Part I Architecture and Algorithms of High Performance Computing High-speed ultra-low-power simulation methodology for material science

Hybrid mathematical principles were introduced to large-scale electronic structure theories andrealized one-hundred-million atom (100-nm-scale) calculations on the K computer. Novellinear-algebraic theories were constructed as Krylov-subspace solvers for generalized shiftedlinear equations ((zS-H)x=b) and were implemented in our order-N calculation code ‘ELSES’(http://www.elses.jp/) with modeled (tight-binding-form) systems. A high parallel efficiencywas obtained with up to the full core calculations on the K computer. The linear-algebraictheory is general and, recently, was applied to ab initio quantum transport calculations on thereal space grid formalism. Moreover, hybrid mathematical priciples are constructed on iInternaleigen-pair calculation with Sylvester's theorem of inertia and hybrid direct solver forgeneralized eigen-value problem. Several application studies were carried out: The nano domainof sp2-sp3 nano-composite carbon solid was calculated for the nano-polycrystalline diamond, aultrahard material and typical domain shapes and boundary structures were obtained. An ionicliquid (N-Methyl-N-propylpiperidinium bis trifluoromethanesulfonyl imide), a battery-relatedmaterial, was calculated and the diffusion mechanism was revealed. Several amourphous-likestructures and bundles of conjugated polymers were calculated and characteristic localized πstates were obtained. The present work was carried out, with several members of ComputicsProject: T. Sogabe, S.-L. Zhang, and T. Miyata (Nagoya U), T. Ono (Tsukuba U), asinter-disciplinary collaborations among physics, mathematics and the high-performancecomputation field or ‘Application-Algorithm-Architecture co-design'. Fig. 1 (a) Concept of Application-Algorithm-Architecture co-design. (b) Theschematic workflow of the order-N algorithm based on the shifted Krylov subspacetheory.