Speaker: Raf Vandebril Affiliation: Department of Computer Science, KU Leuven Time: Monday, 08 April 2019, h. 14:00 Place: Aula Seminari, Dipartimento di Matematica
Title: Structures, Gramians, and Krylov
Krylov methods are typically used for transforming large scale prolems and matrices to matrices of a manageable size. Krylov inspired algorithms are based on the principle: project - solve - lift. In this talk we will focus on the projection step, the orthogonal bases involved, and the structured of the resulting projection.
Two bases are needed for the projection: a basis for a search Krylov subspace, say V, and another basis for the constraint Krylov subspace, name it W. For a matrix A, the projected small to medium sized matrix will be W^*AV. There is now a wide variety of possibilities for constructing the matrices W and V. We could go from classical Krylov to extended and rational Krylov subspaces. But we could also attempt to construct subspaces with variants of the matrix A, such as for instance its inverse, its conjugate transpose, or the inverse of its conjugate transpose.
In this lecture we will examine some of these possibilities. A general framework is proposed in which we will touch upon the following aspects: biorthogonal (rational) Krylov, CMV factorizations, non-unitary CMV factorizations, quasiseparable Hessenberg matrices, Hankel and Toeplitz Gramians.