Catholic University of Leuven

Everybody is familiar with the concept of eigenvalues of a matrix. In this talk, we consider the nonlinear eigenvalue problem. These are problems for which the eigenvalue parameter appears in a nonlinear way in the equation. In physics, the Schroedinger equation for determining the bound states in a semiconductor device, introduces terms with square roots of different shifts of the eigenvalue. In mechanical and civil engineering, new materials often have nonlinear damping properties. For the vibration analysis of such materials, this leads to nonlinear functions of the eigenvalue in the system matrix.

One particular example is the sandwhich beam problem, where a layer of damping material is sandwhiched between two layers of steel. Another example is the stability analysis of the Helmholtz equation with a noise excitation produced by burners in a combustion chamber. The burners lead to a boundary condition with delay terms (exponentials of the eigenvalue).

We often receive the question: “How can we solve a nonlinear eigenvalue problem?” This talk explains the different steps to be taken for using Krylov methods. The general approach works as follows: 1) approximate the nonlinearity by a rational function; 2) rewrite this rational eigenvalue problem as a linear eigenvalue problem and then 3) solve this by a Krylov method. We explain each of the three steps.

The convergence of Krylov subspace methods like conjugate gradients

depends on the eigenvalues of the underlying matrix. In many cases

the exact location of the eigenvalues is unknown, but one has some

information about the distribution of eigenvalues in an asymptotic

sense. This could be the case for linear systems arising from a

discretization of a PDE. The asymptotic behavior then takes place

when the meshsize tends to zero.

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We discuss two possible approaches to study the convergence of

conjugate gradients based on such information.

The first approach is based on a straightforward idea to estimate

the condition number. This method is illustrated by means of a

comparison of preconditioning techniques.

The second approach takes into account the full asymptotic

spectrum. It gives a bound on the asymptotic convergence factor

which explains the superlinear convergence observed in many situations.

This method is mathematically more involved since it deals with

potential theory. I will explain the basic ideas.

Chebyshev polynomials are arguably the most useful orthogonal polynomials for computational purposes. In one dimension they arise from the close relationship that exists between Fourier series and polynomials. We describe how this relationship generalizes to Fourier series on certain symmetric lattices, that exist in all dimensions. The associated polynomials can not be seen as tensor-product generalizations of the one-dimensional case. Yet, they still enjoy excellent properties for interpolation, integration, and spectral approximation in general, with fast FFT-based algorithms, on a variety of domains. The first interesting case is the equilateral triangle in two dimensions (almost). We further describe the generalization of Chebyshev polynomials of the second kind, and many new kinds are found when the theory is completed. Connections are made to Laplacian eigenfunctions, representation theory of finite groups, and the Gibbs phenomenon in higher dimensions.