# Reports of Plasma physics simulation

Research within the project: Plasma physics simulation

*Research within the project: Plasma physics simulation*

*Alternative correction equations in the Jacobi-Davidson method*

M. Genseberger, G.L.G. Sleijpen;

1998, MAS-R9816, ISSN 1386-3703

The correction equation in the Jacobi-Davidson method is effective in a subspace orthogonal to the current eigenvector approximation, while for the continuation of the process only vectors orthogonal to the search subspace are of importance. Such a vector is obtained by orthogonalizing the (approximate) solution of the correction equation against the search subspace. As an alternative, a variant of the correction equation can be formulated that is restricted to the subspace orthogonal to the current search subspace. In this paper, we discuss the effectivity of this variant.

Download the pdf-file*Restarting parallel Jacobi-Davidson with both standard and harmonic Ritz values*M. Nool, A. van der Ploeg

1998, MAS-R9807, ISSN 1386-3703

We study the Jacobi-Davidson method for the solution of large generalized eigenproblems as they arise in MagnetoHydroDynamics. We have combined Jacobi-Davidson (using standard Ritz values) with a shift and invert technique. We apply a complete LU decomposition in which reordering strategies based on a combination of block cyclic reduction and domain decomposition result in a well-parallelizable algorithm. Moreover, we describe a variant of Jacobi-Davidson in which harmonic Ritz values are used. In this variant the same parallel LU decomposition is used, but this time as a preconditioner to solve the `correction` equation.

Download the pdf-file*A parallel Jacobi-Davidson method for solving generalized eigenvalue problems in linear magnetohydrodynamics*

M. Nool, A. van der Ploeg

1997, MAS-R9733, ISSN 1386-3703

We study the solution of generalized eigenproblems generated by a model which is used for stability investigation of tokamak plasmas. The eigenvalue problems are of the form*A x = λ B x*, in which the complex matrices*A*and*B*are block tridiagonal, and*B*is Hermitian positive definite. The Jacobi-Davidson method appears to be an excellent method for parallel computation of a few selected eigenvalues, because the basic ingredients are matrix-vector products, vector updates and inner products. The method is based on solving projected eigenproblems of order typically less than 30. The computation of an approximate solution of a large system of linear equations is usually the most expensive step in the algorithm. By using a suitable preconditioner, only a moderate number of steps of an inner iteration is required in order to retain fast convergence for the JD process. Several preconditioning techniques are discussed.

Download the pdf-file*Reordering strategies and LU-decomposition of block tridiagonal matrices for parallel processing*A. van der Ploeg

1996, NM-R9618, ISSN 0169-0388

Solution of large sparse systems of linear equations continues to be a major research area with widespread application. In many applications, the unknowns appear in groups, and the coefficient matrix has a block structure corresponding to these groups. For example, in discretised incompressible Navier-Stokes equations the velocities and the pressure belonging to the same grid point form such a group.

Download the pdf-file*Jacobi-Davidson methods for generalized MHD-eigenvalue problems*

J.G.L. Booten, D. Fokkema, G.L.G. Sleijpen, H.A. van der Vorst

1995, NM-R9514, ISSN 0169-0388

A Jacobi-Davidson algorithm for computing selected eigenvalues and associated eigenvectors of the generalized eigenvalue problem*Ax*= λ*Bx*is presented. In this paper the emphasis is put on the case where one of the matrices, say the*B*-matrix, is Hermitian positive definite. The method is an inner-outer iterative scheme, in which the inner iteration process consists of solving linear systems to some accuracy. The factorization of either matrix is avoided. Numerical experiments are presented for problems arising in magnetohydrodynamics (MHD).

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