Martin Stoll (Uni Chemnitz) und Melina Freitag (Uni Potsdam)
Zoom 14:00 Martin Stoll (Uni Chemnitz): From PDEs to data science: an adventure with the graph Laplacian
14:45 Tee und Kaffee Pause
15:15 Melina Freitag (Uni Potsdam): Large scale Lyapunov differential equations: relaxation strategies within inexact methods
Martin Stoll (Uni Chemnitz): From PDEs to data science: an adventure with the graph Laplacian
In this talk we briefly review some basic PDE models that are used to model phase separation in materials science. They have since become important tools in image processing and over the last years semi- supervised learning strategies could be implemented with these PDEs at the core. The main ingredient is the graph Laplacian that stems from a graph representation of the data. This matrix is large and typically dense. We illustrate some of its crucial features and show how to efficiently work with the graph Laplacian. We illustrate the performance on several examples.
Melina Freitag (Uni Potsdam): Large scale Lyapunov differential equations: relaxation strategies within inexact methods
Large-scale Lyapunov differential equations (LDEs) arise in many fields like: model reduction, damping optimization, optimal control, numerical solution of stochastic partial differential equations (SPDEs), etc...
We shall briefly describe two useful and powerful tools for the numerical solution of large-scale Lyapunov equations: the Rational Krylov subspace (RKS) and Alternating Direction implicit (ADI) methods. One of the computationally most expensive parts in both methods is that, in each iteration step, shifted linear systems have to be solved. For very large systems this solution is usually implemented using iterative methods, leading to inexact solves. In this talk we will provide theory for a relaxation strategy within these inexact solves, supported by numerical examples. This is joint work with Patrick Kürschner (Leipzig).
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