Studying the optoelectronic structure of materials can require the computation of up to several thousands of the smallest eigenpairs of a pseudo-hermitian Hamiltonian. Iterative eigensolvers may be preferred over direct methods for this task since their complexity is a function of the desired fraction of the spectrum. In addition, they generally rely on highly optimized and scalable kernels such as matrix-vector multiplications that leverage the massive parallelism and the computational power of modern exascale systems. \textit{Chebyshev Accelerated Subspace iteration Eigensolver} (ChASE) is able to compute several thousands of the most extreme eigenpairs of dense hermitian matrices with proven scalability over massive parallel accelerated clusters. This work presents an extension of ChASE to solve for a portion of the spectrum of pseudo-hermitian Hamiltonians as they appear in the treatment of excitonic materials. The new pseudo-hermitian solver achieves similar convergence and performance as the hermitian one. By exploiting the numerical structure and spectral properties of the Hamiltonian matrix, we propose an oblique variant of Rayleigh-Ritz projection featuring quadratic convergence of the Ritz-values with no explicit construction of the dual basis set. Additionally, we introduce a parallel implementation of the recursive matrix-product operation appearing in the Chebyshev filter with limited amount of global communications. Our development is supported by a full numerical analysis and experimental tests.

翻译：研究材料的光电结构可能需要计算伪厄米特哈密顿量中多达数千个最小特征对。对于此任务，迭代特征求解器通常优于直接方法，因为其计算复杂度与所需谱段的比例相关。此外，这类方法普遍依赖高度优化且可扩展的核心计算单元（如矩阵-向量乘法），能够充分利用现代百亿亿次计算系统的大规模并行性和计算能力。\textit{切比雪夫加速子空间迭代特征求解器}（ChASE）已被证明能在超大规模并行加速集群上高效计算稠密厄米特矩阵的数千个极端特征对，并具备良好的可扩展性。本研究提出将ChASE扩展应用于求解激子材料处理中出现的伪厄米特哈密顿量的部分谱段。新型伪厄米特求解器在收敛性和计算性能方面达到了与厄米特版本相当的水平。通过利用哈密顿矩阵的数值结构与谱特性，我们提出了一种斜交瑞利-里兹投影变体，该变体在无需显式构造对偶基组的情况下即可实现里兹值的二次收敛。此外，我们针对切比雪夫滤波器中出现的递归矩阵乘积运算，提出了一种全局通信量受限的并行实现方案。本研究的完整数值分析和实验测试为上述进展提供了支撑。