Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems

Axelsson, Owe; Lukáš, Dalibor

doi:10.1515/jnma-2017-0064

Cited by 30 publications

(18 citation statements)

References 21 publications

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“…A natural extension of our work is to apply the new structured preconditioner to tackle some more difficult problems, such as the time-harmonic eddy current control problem in [7]. In addition, as discussed in [23], the proposed preconditioner can also tackle more general algebraic problems with semidefinite or nonsymmetric matrix blocks.…”

Section: Discussionmentioning

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: Discussionmentioning

confidence: 99%

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

2017

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“…If one prefers to use only real arithmetics, the two-by-two block complex matrix can be reformulated as a four-by-four real-valued block matrix, see e.g. [10] for an example of this.…”

Section: Galerkin Variational Femmentioning

confidence: 99%

“…For the solution with Q and Q * we can use the Conjugate Gradient method preconditioned by some Algebraic Multigrid (AMG) method, see e.g. [20,27] or by some other well-parallelizable preconditioner such as special AMG versions [10,11,13] or such as the gaining popularity Monte Carlo-based approximate inverse preconditioning methods (cf. e.g.…”

Section: Computational and Communication Complexity Of The Proposed Amentioning

confidence: 99%

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Parallel Solution Methods and Preconditioners for Evolution Equations

Axelsson

Neytcheva

Liang

2018

Mathematical Modelling and Analysis

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The recent development of the high performance computer platforms shows a clear trend towards heterogeneity and hierarchy. In order to utilize the computational power, particular attention must be paid to ﬁnding new algorithms or adjust existing ones so that they better match the HPC computer architecture. In this work we consider an alternative to classical time-stepping methods based on use of time-harmonic properties and discuss solution approaches that allow eﬃcient utilization of modern HPC resources. The method in focus is based on a truncated Fourier expansion of the solution of an evolutionary problem. The analysis is done for linear equations and it is remarked on the possibility to use two- or multilevel mesh methods for nonlinear problems, which can enable further, even higher degree of parallelization. The arising block matrix system to be solved admits a two-by-two block form with square blocks, for which a very eﬃcient preconditioner exists. It leads to tight eigenvalue bounds for the preconditioned matrix and, hence, to a very fast convergence of a preconditioned Krylov subspace or iterative reﬁnement method. The analytical background is shown as well as some illustrating numerical examples.

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“…Some other methods such as the preconditioned GSOR (PGSOR) iteration method [14], the complex-value to real-value (C-to-R) preconditioner [15], and so on, also attract a lot of researchers' interest. For more efficient methods, we refer to the works in [16][17][18].…”

Section: Introductionmentioning

confidence: 99%

On C-To-R-Based Iteration Methods for a Class of Complex Symmetric Weakly Nonlinear Equations

Zeng

2020

Mathematics

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To avoid solving the complex systems, we first rewrite the complex-valued nonlinear system to real-valued form (C-to-R) equivalently. Then, based on separable property of the linear and the nonlinear terms, we present a C-to-R-based Picard iteration method and a nonlinear C-to-R-based splitting (NC-to-R) iteration method for solving a class of large sparse and complex symmetric weakly nonlinear equations. At each inner process iterative step of the new methods, one only needs to solve the real subsystems with the same symmetric positive and definite coefficient matrix. Therefore, the computational workloads and computational storage will be saved in actual implements. The conditions for guaranteeing the local convergence are studied in detail. The quasi-optimal parameters are also proposed for both the C-to-R-based Picard iteration method and the NC-to-R iteration method. Numerical experiments are performed to show the efficiency of the new methods.Keywords: weakly nonlinear equations; C-to-R preconditioner; local convergence; Picard iteration; complex symmetric matrix. MSC: 65F10; 65F50; 65N22where A = W + iT ∈ C n×n is a large, sparse, complex symmetric matrix, with W ∈ R n×n and T ∈ R n×n being the real parts and the imaginary parts of the coefficient matrix A, respectively. Here, we assume that W and T are both symmetric positive and semidefinite (SPSD) and at least one of them being symmetric positive and definite (SPD). The right hand vector function φ : D ⊂ C n → C n is a continuously differential function defined on the open convex domain D in the n-dimensional C n . u ∈ C n is an unknown vector. When the linear term Au is strongly dominant over the nonlinear term φ(u) in certain norm [1], we say that the system of nonlinear Equation (1) is weakly nonlinear. Here, and in the sequence, we assume that the Jacobian matrix of the nonlinear function φ(u) at the solution point u ∈ D, denoted as φ (u ), is the non-Hermitian and negative semidefinite.Weakly nonlinear equations of the form (1) arise in many areas of scientific computing and engineering applications, e.g., nonlinear ordinary and partial differential equations, nonlinear integral and integro differential equations, nonlinear optimization and variational problems, saddle point problems from image processing, and so on. For more details, see [2][3][4][5][6][7][8].

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Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems

Cited by 30 publications

References 21 publications

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

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

Parallel Solution Methods and Preconditioners for Evolution Equations

On C-To-R-Based Iteration Methods for a Class of Complex Symmetric Weakly Nonlinear Equations

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