Heinrich Voss

A Jacobi-Davidson-type projection method for nonlinear eigenvalue problems

This paper discusses a projection method for nonlinear eigenvalue problems. The subspace of approximants is constructed by a Jacobi-Davidson-type approach, and the arising eigenproblems of small dimension are solved by safeguarded iteration. The method is applied to a rational eigenvalue problem governing the vibrations of tube bundle immersed in an inviscid compressible fluid.

An A Priori Bound for Automated Multilevel Substructuring

The automated multilevel substructuring (AMLS) method has been developed to reduce the computational demands of frequency response analysis and has recently been proposed as an alternative to iterative projection methods like those of Lanczos or Jacobi--Davidson for computing a large number of eigenvalues for matrices of very large dimension. Based on Schur complements and modal approximations of submatrices on several levels, AMLS constructs a projected eigenproblem which yields good approximations of eigenvalues at the lower end of the spectrum. Rewriting the original problem as a rational eigenproblem of the same dimension as the projected problem and taking advantage of a minmax characterization for the rational eigenproblem, we derive an a priori bound for the AMLS approximation of eigenvalues.

A Maxmin Principle for Nonlinear Eigenvalue Problems with Application to a Rational Spectral Problem in Fluid-Solid Vibration

In this paper we prove a maxmin principle for nonlinear nonoverdamped eigenvalue problems corresponding to the characterization of Courant, Fischer and Weyl for linear eigenproblems. We apply it to locate eigenvalues of a rational spectral problem in fluid-solid interaction.

Fast reconstruction in magnetic particle imaging

Magnetic particle imaging (MPI) is a new tomographic imaging method which is able to capture the fast dynamic behavior of magnetic tracer material. From measured induced signals, the unknown magnetic particle concentration is reconstructed using a previously determined system function, which describes the relation between particle position and signal response. After discretization, the system function is represented by a matrix, whose size can prohibit the use of direct solvers for matrix inversion to reconstruct the image. In this paper, we present a new reconstruction approach, which combines efficient compression techniques and iterative reconstruction solvers. The data compression is based on orthogonal transforms, which extract the most relevant information from the system function matrix by thresholding, such that any iterative solver is strongly accelerated. The effect of the compression with respect to memory requirements, computational complexity and image quality is investigated. With the proposed method, it is possible to achieve real-time reconstruction with almost no loss in image quality using measured 4D MPI data.

A Jacobi–Davidson Method for Nonlinear Eigenproblems

For the nonlinear eigenvalue problem T(λ)x=0 we consider a Jacobi–Davidson type iterative projection method. The resulting projected nonlinear eigenvalue problems are solved by inverse iteration. The method is applied to a rational eigenvalue problem governing damped vibrations of a structure.

A RATIONAL SPECTRAL PROBLEM IN FLUID-SOLID VIBRATION

In this paper we apply a minmax characterization for nonoverdamped nonlinear eigenvalue problems to a rational eigenproblem governing mechanical vibrations of a tube bundle immersed in an inviscid compressible fluid. This eigenproblem is nonstandard in two respects: it depends rationally on the eigenparameter, and it involves non-local boundary conditions. Comparison results are proved comparing the eigenvalues of the rational problem to those of certain linear problems suggesting a way how to construct ansatz vectors for an efficient projection method.

On a quadratic eigenproblem occurring in regularized total least squares

A computational approach for solving regularized total least squares problems via a sequence of quadratic eigenvalue problems has recently been proposed. Taking advantage of a variational characterization of real eigenvalues of nonlinear eigenproblems the existence of a real right-most eigenvalue for each quadratic eigenvalue problem in the sequence is proven. For large problems the approach is improved considerably utilizing information from the previous quadratic problems and early updates in a nonlinear Arnoldi method.

A minmax principle for nonlinear eigenproblems depending continuously on the eigenparameter

Variational characterizations of real eigenvalues of selfadjoint operators on a Hilbert space depending nonlinearly on an eigenparameter usually assume differentiable dependence of the operator on the eigenparameter. In this paper we generalize these results to nonlinear problems that depend only continuously on the parameter. This result is applied to a class of variational eigenvalue problems that in particular contains the vibrations of plates with attached masses. Copyright

The Minimum Eigenvalue of a Symmetric Positive-Definite Toeplitz Matrix and Rational Hermitian Interpolation

A novel method for computing the minimal eigenvalue of a symmetric positive-definite Toeplitz matrix is presented. Similar to the algorithm of Cybenko and Van Loan, it is a combination of bisection and a root finding method. Both phases of the method are accelerated considerably by rational Hermitian interpolation of the secular equation. For randomly generated test problems of dimension 800 the average number of linear systems which must be solved to determine the smallest eigenvalue is 6.6, which reduces the computational cost of the method of Cybenko and Van Loan to approximately 35%. The method includes a rigorous error bound.

Numerical calculation of the electronic structure for three-dimensional quantum dots

In some recent papers Li, Voskoboynikov, Lee, Sze and Tretyak suggested an iterative scheme for computing the electronic states of quantum dots and quantum rings taking into account an electron effective mass which depends on the position and electron energy level. In this paper we prove that this method converges globally and linearly in an alternating way, i.e. yielding lower and upper bounds of a predetermined energy level in turn. Moreover, taking advantage of the Rayleigh functional of the governing nonlinear eigenproblem, we propose a variant which converges even quadratically thereby reducing the computational cost substantially. Two examples of finite element models of quantum dots of different shapes demonstrate the efficiency of the method.