The ESSEX project is an ongoing effort to provide exascale-enabled sparse eigensolvers, especially for quantum physics and related application areas. In this paper we first briefly summarize some key achievements that have been made within this project. Then we focus on a projection-based eigensolver with polynomial approximation of the projector. This eigensolver can be used for computing hundreds of interior eigenvalues of large sparse matrices. We describe techniques that allow using lower-degree polynomials than possible with standard Chebyshev expansion of the window function and kernel smoothing. With these polynomials, the degree, and thus the number of matrix–vector multiplications, typically can be reduced by roughly one half, resulting in comparable savings in runtime.

Methods for the solution of sparse eigenvalue problems that are based on spectral projectors and contour integration have recently attracted more and more attention. Such methods require the solution of many shifted sparse linear systems of full size. In most of the literature concerning these eigenvalue solvers, only few words are said on the solution of the linear systems, but they turn out to be very hard to solve by iterative linear solvers in practice. In this work we identify a row projection method for the solution of the inner linear systems encountered in the FEAST algorithm and introduce a novel hybrid parallel and fully iterative implementation of the eigenvalue solver. Our approach ultimately aims at achieving extreme parallelism by exploiting the algorithm's potential on several levels. We present numerical examples where graphene modeling is one of the target applications. In this application, several hundred or even thousands of eigenvalues from the interior of the spectrum are required, which is a big challenge for state-of-the-art numerical methods.

The ESSEX project investigates computational issues arising at exascale for large-scale sparse eigenvalue problems and develops programming concepts and numerical methods for their solution. The project pursues a coherent co-design of all software layers where a holistic performance engineering process guides code development across the classic boundaries of application, numerical method, and basic kernel library. Within ESSEX the numerical methods cover widely applicable solvers such as classic Krylov, Jacobi-Davidson, or the recent FEAST methods, as well as domain-specific iterative schemes relevant for the ESSEX quantum physics application. This report introduces the project structure and presents selected results which demonstrate the potential impact of ESSEX for efficient sparse solvers on highly scalable heterogeneous supercomputers.