Communication and topology-aware process mapping is a powerful approach to reduce communication time in parallel applications with known communication patterns on large, distributed memory systems. We address the problem as a quadratic assignment problem (QAP) and present algorithms to construct initial mappings of processes to processors and fast local search algorithms to further improve the mappings. By exploiting assumptions that typically hold for applications and modern supercomputer systems such as sparse communication patterns and hierarchically organized communication systems, we obtain significantly more powerful algorithms for these special QAPs. Our multilevel construction algorithms employ perfectly balanced graph partitioning techniques and exploit the given communication system hierarchy in significant ways. We present improvements to a local search algorithm of Brandfass et al. (2013) and further decrease the running time by reducing the time needed to perform swaps in the assignment as well as by carefully constraining local search neighborhoods. We also investigate different algorithms to create the communication graph that is mapped onto the processor network. Experiments indicate that our algorithms not only dramatically speed up local search but also, due to the multilevel approach, find much better solutions in practice.