Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

  author = {G. W. Zumbusch},
  title = {Adaptive Parallel Multilevel Methods for Partial
		  Differential Equations},
  school = {Universit\"at Bonn},
  year = {2001},
  annote = {thesis,parallel,habil},
  type = {Habilitation},
  abstract = {In this text we propose a space-filling curve enumeration
		  scheme for the load balancing problem. It is based on the
		  principles of self-similarity and scaling invariance. It
		  provides even multilevel locality, i.e. as much locality on
		  each scale as possible. We introduce the space-filling
		  curve schemes and prove some of the properties of the
		  partitions. The scheme is cheap, deterministic,
		  incremental, can be parallelised and provides acceptable
		  partitions. However, even more striking, it seems to be one
		  of the few partitioning methods where quasi-optimal
		  estimates can be shown. We are able to derive sharp
		  estimates both on the partition and on the multilevel
		  algorithms on the partition, which is more than is known
		  about competing graph partitioning load balancing methods
		  so far.
		  Furthermore, we give a survey of the three main aspects of
		  the efficient treatment of PDEs, that is, discretisation,
		  multilevel solution and parallelisation. We will treat the
		  main features of each of the three distinct topics and
		  cover the historical background and modern developments. We
		  demonstrate how all three ingredients can be put together
		  to give an adaptive and parallel multilevel approach for
		  the solution of PDEs. Error estimators and adaptive grid
		  refinement techniques for ordinary and for sparse grid
		  discretisations are presented. Different types of additive
		  and multiplicative multilevel solvers are discussed with
		  respect to parallel implementation and application to
		  adaptive refined grids. Efficiency issues are treated both
		  for the sequential multilevel methods and for the parallel
		  version by hash table storage techniques. Furthermore,
		  space-filling curve enumeration for parallel load balancing
		  and processor cache efficiency are discussed. We will apply
		  the method to elliptic boundary value problems.
		  We are able to derive estimates for the quality of the
		  partitions by space-filling curves and the load balancing
		  of the numerical algorithms on the grids. Even for adaptive
		  grid refinement within certain ranges we are able to prove
		  that the partitions are quasi-optimal, i.e. the cut sizes
		  of the dual graph are only a constant factor away from
		  optimum independent of the mesh size. Hence we obtain
		  asymptotic optimality of the parallel algorithms. This
		  seems to be remarkable in comparison to graph based
		  heuristics, where little is known about the quality.
		  Furthermore we were able to demonstrate the performance of
		  the method on a range of the world's largest parallel
		  computers, namely ASCI Blue Pacific and a prototype Cray
		  T3E (now presumably at NSA), which are each larger than any
		  non-US system. We complement this data by simulations run
		  on Parnass2, which was the first non-US self-made cluster
		  in the list of the world's largest 500 computers (TOP500).
		  We also demonstrate that this cluster is able to outperform
		  many other commercial parallel computers on a per processor
		  base. }