Dear all, I am sorry for the delay in this announcement, I was planning
to send it before, but it slipped off my mind. Everybody is welcome to
attend, as usual.
Speaker: Leonardo Robol
Affiliation: Department of Mathematics, University of Pisa
Time: Wednesday, 05/06/2019
Place: Aula Riunioni, Dipartimento di Matematica
Title: Rank structures in matrix equations and matrix
functions
Low-rank matrices (and tensors) appear often in applications, and this
can be exploited in several ways to treat large scale problems that
would be otherwise unfeasible. It has been known since a long time that
in some applications, such as the solution of the Sylvester equation AX
+ XB + C = 0, working with low-rank C often leads to a solution X which
is full rank, but can be efficiently approximated by a low-rank matrix.
This can be used, among other things, in solving PDEs defined by
separable operators, such as the 2D Laplace operator on rectangular domains.
We briefly review some of these results, and the connection with some
rational approximation problems. We show that these structures appear in
a multitude of interesting cases. For instance, the same properties can
be found when modifying the coefficients of linear matrix equation by
performing low-rank updates to their coefficients, and the theory can be
extended to more general matrices which have off-diagonal blocks of
low-rank.
Time permitting, we discuss some recent developments that allow to prove
the existence of a low-rank structures in the action of functions of
matrices that have a Kronecker sum structure, such as x = f (A⊗I +
B^T⊗I)v, for particularly structured vector v. An efficient algorithm
for the approximation of the vector x, reshaped in matrix form, is
given. This founds applications in extending the ideas used for the
Laplace operator and 2D PDEs to more general nonlocal
operators, such as the fractional Laplacian.
