征值问题，是由计算物理系统的激发态（能量）引起的。这样的特征值问题通常是大规模的——一个合理大小的分子的矩阵维数可以很容易地达到数千万。它被认为比对称特征值问题困难得多因为它本质上是非厄米问题。但它在每个子矩阵块中都有内部对称结构，这在很大程度上是以前未开发的。该项目通过充分利用内部对称结构，发展新的理论和先进的计算方法。将进行系统的研究，以揭示新的最小/最大化原则。这些原则应反映对称特征值问题的原则，并将使研究人员有可能并指导研究人员将对称特征值问题的一些现有和成功的技术转化为用于线性响应特征值问题的研究。我们高度期望能够同时计算几个最小正特征值的新有效算法将作为本项目的结果出现。除了推进线性响应特征值问题的研究外，研究者还将招募和培养计算数学和跨学科研究的研究生。线性响应特征值问题是计算电子和分子能量激发态的一个重要工具。该项目将在数学理论、计算方法和软件的背景下，批判性地推进当前对特征值问题的理解和解决技术。随着项目的成功完成，将对通过随机相位近似计算最先进的物理激发能做出重大贡献，这是一种广泛应用于计算量子化学和物理的成熟技术。

The linear response eigenvalue problem, also known as the random phase approximation eigenvalue problem, arises from computing excitation states (energies) of physical systems. Such an eigenvalue problem is usually of large scale -- the matrix dimensions for a molecule of reasonable size can easily get up to tens of millions. It is considered much more difficult than the symmetric eigenvalue problem because it is non-Hermitian in nature. But it has internal symmetric structures in each of the submatrix blocks, which previously have been largely unexploited. This project involves the development of new theory and advanced computational methods through fully exploiting the internal symmetric structures. A systematic study will be thoroughly conducted to uncover new min/maximization principles. These principles should mirror those for the symmetric eigenvalue problem, and will make it possible and guide the investigator to transform some of existing and successful techniques for the symmetric eigenvalue problem for use in the linear response eigenvalue problem research, It is highly expected that new efficient algorithms that are capable of computing several smallest positive eigenvalues simultaneously will emerge as the result of this project. In addition to advancing research in the linear response eigenvalue problem, the investigator will recruit and train graduate students in computational mathematics and interdisciplinary studies. The linear response eigenvalue problem is a major tool in computing energy excitation states of electrons and molecules. This project will critically advance current understanding and solution techniques for the eigenvalue problem in the context of mathematical theory, computational methods, and software. With the successful completion of the project, a significant contribution will be made to the state-of-the-art physical excitation energy computations via random phase approximations, a proven technique that is widely used in computational quantum chemistry and physics.