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Abstract We present the Alternating Anderson–Richardson (AAR) method: an efficient and scalable alternative to preconditioned Krylov solvers for the solution of large, sparse linear systems on high performance computing platforms. Specifically, we generalize the recently proposed Alternating Anderson–Jacobi (AAJ) method (Pratapa et al., 2016) to include preconditioning, discuss efficient parallel implementation, and provide serial MATLAB and parallel C/C++ implementations. In serial applications to nonsymmetric systems, we find that AAR is comparably robust to GMRES, using the same preconditioning, while often outperforming it in time to solution; and find AAR to be more robust than Bi-CGSTAB for the problems considered. In parallel applications to the Helmholtz and Poisson equations, we find that AAR shows superior strong and weak scaling to GMRES, Bi-CGSTAB, and Conjugate Gradient (CG) methods, using the same preconditioning, with consistently shorter times to solution at larger processor counts. Finally, in massively parallel applications to the Poisson equation, on up to 110,592 processors, we find that AAR shows superior strong and weak scaling to CG, with shorter minimum time to solution. We thus find that AAR offers a robust and efficient alternative to current state-of-the-art solvers, with increasing advantages as the number of processors grows. Program summary Program Title: AAR Program Files doi: http://dx.doi.org/10.17632/669xrdcwry.1 Licensing provisions: GNU General Public License 3 (GPL) Programming language: MATLAB for the MATLAB version. C/C++ for the PETSc version. C/C++ for the standalone version. External routines/libraries: MATLAB 2014a or later for the MATLAB version. PETSc 3.5.3 (http://www.mcs.anl.gov/petsc) or later and MVAPICH2 2.1 (http://mvapich.cse.ohio-state.edu/) or later for the PETSc version. MVAPICH2 2.1 or later for the standalone version. Nature of problem: Linear systems of equations. Solution method: Anderson extrapolation at periodic intervals within a preconditioned Richardson iteration. Restrictions: Jacobi preconditioning for the standalone version. [ABSTRACT FROM AUTHOR]

We present SQDFT: a large-scale parallel implementation of the Spectral Quadrature (SQ) method for O ( N ) Kohn–Sham Density Functional Theory (DFT) calculations at high temperature. Specifically, we develop an efficient and scalable finite-difference implementation of the infinite-cell Clenshaw–Curtis SQ approach, in which results for the infinite crystal are obtained by expressing quantities of interest as bilinear forms or sums of bilinear forms, that are then approximated by spatially localized Clenshaw–Curtis quadrature rules. We demonstrate the accuracy of SQDFT by showing systematic convergence of energies and atomic forces with respect to SQ parameters to reference diagonalization results, and convergence with discretization to established planewave results, for both metallic and insulating systems. We further demonstrate that SQDFT achieves excellent strong and weak parallel scaling on computer systems consisting of tens of thousands of processors, with near perfect O ( N ) scaling with system size and wall times as low as a few seconds per self-consistent field iteration. Finally, we verify the accuracy of SQDFT in large-scale quantum molecular dynamics simulations of aluminum at high temperature. Program summary Program Title: SQDFT Program Files doi: http://dx.doi.org/10.17632/h4jnrmh9v3.1 Licensing provisions: GNU General Public License 3 (GPL) Programming language: C/C++ External routines/libraries: MVAPICH2 2.1 ( http://mvapich.cse.ohio-state.edu/ ) or later. Nature of problem: Quantum mechanical calculation of static and dynamic properties of condensed matter at high temperature in the framework of Kohn–Sham Density Functional Theory (DFT). Solution method: High-order finite-difference discretization. Calculation of the electronic ground state using the self-consistent field (SCF) iteration in conjunction with Periodic Pulay extrapolation/mixing. Local reformulation of the electrostatics in terms of the electrostatic potential and pseudocharge densities. Solution of the Poisson equation using the Alternating Anderson–Richardson (AAR) method. Calculation of the electron density, band structure energy, electronic entropy energy, and atomic forces using the infinite-cell Clenshaw–Curtis Spectral Quadrature (SQ) approach, in which results for the infinite crystal are obtained by expressing quantities as bilinear forms or sums of bilinear forms, that are then approximated by spatially localized Clenshaw–Curtis quadrature rules. NVE molecular dynamics using the leapfrog method and NVT molecular dynamics using the Verlet algorithm with the Nose–Hoover thermostat. Extrapolation of electron density between molecular dynamics steps. Parallelization via domain decomposition. Restrictions: Cubical domain. Local Density Approximation (LDA). Troullier–Martins pseudopotentials without relativistic or non-linear core corrections. [ABSTRACT FROM AUTHOR]