06.11.2024, 14:00 - 16:00
– Campus Golm, Building 9, Room 2.22 and via Zoom
Institutskolloquium
Graphon Models for Inhomogeneous Random Graphs
Olga Klopp (Paris), Nicolas Verzelen (Montpellier)
Siobhan Correnty (KTH Stockholm)
We consider parameterized linear systems of the form $A(\mu) x(\mu) = b$ for many different $\mu$, where $A(\mu)$ is large, sparse and nonsingular, with a nonlinear dependence on $\mu$. In this work we propose two methods to compute a partial parameterization $\tilde{x} \approx x(\mu)$, where $\tilde{x}(\mu)$ is cheap to evaluate for many values of $\mu$. Both methods utilize a companion linearization, inspired by [Effenberger, Kressner, 2012], based on an accurate Chebyshev interpolation of $A(\mu)$ on the interval $[-a,a]$, $a \in \mathbb{R}$. We approximate the solution to the linearization in a preconditioned multishift BiCG setting, as proposed in [Ahmad, Szyld, van Gijzen, 2017]. This process leads to two short-term recurrence methods, where one execution of either algorithm produces the approximation $\tilde{x}(\mu)$ for many different $\mu \in [-a,a]$ simultaneously. The first proposed method considers an exact application of a shift-and-invert preconditioner on each iteration, and the second applies an approximation to the same preconditioner in an inexact, flexible setting. The competitiveness of our methods are illustrated with large-scale problems arising from a finite element discretization of a Helmholtz equation with parameterized material coefficient.