BiCGStab(?) for Families of Shifted Linear Systems

Frommer, Andreas

doi:10.1007/s00607-003-1472-6

Cited by 104 publications

(136 citation statements)

References 17 publications

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“…A short-term recurrence alternative to Lanczos-type methods in the nonsymmetric case has been recently proposed by Frommer [139], where the Bi-CGstab( ) method is adapted to handle several shifts simultaneously.…”

Section: Complex Symmetric Matricesmentioning

confidence: 99%

Recent computational developments in Krylov subspace methods for linear systems

Simoncini

Szyld

2006

Numerical Linear Algebra App

312

254

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Abstract. Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.

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Section: Complex Symmetric Matricesmentioning

confidence: 99%

Recent computational developments in Krylov subspace methods for linear systems

Simoncini

Szyld

2006

Numerical Linear Algebra App

312

254

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“…However, in many-electron theory, the dimensions of the intermediate vectors are extremely large and then the number of intermediate vectors is restricted to some tens or hundreds. In the standard loop structure in the Frommer's original implementation, 6) the outermost loop index is the iteration step n, and all the p n , r n , and x n for each energy mesh point , are required in order to perform the calculation at the iteration step n þ 1. Then all the energy mesh points must be fixed during the whole calculation.…”

Section: Shifted Equationsmentioning

confidence: 99%

“…The system is the same as that in Fig. 1 except for 6.03 eV. We must perform seed switching another two times at the 1509th and the 3297th iterations in order to obtain global convergence.…”

Section: Seed Switchingmentioning

confidence: 99%

“…6,12) One is the orthogonality of the residual vectors fr k g with respect to a non-standard inner product ðu; vÞ ¼ u T v, on which the theorem of collinear residuals is based. Note that this orthogonality is different from the well-known Aorthogonality of the searching directions f p k g. Because fr k g k¼0;1;...;nÀ1 is a basis set of K nÀ1 ðA; bÞ, the orthogonality is also represented as follows: r n 2 K n ðA; bÞ and r n ?…”

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confidence: 99%

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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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“…The method is purely mathematical and may also be useful in other physics fields. Research in the case of S ¼ I has been reported, for example, in quantum chromodynamics 6) and in many-body electron theory. 7) The multiple Arnoldi method is used in this study, since it is suitable for molecular-dynamics (MD) simulation, or the calculation of energy and force.…”

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confidence: 99%

Ten-Million-Atom Electronic Structure Calculations on the K Computer with a Massively Parallel Order-NTheory

et al. 2013

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A massively parallel order-N electronic structure theory was developed by interdisciplinary research between physics, applied mathematics, and computer science. (1) A high parallel efficiency with ten-million-atom nanomaterials was realized on the K computer with up to 98,304 processor cores. The mathematical foundation is a novel linear algebraic algorithm for generalized shifted linear equations. The calculation was carried out using our code ''ELSES'' (www.elses.jp) with modelled (tight-binding-form) systems based on ab initio calculations. (2) A postcalculation analysis method, called the -orbital crystalline orbital Hamiltonian population ( -COHP) method, is presented, since this method is ideal for huge electronic structure data distributed among massive nodes. The analysis method is demonstrated in an sp 2 -sp 3 nanocomposite carbon solid, with the original visualization software ''VisBAR''. The present research indicates the general aspects of computational physics with current or next-generation supercomputers.KEYWORDS: order-N electronic structure theory, numerical linear algebra, massive parallel computation, sp 2 -sp 3 nanocomposite carbon solid, crystalline orbital Hamiltonian populationA common issue in current computational physics is the theory for large calculations with modern massively parallel supercomputers, such as the K computer. Large calculations should accompany a large-data analysis theory, as a postcalculation procedure, so as to obtain a physical conclusion from huge numerical data distributed among massive computer nodes. Interdisciplinary research studies between physics, applied mathematics and computer science are thus crucial and such a study is sometimes called ''application-algorithm-architecture co-design''.The present paper is devoted to theories, for both largescale calculation and large-data analysis, as order-N electronic structure theories suitable for modern massively parallel computers.Order-N electronic structure theories are those in which the computational time is proportional to the number of atoms in the system N and are promising for large-scale calculation. The reference list for the order-N theories can be found, for example, in Ref. 1. Recently, several novel linear algebraic algorithms, with Krylov subspace, have been developed for the order-N theory, i.e., the generalized shifted conjugate-orthogonal conjugate-gradient method, [1][2][3] generalized Lanczos method, 1) generalized Arnoldi method, 1) Arnoldi ðM; W; GÞ method, 4) multiple Arnoldi method, 5) and generalized shifted quasi-minimal-residual method. 3) Their common foundation is the ''generalized shifted linear equation'', or the set of linear equationswhere z is a (complex) energy value and H and S denote the Hamiltonian and overlap matrices in the linear-combinationof-atomic-orbital (LCAO) representation, respectively. The matrices are sparse real-symmetric M Â M matrices, and S is positive definite. The vector b is the input and the vector x is the solution vector. Equation (1) is solved, iterativ...

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BiCGStab(?) for Families of Shifted Linear Systems

Cited by 104 publications

References 17 publications

Recent computational developments in Krylov subspace methods for linear systems

Recent computational developments in Krylov subspace methods for linear systems

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

Ten-Million-Atom Electronic Structure Calculations on the K Computer with a Massively Parallel Order-NTheory

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