ESSEX: Equipping Sparse Solvers For Exascale

Abstract

The ESSEX project has investigated programming concepts, data structures, and numerical algorithms for scalable, efficient, and robust sparse eigenvalue solvers on future heterogeneous exascale systems. Starting without the burden of legacy code, a holistic performance engineering process could be deployed across the traditional software layers to identify efficient implementations and guide sustainable software development. At the basic building blocks level, a flexible MPI+X programming approach was implemented together with a new sparse data structure (SELL-C-σ) to support heterogeneous architectures by design. Furthermore, ESSEX focused on hardware-efficient kernels for all relevant architectures and efficient data structures for block vector formulations of the eigensolvers. The algorithm layer addressed standard, generalized, and nonlinear eigenvalue problems and provided some widely usable solver implementations including a block Jacobi-Davidson algorithm, contour-based integration schemes, and filter polynomial approaches. Adding to the highly efficient kernel implementations, algorithmic advances such as adaptive precision, optimized filtering coefficients, and preconditioning have further improved time to solution. These developments

Flexible subspace iteration with moments for an effective contour integration‐based eigensolver

“…Given an estimated number of eigenpairs m = 300 for all test problems, TA B L E 1 Sizes, number of nonzeros per row (nnzr; rounded), smallest (𝜆 min ), and largest (𝜆 max ) eigenvalues (rounded to three significant digits), and the two search intervals (with eigenvalue counts m) for 9 test matrices from graphene modeling and 10 test matrices from the GHS_indef (laser, linverse, brainpc2), PARSEC (SiH4, Si5H12, SiO), ACUSIM (Pres_Poisson), Boeing (bcsstk37), DIMACS10 (rgg_n_2_15_s0), and Andrews (Andrews) groups of the SuiteSparse Matrix Collection. Note: Problems for which all not schemes found all eigenpairs (7,14,20,26,34) are not included in the average.…”

“…To this end, we introduce and compare various strategies, utilizing a software framework for projected subspace iterative eigensolvers, BEAST. 20,21 The results of this article are also used in the dissertation of the first author. 22…”

“…Our goal is to increase the efficiency of the eigensolver by reducing $$$ {\mathrm{RHS}}_{\mathrm{ovl}} $$$ while maintaining accuracy and robustness. To this end, we introduce and compare various strategies, utilizing a software framework for projected subspace iterative eigensolvers, BEAST 20,21 . The results of this article are also used in the dissertation of the first author 22…”