Block preconditioning of real-valued iterative algorithms for complex linear systems

Benzi, Michele; Bertaccini, Daniele

doi:10.1093/imanum/drm039

Cited by 155 publications

(72 citation statements)

References 38 publications

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“…To our knowledge, there are two computational techniques for solving this type of linear systems. The first is the so-called complex-to-real method, i.e., changing the original complex linear system (1.9) to a real one and taking advantages of some real-valued iteration methods, cf., e.g., [6,10,13]. This type of methods avoid complex arithmetic operations and may appear to be effective.…”

Section: J (Ymentioning

confidence: 99%

A robust structured preconditioner for time-harmonic parabolic optimal control problems

2017

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We consider the iterative solution of optimal control problems constrained by the time-harmonic parabolic equations. Due to the time-harmonic property of the control equations, a suitable discretization of the corresponding optimality systems leads to a large complex linear system with special two-by-two block matrix of saddle point form. For this algebraic system, an efficient preconditioner is constructed, which results in a fast Krylov subspace solver, that is robust with respect to the mesh size, frequency, and regularization parameters. Furthermore, the implementation is straightforward and the computational complexity is of optimal order, linear in the number of degrees of freedom. We show that the eigenvalue distribution of the corresponding preconditioned matrix leads to a condition number bounded above by 2. Numerical experiments confirming the theoretical derivations are presented, including comparisons with some other existing preconditioners.

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Section: J (Ymentioning

confidence: 99%

A robust structured preconditioner for time-harmonic parabolic optimal control problems

2017

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“…The choice of µ z is the same as in Table 1. As in the classical CG method, we define the approximate solution w n = w 0 + v n , with v n ∈ V n , by Galerkin's method, or 2) and find that v n = w n − w 0 satisfies…”

Section: )mentioning

confidence: 99%

“…Such equations, with a complex shift z of the positive-definite operator A, need to be solved in a method for discretization in time of parabolic equations, based on Laplace transformation and quadrature, which has been studied recently, as is made [2] Iterative solution of shifted positive-definite linear systems 135 more specific below. We consider a basic Richardson iteration and a conjugate gradient (CG) method for (1.1), as well as preconditioned versions of these methods.…”

Section: Introductionmentioning

confidence: 99%

Iterative Solution of Shifted Positive-Definite Linear Systems Arising in a Numerical Method for the Heat Equation Based on Laplace Transformation and Quadrature

McLean

Thomée

2011

ANZIAM J.

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In earlier work we have studied a method for discretization in time of a parabolic problem, which consists of representing the exact solution as an integral in the complex plane and then applying a quadrature formula to this integral. In application to a spatially semidiscrete finite-element version of the parabolic problem, at each quadrature point one then needs to solve a linear algebraic system having a positive-definite matrix with a complex shift. We study iterative methods for such systems, considering the basic and preconditioned versions of first the Richardson algorithm and then a conjugate gradient method.2010 Mathematics subject classification: primary 65F10; secondary 65M22, 65M60, 65R10.

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“…This class of systems of linear equations arise frequently from science and engineering applications such as computational electrodynamic [18,24], diffuse optical tomography [1], FFT-based solution of certain time-dependent PDEs [13], quantum mechanics [14], molecular scattering [20], structural dynamics [15], and lattice quantum chromodynamics [16]. For more examples and applications see [11] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Two Efficient Inexact Algorithms for a Class of Large Sparse Complex Linear Systems

2015

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Recently Salkuyeh et al. (Int J Comput Math 92:802-815, 2015) studied the generalized SOR (GSOR) iterative method for a class of complex symmetric linear system of equations. In this paper, we present an inexact variant of the GSOR method in which the conjugate gradient and the preconditioned conjugate gradient methods are regarded as its inner iteration processes at each step of the GSOR outer iteration. Moreover, we construct a new method called shifted GSOR iteration method which is obtained from combination of a shiftsplitting iteration scheme and the GSOR iteration method. The convergence analysis of the proposed methods are presented. Some numerical experiments are given to show the performance of the methods and are compared with those of the inexact MHSS method.Mathematics Subject Classification. 65F10, 93C10.

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Block preconditioning of real-valued iterative algorithms for complex linear systems

Cited by 155 publications

References 38 publications

A robust structured preconditioner for time-harmonic parabolic optimal control problems

A robust structured preconditioner for time-harmonic parabolic optimal control problems

Iterative Solution of Shifted Positive-Definite Linear Systems Arising in a Numerical Method for the Heat Equation Based on Laplace Transformation and Quadrature

Two Efficient Inexact Algorithms for a Class of Large Sparse Complex Linear Systems

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