Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems

Abstract

Abstract.In this paper we will show how the Jacobi-Davidson iterative method can be used to solve generalized eigenproblems. Similar ideas as for the standard eigenproblem are used, but the projections, that are required to reduce the given problem to a small manageable size, need more attention. We show that by proper choices for the projection operators quadratic convergence can be achieved. The advantage of our approach is that none of the involved operators needs to be inverted. It turns out that similar projections can be used for the iterative approximation of selected eigenvalues and eigenvectors of polynomial eigenvalue equations. This approach has already been used with great success for the solution of quadratic eigenproblems associated with acoustic problems.

On correction equations and domain decomposition for computing invariant subspaces

“…In practice, this means that we need to solve the correction equation, i.e., the equation which updates the current approximate eigenvector, in a subspace that is orthogonal to the most current approximate eigenvectors. Several methods can be mentioned including the Trace Minimization method [12,11], the Davidson method [4,9] and the Jacobi-Davidson approach [14,13,16]. Most of these methods update an existing approximation by a step of Newton's method and this was illustrated in a number of papers, see, e.g., [8], and in [17].…”

Effect of poloidal inhomogeneity in plasma parameters on edge anomalous transport

“…After discretization, we end up with a cubic matrix eigenvalue problem, whose solution requires an efficient numerical algorithm. For this purpose a variant of the Jacobi-Davidson method [11], in which the convergence is accelerated significantly by combining it with a multilevel approach, has been elaborated [10].The poloidal dependence of equilibrium plasma parameters is determined from the heat and pressure balance equations integrated over the radial width of the edge region where neutrals are mostly ionized. Conditions for the MARFE formation in the Tokamak Experiment for Technology Oriented Research (TEXTOR) [13] are analyzed and the importance to calculate the characteristics of drift instabilities and anomalous transport self-consistently with the poloidal structure of plasma parameters is elucidated.…”

“…After discretization, we end up with a cubic matrix eigenvalue problem, whose solution requires an efficient numerical algorithm. For this purpose a variant of the Jacobi-Davidson method [11], in which the convergence is accelerated significantly by combining it with a multilevel approach, has been elaborated [10].…”

Harmonic and refined Rayleigh–Ritz for the polynomial eigenvalue problem

“…VIII] or twosided Jacobi-Davidson [11,12,9]. Equation (3.2) implies that if u is a first order accurate approximation to the right eigenvector (that is, u = x + d, with d small), and v is a first order accurate approximation to the left eigenvector (v = y + e, with e small), then θ(u, v) is a second order accurate approximation to the eigenvalue (|θ − λ| = O( d e ).…”

“…Subspace expansion for the polynomial eigenvalue problem. We take a new look at the Jacobi-Davidson subspace expansion for the polynomial eigenvalue problem (see [11] and [9] for previous work). Assume that we have an approximate eigenpair (θ, u), where the residual r = p(θ) u is orthogonal to a certain test vector v. We are interested in an update s ⊥ u such that p(λ)(u + s) = 0, that is, u + s is a multiple of the true eigenvector.…”