Kai Bergermann

Kai Bergermann

Randomized sketching is currently introduced into every area of numerical linear algebra. In Krylov subspace methods, it allows runtime savings at the cost of small accuracy reductions. This work offers a different view on sketching in Krylov methods by analyzing what subspace embeddings are obtained by arbitrary sketching matrices. The analysis gives rise to a deterministic sketching approach leveraging row subset selection techniques that yield subspace embeddings holding with probability 1. We propose deterministically sketched Krylov methods for matrix functions, linear systems, and eigenvalue problems that show a similar performance to their randomly sketched counterparts.

Kai Bergermann, Martin Stoll, Toni Volkmer

We generalize a graph-based multiclass semi-supervised classification technique based on diffuse interface methods to multilayer graphs. Besides the treatment of various applications with an inherent multilayer structure, we present a very flexible approach that interprets high-dimensional data in a low-dimensional multilayer graph representation. Highly efficient numerical methods involving the spectral decomposition of the corresponding differential graph operators as well as fast matrix-vector products based on the nonequispaced fast Fourier transform (NFFT) enable the rapid treatment of large and high-dimensional data sets. We perform various numerical tests putting a special focus on image segmentation. In particular, we test the performance of our method on data sets with up to 10 million nodes per layer as well as up to 104 dimensions resulting in graphs with up to 52 layers. While all presented numerical experiments can be run on an average laptop computer, the linear dependence per iteration step of the runtime on the network size in all stages of our algorithm makes it scalable to even larger and higher-dimensional problems.