METT VIII - 8th Workshop on Matrix Equations and Tensor Techniques

Rational Krylov for Stieltjes matrix functions: convergence and pole selection

8 Nov 2019, 11:10

25m

Prigogine (MPI Magdeburg)

Talk Talks Day III

Speaker

Stefano Massei (EPF Lausanne)

Description

Evaluating the action of a matrix function on a vector, that is $x=f(\mathcal{M})v$, is an ubiquitous task in applications. When the matrix $\mathcal{M}$ is large, subspace projection method, such as the rational Krylov method, are usually employed.
In this work, we provide a quasi-optimal pole choice for rational Krylov methods applied to this task when $f(z)$ is either Cauchy-Stieltjes or Laplace-Stieltjes (or, which is equivalent, completely monotonic) for a positive definite matrix $\mathcal{M}$.

Then, we consider the case when the argument $\mathcal{M}$ has the Kronecker structure $\mathcal{M}=I\otimes A-B^T\otimes I$, and is applied to a vector obtained vectorizing a low-rank matrix. This finds application, for instance, in solving fractional diffusion equation on rectangular domains.
We introduce an error analysis for the numerical approximation of $x$. Pole choices and explicit convergence bounds are given also in this case.