Linear algebra research @ NALAG

In NALAG there has been a long tradition of studying fast algorithms for low displacement rank matrices like Toeplitz, Hankel, etc. More recently the focus has shifted to the study of rank structured matrices. This involves the solution of linear systems as wel as eigenvalue problems and the computation of singular value decompositions for matrices and corresponding decompositions in the multilinear case. Also large scale problems, and corresponding iterative methods are investigated. Many of these algorithms need to be adapted to new hardware architectures. The reseach group takes part in Flanders ExaScale Lab project where the group is mainly involved in Krylov type methods and preconditioning for the solution of linear systems.
Large scale matrices with parameters arise in PDE constraint optimization, uncertainty modeling, structured distance problems, and graph applications. In order to reduce the computational cost to work with such matrices and related tensors, suitable model reduction techniques are required. In this context the group also participates in an IAP project. Related is the solution of eigenvalue problems with specific structures, e.g., symmetric matrices, matrices consisting of Kronecker sums, non-linear eigenvalue problems, etc. An important class of applications are structures and vibration.
For the algorithms that are designed, also numerically reliable and robust software is developed. The idea of the GLAS project is to produce C++ code for Generic Linear Algebra Software.
The recursions used in the structured linear algebra problems have explicit links with recursions for (formal) orthogonal polynomials and orthogonal functions, and problems of rational approximation and (discrete) least squares problems.