Preconditioned iterative methods for a class of nonlinear eigenvalue problems

Abstract

This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to nonlinear eigenvalue problems with very large sparse ill-conditioned matrices monotonically depending on the spectral parameter. To compute the smallest eigenvalue of large-scale matrix nonlinear eigenvalue problems, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors, and inner products of vectors. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.

Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Discrete Fourier Transform

“…This allows limiting by investigation of several first points (lines) of branching. To find the branching points (lines) of solutions of (8), it is necessary, as opposed to [3,17], to solve not enough studied multiparametric spectral problem. The offered in this work approaches allow to find the solutions of a nonlinear two-parametric spectral problem for homogeneous integral equations with degenerate kernels analytically dependent on two spectral parameters.…”

“…The obtained there results show that for equations of the type (10) and (14) non-uniqueness and branching of solutions dependent on the size of physical parameter are characteristic. Directly the results [17] cannot be transferred on the two-dimensional two-parametric problem (8) and (14). Here, as unlike the points of branching [17], the branching lines of solutions exist and a problem on finding the lines of branching is a nonlinear two-parametrical spectral problem.…”

“…The problem on finding the set of branching points, in turn, is not adequately explored nonlinear spectral two-parametric problem. The methods of investigation and numerical finding the solutions of one-parametric spectral problems at presence of discrete spectrum [4][5][6][7][8] are most well-developed. The existence of coherent components of spectrum, which are spectral lines for the case of real parameters [9], is essential difference of nonlinear two-parametric spectral problems.…”

A block Newton method for nonlinear eigenvalue problems

“…As a second application, we consider a simple boundary eigenvalue problem, which was also considered in [24]. Find λ > κ and a nonzero function u : A finite element discretization of (27) with linear elements on subintervals of length h = 1/n leads to the nonlinear matrix eigenvalue problem…”

Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. I. Extreme eigenvalues

“…Introduction. Nonlinear Hermitian algebraic eigenproblems of the form T (λ)v = 0 arise naturally in a variety of scientific and engineering applications, such as simulations of the sound radiation from rolling tires [7], time-harmonic acoustic wave equation in bounded domains [9], delay differential equations [20], and modeling of vibrations of certain fluid-solid structures [10], loaded strings [30], and wiresaws [36]. Most of these nonlinear eigenproblems, similar to their linear counterparts, allow for a variational characterization (min-max principle) of some eigenvalues on certain intervals.…”