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.
I forgot to mention the time: the seminar will be at 14:30.
Best, -- Leonardo.
On 03/06/19 10:18, Leonardo Robol wrote:
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.
-
Leonardo Robol