We show how to achieve fast autocompletion for SPARQL queries on very large knowledge graphs. At any position in the body of a SPARQL query, the autocompletion suggests matching subjects, predicates, or objects. The suggestions are context-sensitive and ranked by their relevance to the part of the query already typed. The suggestions can be narrowed down by prefix search on the names and aliases of the desired subject, predicate, or object. All suggestions are themselves obtained via SPARQL queries. For existing SPARQL engines, these queries are impractically slow on large knowledge graphs. We present various algorithmic and engineering improvements of an open-source SPARQL engine such that these queries are executed efficiently. We evaluate a variety of suggestion methods on three large knowledge graphs, including the complete Wikidata. We compare our results with two widely used SPARQL engines, Virtuoso and Blazegraph. Our code, benchmarks, and complete reproducibility materials are available on https://ad.cs.uni-freiburg.de/publications.

In this paper, we propose to use hierarchical matrices as fast and robust preconditioners for a recent formulation of Maxwell's equations. This formulation was introduced by Hiptmair (IEEE Trans. Magn., vol. 44, no. 6, pp. 682-685, Jun. 2008) and remains stable in the stationary limit for nonconductive domains. The preconditioner is constructed based on a decoupling of the magnetic and electric potential. Numerical results are presented for real-life geometries and both sequential and parallel implementations.