Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems

Pratapa, Phanisri P.; Suryanarayana, Phanish; Pask, John E.

doi:10.1016/j.jcp.2015.11.018

Cited by 74 publications

(66 citation statements)

References 25 publications

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“…The matrices X k and R k are defined as in (8) and (9), respectively. The parameter p represents the number of Richardson sweeps separating two consecutive Anderson mixing accelerations.…”

Section: Aar Methodsmentioning

confidence: 99%

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Convergence analysis of Anderson‐type acceleration of Richardson's iteration

Pasini

2019

Numerical Linear Algebra App

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Summary We consider Anderson extrapolation to accelerate the (stationary) Richardson iterative method for sparse linear systems. Using an Anderson mixing at periodic intervals, we assess how this benefits convergence to a prescribed accuracy. The method, named alternating Anderson–Richardson, has appealing properties for high‐performance computing, such as the potential to reduce communication and storage in comparison to more conventional linear solvers. We establish sufficient conditions for convergence, and we evaluate the performance of this technique in combination with various preconditioners through numerical examples. Furthermore, we propose an augmented version of this technique.

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“…The matrices X k and R k are defined as in (8) and (9), respectively. The parameter p represents the number of Richardson sweeps separating two consecutive Anderson mixing accelerations.…”

Section: Aar Methodsmentioning

confidence: 99%

“…Pratapa et al have recently proposed a new way to combine Richardson schemes and Anderson mixing, which aims to reduce the communication by relaxing the number of least squares problem to solve. The method they propose is called the AAR method …”

Section: Stationary Richardsonmentioning

confidence: 99%

Convergence analysis of Anderson‐type acceleration of Richardson's iteration

Pasini

2019

Numerical Linear Algebra App

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“…The Dirichlet boundary condition values are determined using a multipole expansion for isolated systems and a dipole correction for surfaces and nanowires [41,26]. The linear system is solved using the AAR method [42,43] in conjunction with Cholesky preconditioning.…”

Section: Mixingmentioning

confidence: 99%

M-SPARC: Matlab-Simulation Package for Ab-initio Real-space Calculations

Sharma

Suryanarayana

2020

SoftwareX

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We present M-SPARC: Matlab-Simulation Package for Ab-initio Real-space Calculations. It can perform pseudopotential spin-polarized and unpolarized Kohn-Sham Density Functional Theory (DFT) simulations for isolated systems such as molecules as well as extended systems such as crystals, surfaces, and nanowires. M-SPARC provides a rapid prototyping platform for the development and testing of new algorithms and methods in real-space DFT, with the potential to significantly accelerate the rate of advancements in the field. It also provides a convenient avenue for the accurate first principles study of small to moderate sized systems.

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“…We solve the Poisson problem in Eq. 3 using the Alternating Anderson-Richardson (AAR) method [55,56], an approach that outperforms the conjugate gradient method [57] in the context of large-scale parallel computations [56]. We perform NVE (microcanonical) simulations using 7 The Clenshaw-Curtis variant of SQ is chosen here because it is more efficient compared to its Gauss counterpart [50], particularly in the computation of the nonlocal component of the forces [38].…”

Section: Formulation and Implementation Of Sqdftmentioning

confidence: 99%

SQDFT: Spectral Quadrature method for large-scale parallel O(N) Kohn–Sham calculations at high temperature

Suryanarayana

Pratapa

Sharma

et al. 2018

Computer Physics Communications

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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.

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Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems

Cited by 74 publications

References 25 publications

Convergence analysis of Anderson‐type acceleration of Richardson's iteration

Convergence analysis of Anderson‐type acceleration of Richardson's iteration

M-SPARC: Matlab-Simulation Package for Ab-initio Real-space Calculations

SQDFT: Spectral Quadrature method for large-scale parallel O(N) Kohn–Sham calculations at high temperature

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