Studying the optoelectronic structure of materials can require the computation of several thousands of the smallest positive 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. The 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 smallest positive eigenpairs of pseudo-hermitian Hamiltonians as they appear in the treatment of excitonic materials. By exploiting the numerical structure and spectral properties of the Hamiltonian matrix, we preserve the characteristic positive-negative symmetry in the treatment of the eigenvectors and propose an oblique variant of Rayleigh-Ritz projection that features quadratic convergence of the Ritz values with no explicit construction of the dual basis. 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哈密顿量的数千个最小正特征对。对于此任务，迭代本征求解器可能优于直接方法，因其复杂度取决于所需谱段的比例。此外，它们通常依赖高度优化且可扩展的核（如矩阵-向量乘法），这些核利用了现代百亿亿次系统的巨大并行性和计算能力。Chebyshev加速子空间迭代本征求解器（ChASE）能够计算密集厄米矩阵的数千个极端特征对，并在大规模并行加速集群上证明了其可扩展性。本文提出了ChASE的扩展，以求解处理激子材料时出现的伪厄米哈密顿量的部分最小正特征对。通过利用哈密顿量矩阵的数值结构和谱特性，我们在特征向量处理中保留了正负对称性，并提出了一种Rayleigh-Ritz投影的斜变体，该变体在无需显式构造对偶基的情况下实现了Ritz值的二次收敛。此外，我们引入了一种并行实现Chebyshev滤波器中递归矩阵乘积操作的方法，该实现具有有限的全局通信量。我们的开发得到了完整的数值分析和实验测试的支持。