ABSTRACTS AND SLIDES

14:00 - 14:05 : Kees Vuik (Director of DCSE)
Welcome (slides)

14:05 - 14:35 : Mike Botchev (University of Twente)
An SVD-approach for Jacobi-Davidson solution of nonlinear integrated-optics eigenproblems (slides)
We consider nonlinear, nonpolynomial eigenvalue problems stemming from finite element discretization of the Helmholtz equation and arising in simulation of integrated optical devices.  The Helmholtz equation is originally posed in an infinite domain but solved numerically in a bounded domain with artificial boundary conditions.  These boundary conditions, which can be of different types (and called by different names, e.g., transparent, nonreflecting, absorbing), should guarantee transparency of the domain boundary for outgoing waves.  The boundary conditions used in this work are the recently developed TIBCs (transparent-influx boundary conditions [Nicolau & Van Groesen, 2005]). These conditions are obtained by solving the problem in the exterior of the computational domain analytically and have a number of advantages as compared to other known formulations of transparent or nonreflecting boundary conditions.

A finite element discretization of the Helmholtz equation leads to an eigenvalue problem where the transparent-influx boundary conditions cause a nonlinear, nonpolynomial dependence of the matrix on the eigenvalue. Moreover, it is not trivial to express this nonlinearity in an explicit way. However, since the nonlinearity results from the boundary conditions, the nonlinear contributions to the matrix of the eigenvalue problem can be seen a smaller dimensional discrete operator. This allows for a relatively cheap low-rank SVD parametrization of the nonlinear dependence so that it can be approximated by a low-degree matrix polynomial. Thus, we reduce the nonlinear nonpolynomial eigenvalue problem to a nonlinear polynomial one. Once this reduction is done, the Jacobi-Davidson method can readily be applied. Depending on the accuracy requirements of the eigenvalue problem, the polynomial approximation can be refined during the Jacobi-Davidson iterations.

This is a joint work with Ardhasena Sopaheluwakan, Gerard Sleijpen, Brenny van Groesen, and Manfred Hammer.  This research was a work in progress with Gene Golub.

14:35 - 15:05 : Jan Brandts (University of Amsterdam)
Gene Golub's contributions to computation of Google's PageRank (slides)
In this presentation we briefly review the definition of the PageRank concept of Google by means of an elementary game. Then we concentrate on the computation of PageRank, while highlighting Gene Golub's contributions in this area.

15:05 - 15:35 : Jok Tang (Delft University of Technology)
A Generalized Two-Level Preconditioned Conjugate Gradient Method (slides)
For various applications, it is well-known that standard preconditioned conjugate gradient (PCG) methods converge slowly. Instead, more efficient two-level PCG methods can be used, which are proposed in the deflation, domain decomposition and multigrid literature. At first glance, these methods seem to be completely different. However, from an abstract point of view, we can show that some of them are closely related to each other or even equivalent. A generalized two-level PCG method can be formulated that covers all different variants. Finally, numerical experiment will be used to show the efficiency and robustness of the resulting method.

15:50 - 16:30 : Bernd Fischer (Universität zu Lübeck)
Orthogonal polynomials, linear systems and applications (slides)
When I spent a wonderful postdoc year with Gene back in 1988, he introduced me to the beautiful world of orthogonal polynomials and their use for the design of efficient iterative methods for linear systems. In my talk, I will present such a striking connection, hopefully in Gene's spirit:

"We don't need new algorithms, we need better ones".

It has been always a major goal of Gene's scientific work, to bring his findings into real applications. In the second part of my talk I will report on the role of mathematics within a liver surgery, where again, the solution of huge linear system plays a crucial role.

16:30 - 17:00 : Marielba Rojas (Technical University of Denmark)
TRUSTμ: An Interior-Point Method for Large-Scale Non-Negative Regularization (slides1, slides2)
We describe the TRUSTμ method for large-scale quadratic problems with quadratic and non-negativity constraints. Such problems arise for example in image restoration where some of the matrices involved are very ill-conditioned.

TRUSTμ is an interior-point method that requires the solution of a sequence of parameterized, large-scale, and possibly ill-conditioned trust-region subproblems. The current implementation is based on recent techniques for the large-scale trust-region subproblem. We will describe the method and present results on image restoration problems.