Task Parallel Incomplete Cholesky Factorization using 2D Partitioned-Block Layout

Kim, Kyungjoo; Rajamanickam, Sivasankaran; Stelle, George; Edwards, Harold; Olivier, Stephen Lecler

doi:10.48550/arxiv.1601.05871

Cited by 2 publications

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“…Although there are many techniques to compute or to approximate the Z factor, 13,38,39 our method allows us to employ a close initial guess from a previous time step of an MD simulation providing a fast convergence and an overall linear scaling with small overhead. Also, thanks to the XL framework, we avoid any systematic energy drift.…”

Section: ■ Methodologymentioning

confidence: 99%

Recursive Factorization of the Inverse Overlap Matrix in Linear-Scaling Quantum Molecular Dynamics Simulations

Negre

Mniszewski

Cawkwell

et al. 2016

J. Chem. Theory Comput.

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We present a reduced complexity algorithm to compute the inverse overlap factors required to solve the generalized eigenvalue problem in a quantum-based molecular dynamics (MD) simulation. Our method is based on the recursive, iterative refinement of an initial guess of Z (inverse square root of the overlap matrix S). The initial guess of Z is obtained beforehand by using either an approximate divide-and-conquer technique or dynamical methods, propagated within an extended Lagrangian dynamics from previous MD time steps. With this formulation, we achieve long-term stability and energy conservation even under the incomplete, approximate, iterative refinement of Z. Linear-scaling performance is obtained using numerically thresholded sparse matrix algebra based on the ELLPACK-R sparse matrix data format, which also enables efficient shared-memory parallelization. As we show in this article using self-consistent density-functional-based tight-binding MD, our approach is faster than conventional methods based on the diagonalization of overlap matrix S for systems as small as a few hundred atoms, substantially accelerating quantum-based simulations even for molecular structures of intermediate size. For a 4158-atom water-solvated polyalanine system, we find an average speedup factor of 122 for the computation of Z in each MD step.

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Section: ■ Methodologymentioning

confidence: 99%

Recursive Factorization of the Inverse Overlap Matrix in Linear-Scaling Quantum Molecular Dynamics Simulations

Negre

Mniszewski

Cawkwell

et al. 2016

J. Chem. Theory Comput.

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“…Instead, it is often used on each compute node as the subdomain solver in a multilevel domaindecomposition method. Therefore, single-node shared-memory parallelism is crucial for incomplete Cholesky factorization, and such parallel algorithms have been studied extensively [2,4,16,24].…”

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confidence: 99%

RCHOL: Randomized Cholesky Factorization for Solving SDD Linear Systems

Chen¹,

Liang²,

Biros³

2020

Preprint

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We introduce a randomized algorithm, namely rchol, to construct an approximate Cholesky factorization for a given sparse Laplacian matrix (a.k.a., graph Laplacian). The (exact) Cholesky factorization for the matrix introduces a clique in the associated graph after eliminating every row/column. By randomization, rchol samples a subset of the edges in the clique. We prove rchol is breakdown free and apply it to solving linear systems with symmetric diagonallydominant matrices. In addition, we parallelize rchol based on the nested dissection ordering for shared-memory machines. Numerical experiments demonstrated the robustness and the scalability of rchol. For example, our parallel code scaled up to 64 threads on a single node for solving the 3D Poisson equation, discretized with the 7-point stencil on a 1024 × 1024 × 1024 grid, or one billion unknowns.

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Task Parallel Incomplete Cholesky Factorization using 2D Partitioned-Block Layout

Cited by 2 publications

References 0 publications

Recursive Factorization of the Inverse Overlap Matrix in Linear-Scaling Quantum Molecular Dynamics Simulations

Recursive Factorization of the Inverse Overlap Matrix in Linear-Scaling Quantum Molecular Dynamics Simulations

RCHOL: Randomized Cholesky Factorization for Solving SDD Linear Systems

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