In this paper, we propose a Parallel Hybrid Direct-Iterative Eigensolver for Hermitian Eigenvalue Problems without tridiagonalization, denoted byPHIDE, which combines direct and iterative methods.PHIDEfirst reduces a Hermitian matrix to banded form, then applies a spectrum slicing algorithm to the banded matrix, and finally computes the eigenvectors of the original matrix via backtransformation. Compared with conventional direct eigensolvers,PHIDEavoids tridiagonalization, which involves many memory-bound operations. InPHIDE, the banded eigenvalue problem is solved using the contour integral method implemented in FEAST, which may yield slightly lower accuracy than tridiagonalization-based approaches. For sequences of correlated Hermitian eigenvalue problems arising in density functional theory (DFT),PHIDEachieves an average speedup of$1.22\times$over the state-of-the-art direct solver in ELPA when using 1024 processes. Numerical experiments are conducted on dense Hermitian matrices from real applications as well as large sparse matrices from the SuiteSparse and ELSES collections.

Summary Iterative linear algebra methods to solve linear systems and eigenvalue problems with non‐Hermitian matrices are important for both the simulation arising from diverse scientific fields and the applications related to big data, machine learning, and artificial intelligence. The spectral property of these matrices has impacts on the convergence of these solvers. Moreover, with the increase of the size of applications, iterative methods are implemented in parallel on clusters. Analysis of their behaviors with non‐Hermitian matrices on supercomputers is so complex that we need to generate large‐scale matrices with different given spectra for benchmarking. These test matrices should be non‐Hermitian and nontrivial, with high dimension. This paper highlights a scalable matrix generator that constructs large sparse matrices using the user‐defined spectrum, and the eigenvalues of generated matrices are ensured to be the same as the predefined spectrum. This generator is implemented on CPUs and multi‐GPUs platforms, with good strong and weak scaling performance on several supercomputers. We also propose a method to verify its ability to guarantee the given spectra. Finally, we give an example to evaluate the numerical properties and parallel performance of iterative methods using this matrix generator.

Parallel Krylov Subspace Methods are commonly used for solving large-scale sparse linear systems. Facing the development of extreme scale platforms, the minimization of synchronous global communication becomes critical to obtain good efficiency and scalability. This paper highlights a recent development of a hybrid (unite and conquer) method, which combines three computation algorithms together with asynchronous communication to accelerate the resolution of non-Hermitian linear systems and to improve its fault tolerance and reusability. Experimentation shows that our method has an up to 5x speedup and better scalability than the conventional methods for the resolution on hierarchical clusters with hundreds of nodes.