Quasi-Minimal Residual Variants of the COCG and COCR Methods for Complex Symmetric Linear Systems in Electromagnetic Simulations

Self-consistent approaches to superfluid many-fermion systems in three-dimensions (and their subsequent use in time-dependent studies) require a large number of diagonalizations of very large dimension Hermitian matrices, which results in enormous computational costs. We present an approach based on the shifted conjugate-orthogonal conjugate-gradient (COCG) Krylov method for the evaluation of the Green's function, from which we subsequently extract various densities (particle number, spin, current, kinetic energy, anomalous, etc.) of a nuclear system. The approach eschews the determination of the quasiparticle wavefunctions and their corresponding quasiparticle energies, which never explicitly appear in the construction of a single-particle Hamiltonian or needed for the calculation of various static nuclear properties, which depend only on densities. As benchmarks we present calculations for nuclei with axial symmetry, including the ground state of spherical (magic or semi-magic) and axially deformed nuclei, the saddle-point in the 240 Pu constrained fission path, and a vortex in the neutron star crust and demonstrate the superior efficiency of the shifted COCG Krylov method over traditional approaches.

show abstract

“…(55), and using the axial symmetry we cover all the equivalent points in Cartesian space via reverse transformation, using Eqs. (56).…”

Section: Exploiting Symmetriesmentioning

confidence: 99%

Coordinate-space solver for superfluid many-fermion systems with the shifted conjugate-orthogonal conjugate-gradient method

et al. 2017

View full text Add to dashboard Cite

show abstract

“…However, rank deficiency is a common problem that can lead block Krylov subspace methods to breakdown. The main reason is that the block search direction vectors may be linearly dependent on the existing basis by the increasing of the iteration number [9,10]. Consequently, some useless information will affect the accuracy of the solution and the numerical stability.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, it is valuable to solve the rank deficiency problem, and finally to enhance the numerical stability of BCOCG and BCOCR. Motivated by [10], in this paper, we propose a breakdown-free block COCG algorithm (BFBCOCG) and a breakdown-free block COCR algorithm (BFBCOCR) that can efficiently solve the rank deficiency problem of BCOCG and BCOCR, respectively.…”

Section: Introductionmentioning

confidence: 99%

A Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides

Zhong

Zhang

2019

Symmetry

Self Cite

View full text Add to dashboard Cite

The block conjugate orthogonal conjugate gradient method (BCOCG) is recognized as a common method to solve complex symmetric linear systems with multiple right-hand sides. However, breakdown always occurs if the right-hand sides are rank deficient. In this paper, based on the orthogonality conditions, we present a breakdown-free BCOCG algorithm with new parameter matrices to handle rank deficiency. To improve the spectral properties of coefficient matrix A, a precondition version of the breakdown-free BCOCG is proposed in detail. We also give the relative algorithms for the block conjugate A-orthogonal conjugate residual method. Numerical results illustrate that when breakdown occurs, the breakdown-free algorithms yield faster convergence than the non-breakdown-free algorithms.

show abstract

“…Additionly, there are also other solution techniques for solving complex symmetric linear system (1) which are extensions of Krylov subspace methods. For example, the well-known conjugate orthogonal conjugate gradient (COCG) [13], quasi-minimal residual (QMR) method [14], conjugate A-orthogonal conjugate residual (COCR) [15], symmetric complex bi-conjugate gradient (SCBiCG) method [16] and Quasi-minimal residual variants of the COCG and COCR methods [17].…”

Section: Introductionmentioning

confidence: 99%

A new iterative method for solving a class of complex symmetric system of linear equations

2016

View full text Add to dashboard Cite

We present a new stationary iterative method, called Scale-Splitting (SCSP) method, and investigate its convergence properties. The SCSP method naturally results in a simple matrix splitting preconditioner, called SCSP-preconditioner, for the original linear system. Some numerical comparisons are presented between the SCSP-preconditioner and several available block preconditioners, such as PGSOR (Hezari et al. Numer. Linear Algebra Appl. 22, 761-776, 2015) and rotate block triangular preconditioners (Bai Sci. China Math. 56, 2523-2538, 2013, when they are applied to expedite the convergence rate of Krylov subspace iteration methods for solving the original complex system and its block real formulation, respectively. Numerical experiments show that the SCSP-preconditioner can compete with PGSOR-preconditioner and even more effective than the rotate block triangular preconditioners.

show abstract

Quasi-Minimal Residual Variants of the COCG and COCR Methods for Complex Symmetric Linear Systems in Electromagnetic Simulations

Cited by 24 publications

References 26 publications

Coordinate-space solver for superfluid many-fermion systems with the shifted conjugate-orthogonal conjugate-gradient method

Coordinate-space solver for superfluid many-fermion systems with the shifted conjugate-orthogonal conjugate-gradient method

A Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides

A new iterative method for solving a class of complex symmetric system of linear equations

Contact Info

Product

Resources

About