16.447 TFlops and 159-Billion-dimensional Exact-diagonalization for Trapped Fermion-Hubbard Model on the Earth Simulator

Yamada, S.; Imamura, Toshiyuki; Machida, Masahiko

doi:10.1109/sc.2005.1

Cited by 24 publications

(32 citation statements)

References 14 publications

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“…This partition of the vector space leads to a more simplified partition of the operator than eq. (B·2), H ¼ H "" 1 þ 1 H ## þ ðresidual partÞ, 13) where H "" and H ## include the hopping terms and part of the on-site Coulomb interaction. As a result, the memory required to store the Hamiltonian reduced to 1 GB.…”

Section: Discussion and Summarymentioning

confidence: 99%

Shifted Conjugate-Orthogonal–Conjugate-Gradient Method and Its Application to Double Orbital Extended Hubbard Model

et al. 2008

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We discuss the shifted conjugate-orthogonal-conjugate-gradient (COCG) method, which can be used to solve a series of linear equations generated by a number of scalar shifts, without performing timeconsuming matrix-vector operations, except at a single reference energy. This is a type of CG method and is robust and has an expression of the accuracy of the calculated result that requires no additional calculation cost. The shifted COCG is useful for calculating the Green's function of a many-electron Hamiltonian, which has a very large dimension. We applied the shifted COCG method to the double orbital extended Hubbard model with twelve electrons on a periodic ffiffi ffi 8 p Â ffiffi ffi 8 p site system, with the dimension of the Hamiltonian equal to 64,128,064, and we found that the ground state is an insulator. We also discuss the main points of reducing the amount of memory required to apply COCG algorithm.

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Section: Discussion and Summarymentioning

confidence: 99%

Shifted Conjugate-Orthogonal–Conjugate-Gradient Method and Its Application to Double Orbital Extended Hubbard Model

et al. 2008

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“…Although the dimension partly decreases by eliminating clear irrelevant states, the calculation solving the ground state of the superblock Hamiltonian matrix is obviously the most time-and memory-consuming. Since the Hamiltonian matrix is a sparse symmetric matrix, an iterative method, such as Lanczos method [1] or conjugate gradient method [5,6,19,20], is suitable for solving the ground state. Inside these iterative calculation steps, the heaviest operation is a multiplication of the superblock Hamiltonian matrix and a vector.…”

Section: Algorithmmentioning

confidence: 99%

“…Among them, the most accurate one is the so-called "exact diagonalization method", which solves the ground state of the model Hamiltonian matrix. It is mathematically equivalent to calculating an eigenvector whose eigenvalue is the lowest for a symmetric Hamiltonian matrix (see, e.g., [19,20] for the parallelization of the exact diagonalization and [7][8][9] for the calculation results). However, since the Hamiltonian matrix size increases almost exponentially with the system size, i.e., the numbers of electrons and lattice sites, the truncation of irrelevant states has been an important issue in this research field.…”

Section: Introductionmentioning

confidence: 99%

Direct extension of the density-matrix renormalization group method toward two-dimensional large quantum lattices and related high-performance computing

Yamada

Okumura

Imamura

et al. 2011

Japan J. Indust. Appl. Math.

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The density-matrix renormalization group (DMRG) method is widely used by computational physicists as a high accuracy tool to explore the ground state in large quantum lattice models, e.g., Heisenberg and Hubbard models, which are well-known standard models describing interacting spins and electrons, respectively, in solid states. After the DMRG method was originally developed for 1-D lattice/chain models, some specific extensions toward 2-D lattice (n-leg ladder) models have been proposed. However, high accuracy as obtained in 1-D models is not always guaranteed in their extended versions because the original exquisite advantage of the algorithm is partly lost. Thus, we choose an alternative way. It is a direct 2-D extension of DMRG method which instead demands an enormously large memory space, but the memory explosion is resolved by parallelizing the DMRG code with performance tuning. The parallelized direct extended DMRG shows a good accuracy like 1-D models and an excellent parallel efficiency as the number of states kept increases. This success promises accurate analysis on large 2-D (n-leg ladder) quantum lattice models in the near future when peta-flops parallel supercomputers are available.

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“…The most accurate one of them is the exact diagonalization method, which solves the ground state (the smallest eigenvalue and the corresponding eigenvector) of the Hamiltonian matrix derived from the systems. We have actually parallelized the exact diagonalization method and obtained some novel physical results [4][5][6][7]. However, the dimension of the Hamiltonian matrix for the exact diagonalization method increases almost exponentially with the number of the lattice sites.…”

Section: Introductionmentioning

confidence: 99%

“…However, the dimension of the Hamiltonian matrix for the exact diagonalization method increases almost exponentially with the number of the lattice sites. Thus, the limit of the simulation on a supercomputer with a terabyte memory system is an about-20-site system [6,7].…”

Section: Introductionmentioning

confidence: 99%

High Performance Computing for Eigenvalue Solver in Density-Matrix Renormalization Group Method: Parallelization of the Hamiltonian Matrix-Vector Multiplication

Yamada

Okumura

Machida

2008

High Performance Computing for Computational Science - VECPAR 2008

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Abstract. The Density Matrix Renormalization Group (DMRG) method is widely used by computational physicists as a high accuracy tool to obtain the ground state of large quantum lattice models. Since the DMRG method has been originally developed for 1-D models, many extended method to a 2-D model have been proposed. However, some of them have issues in term of their accuracy. It is expected that the accuracy of the DMRG method extended directly to 2-D models is excellent. The direct extension DMRG method demands an enormous memory space. Therefore, we parallelize the matrix-vector multiplication in iterative methods for solving the eigenvalue problem, which is the most time-and memoryconsuming operation. We find that the parallel efficiency of the direct extension DMRG method shows a good one as the number of states kept increases.

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16.447 TFlops and 159-Billion-dimensional Exact-diagonalization for Trapped Fermion-Hubbard Model on the Earth Simulator

Cited by 24 publications

References 14 publications

Shifted Conjugate-Orthogonal–Conjugate-Gradient Method and Its Application to Double Orbital Extended Hubbard Model

Shifted Conjugate-Orthogonal–Conjugate-Gradient Method and Its Application to Double Orbital Extended Hubbard Model

Direct extension of the density-matrix renormalization group method toward two-dimensional large quantum lattices and related high-performance computing

High Performance Computing for Eigenvalue Solver in Density-Matrix Renormalization Group Method: Parallelization of the Hamiltonian Matrix-Vector Multiplication

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