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@inproceedings{Bruaset.Langtangen.Zumbusch:1998,
author = {A. M. Bruaset and H. P. Langtangen and G. W. Zumbusch},
title = {Domain Decomposition and Multilevel Methods in
{D}iffpack},
booktitle = {Proceedings of Domain Decomposition Methods 9, DD9},
pages = {655--662},
year = {1998},
editor = {P. E. Bj{\o}rstad and M. S. Espedal and D. E. Keyes},
publisher = {Domain Decomposition Press},
address = {Bergen, Norway},
note = {also as report STF42 F96017, Sintef Applied Mathematics,
Oslo, 1996 },
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/zumbusch/dd9book.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/dd9book.pdf},
abstract = {... Domain decomposition and multilevel methods contain a
variety of more standard numerical building blocks (linear
solvers, matrix assembly, interpolation of fields etc.).
Successful software for complicated applications must offer
the user a flexible run-time combination of all these
different components. The purpose of the present paper is
to describe how one can achieve such flexible software. In
particular, we present a unified framework for domain
decomposition and multilevel methods, and show how this
framework can be efficiently implemented in existing
software packages for PDEs. \\ The unified framework about
to be presented is in part well known from the analysis of
overlapping and non-overlapping methods [M. Dryja O.B.
Widlund 1990 ], as well as from theory for overlapping and
multilevel schemes [J. Xu 1992]. In this context, the goal
of this paper is to extend the known framework to cover
even more methods in common use, especially some Schur
complement and nonlinear schemes. We will formulate the
framework in a novel way that encourages systematic
implementation of a wide class of domain decomposition and
multilevel methods. Finally, we report on the experiences
gathered from a particular implementation in the Diffpack
software.}
}
@incollection{Cai.Bruaset.Langtangen.ea:2003,
author = {X. Cai and A. M. Bruaset and H. P. Langtangen and G. T.
Lines and K. Samuelsson and W. Shen and A. Tveito and G.
Zumbusch},
title = {Performance Modeling of PDE Solvers},
booktitle = {Advanced Topics in Computational Partial Differential
Equations},
pages = {361--400},
publisher = {Springer},
address = {Berlin, Germany},
year = {2003},
editor = {H. P. Langtangen and A. Tveito},
volume = {33},
series = {Lecture Notes in Computational Science and Engineering},
chapter = {9},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/cpu_measurements.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/cpu_measurements.pdf},
abstract = {The primary purpose of this report is to collect some
information on the CPU-time consumption in a series of
typical numerical simulations for solving partial
differential equations (PDEs). We would like to establish,
through analyzing the CPU-measurements, the performance
model for a number of numerical methods when applied in
different model problems. All the simulators, i.e. the
software programs carrying out the simulations, have been
developed using Diffpack which is a generic C++ library
based on object-oriented programming techniques. Therefore,
the established performance models may offer a rough
prediction of real CPU-time consumption by actual Diffpack
simulators for practical problems. Additionally, the report
can also be regarded as an investigation of the
computational efficiency of the current Diffpack
implementations. Last but not least, we wish to point out
to Diffpack programmers some implementation issues that may
affect the performance of the simulators. }
}
@article{Griebel.Kiefer.Zumbusch:2000,
author = {M. Griebel and F. Kiefer and G. Zumbusch},
title = {{V}org{\"a}nge m{\"o}glichst realit{\"a}tsnah simulieren.
{W}issenschaftliches {R}echnen als neue {D}imension in der
{F}orschung},
journal = {Bonner Universit\"{a}tsnachrichten},
address = {Bonn, Germany},
year = {2000},
volume = {217},
pages = {48--49},
month = {January},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/bun217.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/bun217.pdf},
http = {http://cse.mathe.uni-jena.de/pub/zumbusch/bun217.html},
annote = {editorial},
note = {also as `Simulating Processes as Realistically as
Possible', Bonn University News International (BUNI),
7:28--29, November 2000},
abstract = {Computer werden immer schneller und leistungsfähiger. Das
gilt nicht nur für den Personalcomputer im Büro, sondern
auch für die weltweit schnellsten
Hochleistungs-Parallellrechner, die heute schon mehr als
eine Billion Rechen-Operationen in nur einer Sekunde
ausführen können. Insbesondere durch sie hat sich mit dem
"Wissenschaftlichen Rechnen" und der "Numerischen
Simulation" in den Naturwissenschaften neben dem
traditionellen praktischen Weg (Experiment und Beobachtung)
und dem theoretischen Ansatz (mathematische Formulierung)
ein vielversprechender dritter Weg herausgebildet, um die
Wirklichkeit zu beschreiben. In Bonn arbeiten die
Spezialisten dieses Fachgebiets in der Abteilung für
Wissenschaftliches Rechnen und Numerische Simulation im
Institut für Angewandte Mathematik.}
}
@book{Griebel.Knapek.Zumbusch:2007,
author = {M. Griebel and S. Knapek and G. Zumbusch},
title = {Numerical Simulation in Molecular Dynamics: Numerics, Algorithms, Parallelization, Applications},
publisher = {Springer},
address = {Berlin, Heidelberg},
year = {2007},
series = {Texts in Computational Science and Engineering},
volume = {5},
note = {},
annote = {editorial},
amazon = {http://www.amazon.de/Numerical-Simulation-Molecular-Dynamics-Parallelization/dp/3540680942/ref=sr_1_6/028-3794171-7733364?ie=UTF8&s=books&qid=1182883588&sr=1-6},
http = {http://www.springer.com/dal/home/generic/search/results?SGWID=1-40109-22-173713766-0},
abstract = {Particle models play an important role in many applications in physics, chemistry and biology. These can be studied on the computer with the help of molecular dynamics simulations. This book presents in detail the necessary numerical methods, the theoretical background and foundations and the techniques involved, including linked-cell method, SPME-method, tree codes, amd multipol technique. It illustrates such aspects as modeling, discretization, algorithms and their parallel implementation with MPI on computer systems with distributed memory. The text goes on to offer detailed explanations of the different steps of numerical simulation, providing illustrative code examples. With the description of the algorithms and the presentation of the results of various simulations from fields such as material science, nanotechnology, biochemistry and astrophysics, the reader of this book will learn step by step how to write programs capable of running successful experiments for molecular dynamics.}
}
@book{Griebel.Knapek.Zumbusch.ea:2004,
author = {M. Griebel and S. Knapek and G. Zumbusch and A. Caglar},
title = {{N}umerische {S}imulation in der {M}olek\"uldynamik.
{N}umerik, {A}lgorithmen, {P}arallelisierung,
{A}nwendungen},
publisher = {Springer},
address = {Berlin, Heidelberg},
year = {2004},
note = {},
annote = {editorial},
amazon = {http://www.amazon.de/exec/obidos/ASIN/3540418563/comicgeschrei-21},
http = {http://www.ins.uni-bonn.de/info/md},
abstract = {Das Lehrbuch führt in die wichtigsten Simulationstechniken
zur numerischen Behandlung der Newtonschen
Bewegungsgleichungen ein. Der Schwerpunkt liegt hierbei auf
der schnellen Auswertung kurz- und langreichweitiger Kräfte
mittels Linked Cell-, P3M-, Baum- und Multipol-Verfahren
sowie deren paralleler Implementierung und Lastbalancierung
auf Rechensystemen mit verteiltem Speicher. Die einzelnen
Kapitel bieten detaillierte Hinweise, um die Verfahren
Schritt für Schritt in ein Programmpaket umzusetzen.
Zahlreiche farbige Abbildungen enthalten
Simulationsergebnisse für eine Reihe von Anwendungen.}
}
@inproceedings{Caglar.Griebel.Schweitzer.ea:1999,
author = {A. Caglar and M. Griebel and M. A. Schweitzer and G.
Zumbusch},
title = {Dynamic Load-Balancing of Hierarchical Tree Algorithms on
a Cluster of Multiprocessor {PC}s and on the {C}ray
{T}3{E}},
booktitle = {Proceedings 14th Supercomputer Conference, Mannheim},
year = {1999},
editor = {H. W. Meuer},
series = {ISBN 3-932178-08-4},
publisher = {Mateo},
address = {Mannheim, Germany},
note = {SuParCup '99 Award Winning Paper, also as SFB 256 report
27},
annote = {refereed},
ps = {http://wissrech.iam.uni-bonn.de/research/pub/zumbusch/suparcup99.ps.gz},
pdf = {http://wissrech.iam.uni-bonn.de/research/pub/zumbusch/suparcup99.pdf},
abstract = {The solution of many problems in science and engineering
is based on computational kernels for the numerical
treatment of partial differential equations (PDEs) or
N-body problems. Traditional solution methods however
reduce these to linear algebra or brute force algorithms on
structured data sets. Larger and larger simulations require
smarter algorithms to be tractable. Hierarchical tree
algorithms represent such a class, both for PDEs and for
N-body problems. However, their efficient parallelization
is not straightforward. Some difficulties can be removed,
if one can provide a fast dynamic load-balancing scheme to
cope with the tree variations of the unstructured data
sets. In this paper we propose a very cheap yet very
efficient load-balancing scheme for tree algorithms based
on space-filling curves. Furthermore we present the
Parnass2 cluster, on which such parallel codes perform
extremely well. The cluster consists of SMP PCs and a
Myrinet network at Gigabit/s speed configured with full
bisection bandwidth. It turns out that it does not only has
the obvious price/performance advantage, but also an
absolute performance, which is comparable to the latest
commercial Cray T3E.}
}
@inproceedings{Griebel.Zumbusch:2002,
author = {M. Griebel and G. Zumbusch},
title = {Hash based adaptive parallel multilevel methods with
space-filling curves},
booktitle = {NIC Symposium 2001},
year = {2002},
editor = {Horst Rollnik and Dietrich Wolf},
series = {NIC Series, ISBN 3-00-009055-X},
publisher = {Forschungszentrum J\"ulich},
address = {Germany},
volume = {9},
pages = {479--492},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/juelich01.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/juelich01.pdf},
abstract = {The solution of partial differential equations on a
parallel computer usually follows the data parallel
paradigm. The grid is partitioned and mapped onto the
processors. In this paper a parallelisable and cheap method
based on space-filling curves is proposed. The partitioning
is embedded into the parallel solution algorithm using
multilevel iterative solvers and adaptive grid refinement.
Numerical experiments on two massively parallel computers
prove the efficiency of this approach. }
}
@book{Griebel.Zumbusch:2000,
editor = {M. Griebel and G. Zumbusch},
title = {Computing},
publisher = {Springer},
address = {Vienna, Austria},
year = {2000},
volume = {64(4)},
annote = {editorial},
note = {(guest editors) special issue multigrid methods},
http = {http://link.springer.de/link/service/journals/00607/tocs/t0064004.htm},
abstract = {In October 1998 the tenth workshop in a series of biannual
GAMM seminars on multigrid methods was held. Almost two
decades have passed since the first one, and the topics of
the seminars provide a good insight into the progress
during this period. The series began in the former GDR and
is nowadays organised in cooperation with the
GAMM-Committees for ``Discretization Methods in Solid
Mechanics'' and ``Efficient Numerical Methods for PDEs''.
As this was the tenth anniversary of the series, the rather
general title was chosen ``International GAMM-Workshop on
Multigrid Methods''...}
}
@inproceedings{Griebel.Zumbusch:1997,
author = {M. Griebel and G. W. Zumbusch},
title = {Parnass: Porting gigabit-{LAN} components to a workstation
cluster},
booktitle = {Proceedings of the 1st Workshop Cluster-Computing},
pages = {101--124},
year = {1997},
editor = {W. Rehm},
number = {CSR-97-05},
series = {Chemnitzer Informatik Berichte},
organization = {TU Chemnitz},
address = {Chemnitz, Germany},
note = {also as Techn. Report No 19, SFB 256, Univ. Bonn},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/cluster97.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/cluster97.pdf},
abstract = {We will report on a cluster of workstations at our
department, called Parnass. It is based on different types
of MIPS processor workstations and servers, connected by a
Myrinet, a Gigabit per second switched LAN, and
additionally a Fast Ethernet. We have ported some low level
message passing libraries as well as MPI to the Myrinet. A
comparison of the performance of various communication
libraries on different networks will be presented.}
}
@inproceedings{Griebel.Zumbusch:1998,
author = {M. Griebel and G. W. Zumbusch},
title = {Hash-Storage Techniques for Adaptive Multilevel Solvers
and their Domain Decomposition Parallelization},
booktitle = {Proceedings of Domain Decomposition Methods 10, DD10
(1997)},
pages = {271--278},
year = {1998},
editor = {J. Mandel and C. Farhat and X.-C. Cai},
number = {218},
series = {Contemporary Mathematics},
publisher = {AMS},
address = {Providence, Rhode Island},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/dd10.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/dd10.pdf},
abstract = {Partial differential equations can be solved efficiently
by adaptive multigrid methods on a parallel computer. We
report on the concepts of hash-table storage techniques and
space-filling curves to set up such a code. The hash-table
storage requires substantial less amount of memory and is
easier to code than tree data structures used in
traditional adaptive multigrid codes, already for the
sequential case. The parallelization takes place by a
domain decomposition by space filling curves, which are
intimately connected to the hash table. The new data
structure simplifies the parallel version of the code
substantially and introduces a cheap way to solve the load
balancing and mapping problem....}
}
@inproceedings{Griebel.Zumbusch:1998*1,
author = {M. Griebel and G. W. Zumbusch},
title = {Parallel multigrid in an adaptive {PDE} solver based on
hashing},
booktitle = {Parallel Computing: Fundamentals, Applications and New
Directions},
pages = {589--600},
editor = {E. D'Hollander and G.R. Joubert and F.J. Peters and U.
Trottenberg},
publisher = {Elsevier},
series = {Advances in Parallel Computing},
number = {12},
address = {Amsterdam, The Netherlands},
year = {1998},
note = {Proceedings of ParCo 97, Bonn, Germany},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/parco97.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/parco97.pdf},
abstract = {Partial differential equations can be solved efficiently
by adaptive multigrid methods on a parallel computer. We
report on the concept of hash-table storage techniques to
set up such a code. The code requires substantial less
amount of memory and is easier to code in the sequential
case. The parallelization takes place by a space filling
curve domain decomposition intimately connected to the hash
table. The new data structure simplifies the parallel
version of the code substantially way and introduces a
cheap way to solve the load balancing and mapping
problem.}
}
@article{Griebel.Zumbusch:1999,
author = {M. Griebel and G. W. Zumbusch},
title = {Parallel Multigrid in an Adaptive {PDE} Solver based on
Hashing and Space-Filling Curves},
journal = {Parallel Computing},
publisher = {Elsevier},
address = {Amsterdam, The Netherlands},
year = {1999},
volume = {25},
pages = {827--843},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/parco98.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/parco98.pdf},
abstract = {Partial differential equations can be solved efficiently
by adaptive multigrid methods on a parallel computer. We
report on the concept of hash-table storage techniques to
set up such a program. The code requires substantial less
amount of memory than implementations based on tree type
data structures and is easier to program in the sequential
case. The parallelization takes place by a space-filling
curve domain decomposition intimately connected to the hash
table. The new data structure simplifies the
parallelization of the code substantially and introduces a
cheap way to solve the load balancing and mapping problem.
We report on the main features of the method and give the
results of numerical experiments with the new parallel
solver on a cluster of 64 Pentium II/400MHz connected by a
Myrinet in a fat tree topology.}
}
@inproceedings{Griebel.Zumbusch:1999*1,
author = {M. Griebel and G. W. Zumbusch},
title = {Adaptive Sparse Grids for Hyperbolic Conservation Laws},
booktitle = {Hyperbolic Problems: Theory, Numerics, Applications. 7th
International Conference in Z\"{u}rich, February 1998},
editor = {M. Fey and R. Jeltsch},
volume = {1},
pages = {411--422},
series = {International Series of Numerical Mathematics 129},
year = {1999},
publisher = {Birkh\"{a}user},
address = {Basel, Switzerland},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/hyp7.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/hyp7.pdf},
abstract = {We report on numerical experiments using adaptive sparse
grid discretization techniques for the numerical solution
of scalar hyperbolic conservation laws. Sparse grids are an
efficient approximation method for functions. Compared to
regular, uniform grids of a mesh parameter $h$ contain
$h^{-d}$ points in $d$ dimensions, sparse grids require
only $h^{-1}|{\mathrm log}h|^{d-1}$ points due to a
truncated, tensor-product multi-scale basis representation.
\\ For the treatment of conservation laws two different
approaches are taken: First an explicit time-stepping
scheme based on central differences is introduced. Sparse
grids provide the representation of the solution at each
time step and reduce the number of unknowns. Further
reductions can be achieved with adaptive grid refinement
and coarsening in space. Second, an upwind type sparse grid
discretization in $d+1$ dimensional space-time is
constructed. The problem is discretized both in space and
in time, storing the solution at all time steps at once,
which would be too expensive with regular grids. In order
to deal with local features of the solution, adaptivity in
space-time is employed. This leads to local grid refinement
and local time-steps in a natural way.}
}
@article{Griebel.Zumbusch:2000*1,
author = {M. Griebel and G. W. Zumbusch},
title = {Parallel Adaptive Subspace Correction Schemes with
Applications to Elasticity},
journal = {Computer Methods in Applied Mechanics and Engineering},
publisher = {Elsevier},
address = {Amsterdam, The Netherlands},
volume = {184},
year = {2000},
pages = {303--332},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/cmame.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/cmame.pdf},
abstract = {In this paper, we give a survey on the three main aspects
of the efficient treatment of PDEs, i.e. adaptive
discretization, multilevel solution and parallelization. We
emphasize the abstract approach of subspace correction
schemes and summarize its convergence theory. Then, we give
the main features of each of the three distinct topics and
treat the historical background and modern developments.
Furthermore, we demonstrate how all three ingredients can
be put together to give an adaptive and parallel multilevel
approach for the solution of elliptic PDEs and especially
of linear elasticity problems. We report on numerical
experiments for the adaptive parallel multilevel solution
of some test problems, namely the Poisson equation and
Lam{\'e}'s equation. Here, we emphasize the parallel
efficiency of the adaptive code even for simple test
problems with little work to distribute, which is achieved
through hash storage techniques and space-filling curves.}
}
@article{Griebel.Zumbusch:2000*2,
author = {M. Griebel and G. W. Zumbusch},
title = {Preface},
journal = {Computing},
publisher = {Springer},
address = {Vienna, Austria},
year = {2000},
volume = {64},
number = {4},
pages = {287},
note = {(guest editors) special issue multigrid methods},
annote = {editorial},
http = {http://link.springer.de/link/service/journals/00607/tocs/t0064004.htm},
abstract = {In October 1998 the tenth workshop in a series of biannual
GAMM seminars on multigrid methods was held. Almost two
decades have passed since the first one, and the topics of
the seminars provide a good insight into the progress
during this period. The series began in the former GDR and
is nowadays organised in cooperation with the
GAMM-Committees for ``Discretization Methods in Solid
Mechanics'' and ``Efficient Numerical Methods for PDEs''.
As this was the tenth anniversary of the series, the rather
general title was chosen ``International GAMM-Workshop on
Multigrid Methods''...}
}
@article{Hochmuth.Knapek.Zumbusch:2000,
author = {R.~Hochmuth and S.~Knapek and G.~Zumbusch},
title = {Tensor products of {S}obolev spaces and applications},
journal = {submitted},
year = {2000},
note = {also as Technical Report 685, SFB 256, Univ.~Bonn},
annote = {refereed},
abstract = {In many cases the approximation of solutions to
variational problems involving isotropic Sobolev spaces has
a complexity which depends exponentially on the dimension.
However, if the solutions possess dominating mixed
derivatives one can find discretizations to the
corresponding variational problems with a lower complexity
-- sometimes even independent of the dimension. In order to
analyse these effects, we relate tensor products of Sobolev
spaces with spaces with dominating mixed derivatives. Based
on these considerations we construct families of finite
dimensional anisotropic approximation spaces which
generalize in particular sparse grids. The obtained
estimates demonstrate, in which cases a complexity
independent or nearly independent of the dimension can be
expected. Finally numerical experiments demonstrate the
usefulness of the suggested approximation spaces.},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/tensor.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/tensor.pdf}
}
@inproceedings{Korzen.Schriever.Ziener.ea:1996,
author = {M. Korzen and R. Schriever and K.-U. Ziener and O. Paetsch
and G. W. Zumbusch},
title = {Real-Time 3-D Visualization of Surface Temperature Fields
Measured by Thermocouples on Steel Structures in Fire
Engineering},
booktitle = {Proceedings of International Symposium Local Strain and
Temperature Measurements in Non-Uniform Fields at Elevated
Temperatures},
pages = {253--262},
year = {1996},
editor = {J. Ziebs and J. Bressers and H. Frenz and D. R. Hayhurst
and H. Klingelh\"{o}ffer and S. Forest},
publisher = {Woodhead Pub},
address = {Camridge, UK},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/proceedings_2.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/proceedings_2.pdf},
abstract = { The aim of this paper is to present some advanced
techniques for monitoring thermocouples in the field of
fire engineering. In a fire test structural elements like
columns, beams or slabs, which are insulated by a fire
protection material, are subjected to mechanical as well as
thermal loadings. Whereas the mechanical loading is
constant the mean temperature in the furnace varies - due
to oil or gas burners - nearly monotonically as a function
of time within 90 minutes between room temperature and 1000
$^0C$. New technical standards as well as research purposes
require the monitoring of 30 to 60 thermocouples and more.
Although versatile computer based data acquisition systems
including necessary signal condition front-ends exist for
handling such an amount of data at any required rate, there
is a lack in representing these data during the test in
their geometrical context, i.e. as a property of the steel
surface. The method, which is proposed by the authors, uses
some recent developments in computer graphics and numerical
mathematics. By this method the monitoring of the
thermocouples is understood as a representation of a
time-dependent 1-dimensional field, which is based on
discrete measured values on a curved surface in 3-D space.
For this solution CAD and data visualization tools are
under testing, which are originally designed for other
purposes. In praxis a geometry file has to be created
before the fire test for the structural element under
consideration including the information on the position of
the thermocouples. This file is used as an appropriate
triangulation of the surface of the specimen. The
corresponding grid together with the actual temperature
readings are the basis for the real time visualization of
the temperature field by continuous colors or iso-lines.}
}
@incollection{Mardal.Zumbusch.Langtangen:2003,
author = {K.-A. Mardal and G. W. Zumbusch and H. P. Langtangen},
title = {Software Tools for Multigrid Methods},
booktitle = {Advanced Topics in Computational Partial Differential
Equations},
pages = {97--152},
publisher = {Springer},
address = {Berlin, Germany},
year = {2003},
editor = {H. P. Langtangen and A. Tveito},
volume = {33},
series = {Lecture Notes in Computational Science and Engineering},
chapter = {3},
annote = {unrefereed},
abstract = {}
}
@inproceedings{Schiekofer.Zumbusch:1998,
author = {T. Schiekofer and G. W. Zumbusch},
title = {Software Concepts of a Sparse Grid Finite Difference
Code},
booktitle = {Proceedings of the 14th GAMM-Seminar Kiel on Concepts of
Numerical Software},
year = {1998},
editor = {W. Hackbusch and G. Wittum},
series = {Notes on Numerical Fluid Mechanics},
publisher = {Vieweg},
address = {Wiesbaden, Germany},
pages = {11},
note = {submitted},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/kiel98.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/kiel98.pdf},
abstract = {Sparse grids provide an efficient representation of
discrete solutions of PDEs and are mainly based on specific
tensor products of one-dimensional hierarchical basis
functions. They easily allow for adaptive refinement and
compression. We present special finite difference operators
on sparse grids that possess nearly the same properties as
full grid operators. Using this approach, partial
differential equations of second order can be discretized
straightforwardly. We report on an adaptive finite
difference research code implementing this on sparse grids.
It is structured in an object oriented way. It is based on
hash storage techniques as a new data structure for sparse
grids. Due to the direct access of arbitrary data
traditional tree like structures can be avoided. The above
techniques are employed for the solution of parabolic
problems. We present a simple space-time discretization.
Furthermore a time-stepping procedure for the solution of
the Navier Stokes equations in 3D is presented. Here we
discretize by a projection method and obtain Poisson
problems and convection-diffusion problems.}
}
@techreport{Schutte.Dinand.Zumbusch.ea:1995,
author = {Ch. Sch\"{u}tte and M. Dinand and G. W. Zumbusch and R.
Brinkmann},
title = {Dynamics of {E}rbium-doped Waveguide Lasers: Modelling,
Reliable Simulation, and Comparison with Experiments},
institution = {Konrad-Zuse-Zentrum},
address = {Berlin, Germany},
year = {1995},
number = {SC-95-19},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/SC-95-19.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/SC-95-19.pdf},
abstract = { A theoretical investigation of the dynamic properties of
integrated optical Erbium doped waveguide lasers is
presented. It includes the construction of a physical model
and of numerical techniques which allow reliable
simulations of the dynamical behaviour of the laser signal
depending on essential parameters of the laser device and
on its external, time-dependent pump radiation. Therefore,
a physical theory is developed which describes the
propagation of light and its interaction with the active
substrate in the laser cavity. This is realized in two
steps. First, a fundamental model based on Maxwell's
equations and on rate equations for the transitions in the
active medium is constructed. Since this turns out to
prohibit reliable simulations, it is, in a second step,
reformulated via averaging in time and space which
suppresses the fluctuations on the fastest time scales but
represents them correctly. For this reduced model reliable
and efficient simulation techniques using adaptive control
schemes are designed and implemented. We apply the linear
implicit Euler discretization with extrapolation in time
and a multilevel quadrature scheme in space. Finally the
model is justified in comparison with experimental
observations in four cases of technological relevance. }
}
@inproceedings{Schweitzer.Zumbusch.Griebel:1999,
author = {M. A. Schweitzer and G. W. Zumbusch and M. Griebel},
title = {Parnass2: {A} Cluster of Dual-Processor {PC}s},
booktitle = {Proceedings of the 2nd Workshop Cluster-Computing,
Karlsruhe},
year = {1999},
editor = {W. Rehm and T. Ungerer},
number = {CSR-99-02},
series = {Chemnitzer Informatik Berichte},
organization = {TU Chemnitz},
address = {Chemnitz, Germany},
pages = {45--54},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/cluster99.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/schweitz/cluster99.pdf},
abstract = {We report on a cluster with 96 CPUs (48 Dual-Processor PCs
running Linux 2.1.127) at our department, called Parnass2.
The computing nodes are connected by a Myrinet, a Gigabit
per second switched LAN, and additionally by a
Fast-Ethernet. A comparison of different message passing
(MPI) libraries for the Myrinet is given. Furthermore, we
present the performance results of Parnass2 for some of the
parallel codes developed at our department, namely a sparse
grid code for PDEs, a particle code for molecular dynamics,
a finite difference code for the Navier-Stokes equation,
and a parallel adaptive finite element / finite difference
multi-grid code. We compare these results with the
performance of these codes running on a Cray T3E-1200, a
Cray T3E-600, a SGI Origin 200/2000 and Parnass
\cite{MGriebel:GWZumbusch:1997a}, a cluster of SGI O2
workstations.}
}
@article{Zumbusch:2001,
author = {G. Zumbusch},
title = {Load Balancing for Adaptively Refined Grids},
year = {2002},
journal = {Proc. Appl. Math. Mech.},
number = {1},
pages = {534--537},
note = {also as report 722 SFB 256, University Bonn},
annote = {unrefereed},
abstract = {The solution of partial differential equations on a
parallel computer is usually done by a data parallel
approach. The grid is partitioned and mapped onto the
processors. However, partitioning of unstructured meshes
and adaptively refined meshes in general is an $NP$-hard
problem and heuristics are needed. In this paper a
parallelisable and cheap method based on space-filling
curves is analysed. Quasi-optimal estimates are derived for
partitions of adaptively refined grids.},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/gamm01.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/gamm01.pdf}
}
@inproceedings{Zumbusch:1999,
author = {G. Zumbusch},
title = {Dynamic loadbalancing in a lightweight adaptive parallel
multigrid {PDE} solver},
booktitle = {Proceedings of 9th SIAM Conference on Parallel Processing
for Scientific Computing (PP 99), San Antonio, Texas},
year = {1999},
editor = {B. Hendrickson and K. Yelick and C. Bischof and I. Duff
and A. Edelman and G. Geist and M. Heath and M. Heroux and
C. Koelbel and R. Schrieber and R. Sinovec and M. Wheeler},
publisher = {SIAM},
address = {Philadelphia, PA},
series = {ISBN 0-89871-435-4},
pages = {10},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/pp99.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/pp99.pdf},
abstract = {A parallel version of an adaptive multigrid solver for
partial differential equations is considered. The main
emphasis is put on the load balancing algorithm to
distribute the adaptive grids at runtime. The background
and some applications of space-filling curves are
discussed, which are later on used as the basic principle
of the load-balancing heuristic. A tight integration of
space-filling curves as a memory addressing scheme into the
numerical algorithm is proposed. Some experiments on a
cluster of PCs demonstrates the parallel efficiency and
scalability of the approach. }
}
@book{Zumbusch:2003,
author = {G. Zumbusch},
title = {Parallel Multilevel Methods. Adaptive Mesh Refinement and
Loadbalancing},
publisher = {Teubner},
year = {2003},
amazon = {http://www.amazon.de/Parallel-Multilevel-Methods-Gerhard-Zumbusch/dp/3519004518/ref=sr_1_3/028-3794171-7733364?ie=UTF8&s=books&qid=1182883588&sr=1-3},
annote = {editorial},
abstract = {Main aspects of the efficient treatment of partial
differential equations are discretisation,
multilevel/multigrid solution and parallelisation. These
distinct topics are covered from the historical background
to modern developments. It is demonstrated how the
ingredients can be put together to give an adaptive and
parallel multilevel approach for the solution of elliptic
boundary value problems. 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. Finally,
space-filling curve enumeration for parallel load balancing
and processor cache efficiency are discussed.},
series = {Advances in Numerical Mathematics},
http = {http://cse.mathe.uni-jena.de/pub/zumbusch/teubner03.htm}
}
@techreport{Zumbusch:1991,
author = {G. W. Zumbusch},
title = {Adaptive parallele {M}ultilevel-{M}ethoden zur
{L}\"{o}sung elliptischer {R}andwertprobleme},
institution = {SFB 342, TU M\"{u}nchen},
address = {Munich, Germany},
year = {1991},
number = {342/19/91 A},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/p1.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/p1.pdf},
abstract = { Multilevel Methoden sind die zur Zeit effizientesten
Verfahren zur L\"{o}sung gro{\ss}er linearer
Gleichungssysteme, die aus der Diskretisierung elliptischer
Randwertprobleme entstehen. Der Aufwand, mit
Mehrgitterverfahren oder auch mit multilevel
vorkonditionierten Verfahren der konjugierten Gradienten
(CG) im symmetrisch positiv definiten Fall ein
Gleichungssystem bis auf Diskretisierungsgenauigkeit zu
l\"{o}sen, ist unter geeigneten Voraus- setzungen
proportional zur Zahl der Unbekannten oder nur um einen
logarithmischen Term h\"{o}her. ... \\ Die
Ausf\"{u}hrungsgeschwindigkeit kann durch adaptive
Verfeinerungs- techniken, die die Zahl der notwendigen
Unbekannten reduzieren, erh\"{o}ht werden. Dazu existieren
vollst\"{a}ndige Programmpakete, wie PLTMG [Bank] und
Kaskade [Leinen], [Deuflhard, Leinen, Yserentant], die die
Ordnung des eingebauten iterativen L\"{o}sers durch
Gitterverwaltung, Verfeinerung und Gittermanipulation nicht
verschlechtern. Die Ordnung des L\"{o}sungsverfahrens kann
nur durch parallele Ausf\"{u}hrung gesenkt werden. In
Hinblick auf sehr gro{\ss}e lineare Gleichungssysteme, wie
sie insbesondere auch durch Randwertprobleme in drei
Raumdimensionen entstehen, liegt es nahe, beide Techniken
zu verbinden. Bei der Lastverteilung adaptiv, also
dynamisch erzeugter Strukturen, k\"{o}nnen allerdings nicht
mehr alle Vorraussetzungen an die
Finiten-Elemente-R\"{a}ume und alle Algorithmen zur
Gittermanipulation vom sequentiellen Programm
\"{u}bernommen werden. Existierende Ans\"{a}tze, wie [Fox &
Otto], [Berger & Bokhari] und [Bastian] f\"{u}hren zu
Verfahren, deren Ordnung h\"{o}her als die des iterativen
L\"{o}sers ist, und nutzen die Multilevel-Struktur der
Gitter nicht aus. Ans\"{a}tze zur Parallelisierung von
Standard-Mehrgitterverfahren wie [Briggs, Hart, McCormick &
Quinlan] oder von adaptiven Mehrgitterverfahren wie
[Mierendorff] k\"{o}nnen in dieser Form nicht auf adaptive
Verfahren angewendet werden, obwohl sie f\"{u}r regul\"{a}r
verfeinerte Gitter optimale Ergebnisse liefern. ... \\ Wir
werden im folgenden Parallelrechner mit verteiltem Speicher
und Message-Passing-Kommunikation und Parallelrechner mit
gemeinsamem Speicher und Semaphor-Synchronisation
verwenden, um ein multilevel- vorkonditioniertes
CG-Verfahren so zu implementieren, da{\ss} die
Eigenschaften des sequentiellen Programms, soweit
m\"{o}glich, erhalten bleiben, und gleichzeitig eine
effiziente Parallelisierung erreicht wird. }
}
@mastersthesis{Zumbusch:1992,
author = {G. W. Zumbusch},
title = {Adaptive parallele {M}ultilevel-{M}ethoden zur
{L}\"{o}sung elliptischer {R}andwertprobleme},
school = {Mathematisches Institut, TU M\"{u}nchen},
address = {Munich, Germany},
year = {1992},
type = {Diplomarbeit},
annote = {thesis},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/d1.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/d1.pdf},
abstract = {Multilevel Methoden sind die zur Zeit effizientesten
Verfahren zur L\"{o}sung gro{\ss}er symmetrischer
schwachbesetzter linearer Gleichungssysteme, die aus der
Diskretisierung selbstadjungierter elliptischer
Randwertprobleme mit Finiten-Elementen entstehen. Im
folgenden wird die Parallelisierung eines von Bramble,
Pasciak und Xu vorgeschlagenen vorkonditionierten
Verfahrens der konjugierten Gradienten, eingebettet in ein
adaptives Finite-Elemente-Programm wie etwa Kaskade,
diskutiert. Dabei m\"{u}ssen zur effizienten Lastverteilung
zus\"{a}tzliche Forderungen an Triangulierungen,
Finite-Elemente-R\"{a}ume und
Gittermanipulationsalgorithmen gestellt werden. Es werden
Standardverfahren der Lastverteilung mit einem neuen
Aufteilungsverfahren, das auf einem statistischen Ansatz
beruht, verglichen. Mit einer hier vorgestellten gemischten
Strategie der Aufteilung von Gitterpunkten und Dreiecken
kann ein Gesamtverfahren von optimaler Ordnung erreicht
werden. Die experimentellen Ergebnisse auf einigen
Parallelrechnern zeigen eine hohe \"{U}bereinstimmung mit
einem hergeleiteten Kostenfunktional und eine ideale
Parallelisierbarkeit des Verfahrens. Unterschiede zu
anderen bekannten Multilevelverfahren werden in den
einzelnen Abschnitten herausgestellt.}
}
@techreport{Zumbusch:1993,
author = {G. W. Zumbusch},
title = {Symmetric Hierarchical Polynomials for the h-p-Version of
Finite Elements},
institution = {Konrad-Zuse-Zentrum},
address = {Berlin, Germany},
year = {1993},
number = {SC-93-32},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/SC-93-32.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/SC-93-32.pdf},
abstract = {Adaptive numerical methods using the $h$-$p$-version of
finite elements require special kinds of shape functions.
Desirable properties of them are symmetry, hierarchy and
simple coupling. In a first step it is demonstrated that
for standard polynomial vector spaces not all of these
features can be obtained simultaneously. However, this is
possible if these spaces are extended. Thus a new class of
polynomial shape functions is derived, which is well-suited
for the $p$- and $h$-$p$-version of finite elements on
unstructured simplices. The construction is completed by
minimizing the condition numbers of the arising finite
element matrices. The new shape functions are compared with
standard functions widely used in the literature.}
}
@techreport{Zumbusch:1994,
author = {G. W. Zumbusch},
title = {Visualizing Functions of the h-p-version of finite
elements},
institution = {Konrad-Zuse-Zentrum},
address = {Berlin, Germany},
year = {1994},
number = {TR-94-05},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/TR-94-05.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/TR-94-05.pdf},
abstract = {Results from finite-element-calculations are usually
visualized by colored surface- and contour-line-plots or
polygonal patches or simply displaced lines and grids. In
computer graphics however more advanced techniques like
texture-mapping and NURBS are well established and there
exist efficient algorithms and implementations. We show
that these techniques are not only easy to use, but form a
very natural and efficient approach for visualization of
higher order finite-element's solutions like in $p$- and
$h$-$p$-version. Texture-mapping is useful for displaying
vector-valued data, too.}
}
@inproceedings{Zumbusch:1995,
author = {G. W. Zumbusch},
title = {Adaptive h-p approximation procedures, graded meshes and
anisotropic refinement for Numerical Quadrature},
booktitle = {Proceedings of The First European Conference on Numerical
Mathematics and Advanced Applications, ENUMATH 95},
editor = {F. Brezzi and J. Periaux and R. Glowinski and R. Rannacher
and Yu. Kuznetsov},
year = {1995},
note = {accepted, also as report SC-95-24 ZIB, Berlin},
pages = {12},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/SC-95-24.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/SC-95-24.pdf},
abstract = {A set of adaptive algorithms for quadrature on
multi-dimensional polyhedral domains is presented. Several
kinds of refinement are discussed, covering local
improvement of quadrature order and splitting the domain
into sub-domains, resulting in isotropic, graded or
anisotropic grids. The algorithms are pure local heuristics
using no a priori knowledge or tuning parameters. This
approach was motivated by results from finite element
theory for optimal approximation results. Numerical
experiments show the optimality of pure local greedy-like
algorithms for singularity-type functions typically
occurring in finite element computations.}
}
@phdthesis{Zumbusch:2001*1,
author = {G. W. Zumbusch},
title = {Adaptive Parallel Multilevel Methods for Partial
Differential Equations},
school = {Universit\"at Bonn},
year = {2001},
type = {Habilitation},
annote = {thesis},
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. }
}
@phdthesis{Zumbusch:1995*1,
author = {G. W. Zumbusch},
title = {Simultanous h-p Adaptation in Multilevel Finite Elements},
school = {Fachbereich Mathematik und Informatik, FU Berlin},
year = {1995},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/diss.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/diss.pdf},
type = {Doktorarbeit},
annote = {thesis},
abstract = {An important tool in engineering is the finite element
method.... The combination of both methods, the $h$--$p$
version, supplies the pre-asym\-ptotic exponentially
convergent $p$--version continuously with properly adapted
grids. Hence it achieves the superior exponential
convergence asymptotically, too, instead of algebraic
convergence of its ingredients the $h$--version and the
$p$--version. Although the first theoretical results
claiming these convergence rates are quite classic, the
number of codes using the $h$--$p$--version of finite
elements is still rather limited. Reasons for that are the
pure implementational complexity and the details, in
conjunction with the rumor of engineers' low precision
requirements. But the major reason is the lack of a robust
(self-) adaptive control delivering the desired exponential
convergence. ... \\ In the this thesis we present some
steps towards an efficient implementation of the
theoretically known exponential convergence. As it turns
out, an efficient implementation requires additional
theoretical considerations, which play a major role there
as well. This includes both the fully automatic
$h$--$p$--version and as a subset the $p$--version on
suitable grids. We present some details concerning our
approach implementing an adaptive $h$--$p$--version based
on an adaptive multilevel $h$--version code named {\sc
Kaskade}. This software package uses unstructured grids of
triangles in two dimensions and tetrahedra in three
dimensions.}
}
@techreport{Zumbusch:1996,
author = {G. W. Zumbusch},
title = {Multigrid methods in {D}iffpack},
institution = {Sintef Applied Mathematics},
year = {1996},
number = {STF42 F96016},
address = {Oslo, Norway},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/mg.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/mg.pdf},
abstract = {The report gives an introduction to the multigrid
iterative solvers in Diffpack. It is meant as a tutorial
for the use of iterative solvers, preconditioners and
nonlinear solvers based on multigrid methods. The first
steps towards this efficient equation solvers are guided by
a couple of examples and exercises. Since multigrid is a
recipe to construct solution algorithms rather than
black-box algorithms itself, there is lots of freedom for
the user to tailor the actual solver. Reflecting this fact
there are lots of possibilities to use the appropriate
classes in Diffpack. Hence there is much advice needed not
to get started, but also to use the methods efficiently.
The exercises are meant to give some experience needed for
applications and questions not covered in this introductory
report.}
}
@techreport{Zumbusch:1996*1,
author = {G. W. Zumbusch},
title = {Overlapping Domain Decomposition Methods in {D}iffpack},
institution = {Sintef Applied Mathematics},
year = {1996},
address = {Oslo, Norway},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/ddo.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/ddo.pdf},
abstract = {The report gives an introduction to the overlapping domain
decomposition solvers of Schwarz type in Diffpack. It is
meant as a tutorial for the use of iterative solvers,
preconditioners and nonlinear solvers based on overlapping
Schwarz methods for partial differential equations.
Additive Schwarz methods serve as a standard method for
solving equation systems on parallel computers. They are
also useful for computations on complicated domains
constructed from simple domains where efficient equations
solvers are available. We provide an introduction to the
implementation and use of such methods in Diffpack. The
first steps are guided by a couple of examples and
exercises. We also want to refer to an accompanied tutorial
on multigrid methods in Diffpack, which methods and codes
are quite related.}
}
@techreport{Zumbusch:1996*2,
author = {G. W. Zumbusch},
title = {Schur Complement Domain Decomposition Methods in
{D}iffpack},
institution = {Sintef Applied Mathematics},
year = {1996},
address = {Oslo, Norway},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/ddn.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/ddn.pdf},
abstract = {The report gives an introduction to the Schur complement
domain decomposition solvers in Diffpack. It is meant as a
tutorial for the use of iterative solution methods of
equation systems arising in the discretization of partial
differential equations. Schur complement iterative solvers
are discussed, without and with preconditioners. They are
also referred to as iterative sub-structuring methods or
non-overlapping domain decomposition methods. Domain
decomposition methods are well suited and efficient
equation solvers on parallel computers. Schur complement
methods are also advantageous if there are abrupt changes
in the coefficients of the differential operator due to
abrupt changes in material properties. We provide an
introduction to the implementation and use of such methods
in Diffpack. We cover the basic Schur complement method
along with preconditioners of eigen-decomposition, BPS,
wire-basket and Neumann-Neumann type (with coarse grid).
The first steps are guided by a couple of examples and
exercises. We also want to refer to the related tutorials
on overlapping domain decomposition \cite{GWZumbusch:1996b}
and on multigrid \cite{GWZumbusch:1996a} methods in
Diffpack.}
}
@techreport{Zumbusch:1996*3,
author = {G. W. Zumbusch},
title = {Multigrid Applied to Different Partial Differential
Operators},
institution = {Sintef Applied Mathematics},
year = {1996},
address = {Oslo, Norway},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/mgOp.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/mgOp.pdf},
abstract = {The report is a continuation of an introductory report on
the multigrid iterative solvers in Diffpack. We consider
the solution of systems of equations as arising in linear
elasticity, non-symmetric equations as in
convection-diffusion problems, anisotropic operators and
bad conditioned equations as for jumping coefficients
problems. In the introductory report only the Laplacian and
smooth coefficients were treated. The first steps are
guided by a couple of examples and exercises.}
}
@techreport{Zumbusch:1996*4,
author = {G. W. Zumbusch},
title = {Multigrid on Different Grids},
institution = {Sintef Applied Mathematics},
year = {1996},
address = {Oslo, Norway},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/mgGrid.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/mgGrid.pdf},
abstract = {The report is a continuation of an introductory report on
the multigrid iterative solvers in Diffpack. We consider
the solution of equation systems arizing in the finite
element discretization of partial differential equations on
different grids. In the introductory report only uniform
partitions of the unit square and unit cube were treated.
Now we consider also multigrid for mapped elements, grids
generated by the meshing of super elements and unstructured
(and non nested) grids. The first steps are guided by a
couple of examples and exercises.}
}
@techreport{Zumbusch:1996*5,
author = {G. W. Zumbusch},
title = {Multigrid for Different Finite Difference Equations},
institution = {Sintef Applied Mathematics},
year = {1996},
address = {Oslo, Norway},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/mgFdOp.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/mgFdOp.pdf},
abstract = {The report is a continuation of an introductory report on
the multigrid iterative solvers for finite differences in
Diffpack. We consider the solution of partial differential
equations discretized by finite differences. We consider
varying coefficient and anisotropic operators and a variety
of strategies for the convection-diffusion equation and the
biharmonic equation. In the introductory report only the
Laplacian was treated. We also discuss different multigrid
restriction and prolongation operators arising in some
special multigrid versions. The first steps are guided by a
couple of examples.}
}
@techreport{Zumbusch:1996*6,
author = {G. W. Zumbusch},
title = {Multigrid Methods for Finite Differences},
institution = {Sintef Applied Mathematics},
year = {1996},
address = {Oslo, Norway},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/mgFdm.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf},
abstract = {The report serves as an alternative introductory report on
the multigrid iterative solvers in Diffpack using finite
differences instead of finite elements covered previously.
We consider the solution of elliptic partial differential
equations on different domains. We solve the resulting
linear equation systems with a multigrid iteration or a
Krylov iteration with a multigrid preconditioner. The
multigrid restriction and prolongation are also implemented
using finite ``difference'' type stencils. The first steps
are guided by a couple of examples and exercises.}
}
@book{Zumbusch:1996*7,
author = {G. W. Zumbusch},
title = {Simultanous h-p Adaptation in Multilevel Finite Elements},
publisher = {Shaker},
address = {Aachen, Germany},
year = {1996},
series = {ISBN 3-8265-1136-0},
annote = {editorial},
note = {Informatik, FU Berlin, 1995},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/diss.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/diss.pdf},
abstract = {An important tool in engineering is the finite element
method.... The combination of both methods, the $h$--$p$
version, supplies the pre-asym\-ptotic exponentially
convergent $p$--version continuously with properly adapted
grids. Hence it achieves the superior exponential
convergence asymptotically, too, instead of algebraic
convergence of its ingredients the $h$--version and the
$p$--version. Although the first theoretical results
claiming these convergence rates are quite classic, the
number of codes using the $h$--$p$--version of finite
elements is still rather limited. Reasons for that are the
pure implementational complexity and the details, in
conjunction with the rumor of engineers' low precision
requirements. But the major reason is the lack of a robust
(self-) adaptive control delivering the desired exponential
convergence. ... \\ In the this thesis we present some
steps towards an efficient implementation of the
theoretically known exponential convergence. As it turns
out, an efficient implementation requires additional
theoretical considerations, which play a major role there
as well. This includes both the fully automatic
$h$--$p$--version and as a subset the $p$--version on
suitable grids. We present some details concerning our
approach implementing an adaptive $h$--$p$--version based
on an adaptive multilevel $h$--version code named {\sc
Kaskade}. This software package uses unstructured grids of
triangles in two dimensions and tetrahedra in three
dimensions.}
}
@article{Zumbusch:1996*8,
author = {G. W. Zumbusch},
title = {Symmetric Hierarchical Polynomials and the Adaptive
h-p-Version},
journal = {Houston Journal of Mathematics},
address = {Houston, Texas},
year = {1996},
pages = {529--540},
editor = {A.V. Ilin and L. R. Scott},
note = {Proceedings of the Third International Conference on
Spectral and High Order Methods, ICOSAHOM'95, also as
report SC-95-18 ZIB, Berlin},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/SC-95-18.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/SC-95-18.pdf},
abstract = {The $h$-$p$-version of finite-elements delivers a
sub-exponential convergence in the energy norm. A step
towards a full adaptive implementation is taken in the
context of unstructured meshes of simplices with variable
order $p$ in space. Both assumptions lead to desirable
properties of shape functions like symmetry, $p$-hierarchy
and simple coupling of elements.\\ In a first step it is
demonstrated that for standard polynomial vector spaces on
simplices not all of these features can be obtained
simultaneously. However, this is possible if these spaces
are slightly extended or reduced. Thus a new class of
polynomial shape functions is derived, which are especially
well suited for three dimensional tetrahedra.\\ The
construction is completed by directly minimizing the
condition numbers of the arising preconditioned local
finite element matrices. The preconditioner is based on
two-step domain decomposition techniques using a multigrid
solver for the global linear problem $p=1$ and direct
solvers for local higher order problems.\\ Some numerical
results concerning an adaptive (feedback) version of
$h$-$p$ finite elements are presented.}
}
@inproceedings{Zumbusch:1999*1,
author = {G. W. Zumbusch},
title = {A Parallel Adaptive Multigrid Method},
booktitle = {Proceedings of the 15th GAMM-Seminar Kiel on Numerical
Techniques for Composite Materials},
year = {1999},
editor = {W. Hackbusch and S. Sauter},
series = {Notes on Numerical Fluid Mechanics},
publisher = {Vieweg},
address = {Wiesbaden, Germany},
note = {submitted},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/kiel99.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/kiel99.pdf},
abstract = {A parallel version of an adaptive multigrid solver for
elliptic partial differential equations is described. It
operates on a finite difference discretization on quad-tree
and oct-tree meshes, which are obtained by adaptive mesh
refinement. A fast parallel load balancing strategy for the
parallel multigrid equation solver is proposed that is
defined by a space-filling Hilbert curve and is applicable
to arbitrary shaped domains. Some numerical experiments
demonstrate the parallel efficiency and scalability of the
approach.}
}
@incollection{Zumbusch:2000,
author = {G. W. Zumbusch},
title = {A Sparse Grid {PDE} Solver},
booktitle = {Advances in Software Tools for Scientific Computing},
pages = {133--177},
publisher = {Springer},
address = {Berlin, Germany},
year = {2000},
editor = {H. P. Langtangen and A. M. Bruaset and E. Quak},
volume = {10},
series = {Lecture Notes in Computational Science and Engineering},
chapter = {4},
note = {(Proceedings SciTools '98)},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/scitools98.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/scitools98.pdf},
zmath = {http://www.emis.de/cgi-bin/zmen/ZMATH/en/zmath.html?first=1&maxdocs=20&type=html&an=943.65111&format=complete},
abstract = {Sparse grids are an efficient approximation method for
functions, especially in higher dimensions $d \ge 3$.
Compared to regular, uniform grids of a mesh parameter $h$,
which contain $h^{-d}$ points in $d$ dimensions, sparse
grids require only $h^{-1}|\log h|^{d-1}$ points due to a
truncated, tensor-product multi-scale basis representation.
The purpose of this paper is to survey some activities for
the solution of partial differential equations with methods
based sparse grid. Furthermore some aspects of sparse grids
are discussed such as adaptive grid refinement, parallel
computing, a space-time discretization scheme and the
structure of a code to implement these methods.}
}
@incollection{Zumbusch:2000*1,
author = {G. W. Zumbusch},
title = {Parallel Adaptively Refined Sparse Grids},
booktitle = {Multigrid Methods {VI}},
publisher = {Springer},
address = {Berlin, Germany},
year = {2000},
pages = {285--292},
editor = {E. Dick and K. Riemslagh and J. Vierendeels},
volume = {14},
note = {(Proceedings EMG 6)},
series = {Lecture Notes in Computational Science and Engineering},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/emg99.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/emg99.pdf},
abstract = {A parallel version of a finite difference discretization
of PDEs on sparse grids is proposed. Sparse grids or
hyperbolic crosspoints can be used for the efficient
representation of solutions of a boundary value problem,
especially in high dimensions, because the number of grid
points depends only weakly on the dimension. So far only
the `combination' technique for regular sparse grids was
available on parallel computers. However, the new approach
allows for arbitrary, adaptively refined sparse grids. The
efficient parallelisation is based on a dynamic
load-balancing approach with space-filling curves.}
}
@article{Zumbusch:2001*2,
author = {G. W. Zumbusch},
title = {On the Quality of Space-filling Curve Induced Partitions},
journal = {Z. Angew. Math. Mech.},
volume = 81,
pages = {25--28},
note = {Suppl. 1, also as report SFB 256, University Bonn, no.
674, 2000},
institution = {SFB 256, University Bonn},
year = {2001},
annote = {refereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/gamm00.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/gamm00.pdf},
abstract = {The solution of partial differential equations on a
parallel computer is usually done by a domain decomposition
approach. The mesh is split into several partitions mapped
onto the processors. However, partitioning of unstructured
meshes and adaptive refined meshes in general is an
$NP$-hard problem and heuristics are used. In this paper
space-filling curve based partition methods are analysed
and bounds for the quality of the partitions are given.
Furthermore estimates for parallel numerical algorithms
such as multigrid and wavelet methods on these partitions
are derived.}
}
@techreport{degrd.Zumbusch:1996,
author = {{\AA}. {\O}deg{\aa}rd and G. W. Zumbusch},
title = {GraphView - En Java og CGI-pakke for 3D Grafikk},
institution = {Sintef Applied Mathematics},
year = {1996},
address = {Oslo, Norway},
number = {9},
note = {STIM working notes},
annote = {unrefereed},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/stimwn0009.ps.gz},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/stimwn0009.pdf},
abstract = { I v{\aa}re dager er det viktig {\aa} profilere seg
gjennom Internett. En av de siste nyvinningene i denne
forbindelse, er muligheten for {\aa} legge inn
applikasjoner p{\aa} Web-sider, som gir mulighet for
interaksjon mellom Websiden og publikum. Dette har
tidligere v{\ae}rt mulig ved hjelp av CGI, The Common
Gateway Interface. Ved hjelp av denne teknologien kan man
kj{\o}re script, programmer etc. p{\aa} Server, det vil si
``hos seg selv,'' relativi til hvem som har laget Websiden.
Ved hjelp av slike verkt{\o}y, kan man hente ut
informasjon, lage datasett ved hjelp av programmer ol. Den
siste nyskapningen er imidlertid Java. Dette er et
programeringsspr{\aa}k spesialdesignet for {\aa} ha en
dynamisk interaksjon med publikum p{\aa} Web-sider. I
motsetning til CGI, vil et Java program kj{\o}re hos
klient, det vil si lokalt hos den som ser p{\aa} Web-siden.
Man er dermed ikke begrenset av at data skal sendes over
nettet. Dette gj{\o}r det for eksempel mulig {\aa} ha
mus-interaksjon p{\aa} Websider. Bakdelen er at man ikke
kan ha optimalisert kode, med hensyn p{\aa} arkitektur,
siden koden skal kunne kj{\o}res p{\aa} mange forskjellige
arkitekturer. Koden som kj{\o}res er
derfor``halv-kompilert'', slik at den m{\aa} tolkes endelig
av Web-brouseren hos klienten. Dette gj{\o}r at hastigheten
ikke er sv{\ae}rt stor, men absolutt brukbar. ... }
}
@inproceedings{Zumbusch:2006*1,
author = {G. Zumbusch},
title = {Data Parallel Iterators for Hierarchical Grid and Tree Algorithms},
booktitle = {Euro-Par 2006 Parallel Processing},
editor = {W. E. Nagel and W. V. Walter and W. Lehner},
volume = {4128},
pages = {625--634},
series = {LNCS},
year = {2006},
publisher = {Springer},
address = {Heidelberg},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/europar06.pdf},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/europar06.ps.gz},
http = {http://www.springerlink.com/content/978-3-540-37783-2/},
annote = {refereed},
note = {DOI 10.1007/11823285_65},
abstract = { The data parallel programming language construct of a ``for\-each'' loop
is proposed in the context of hierarchically nested arrays and unbalanced
k-ary trees used in high performance applications. In order perform an
initial evaluation, an implementation of an automatic parallelization
system for C++ programs is introduced, which consists of a preprocessor
and a matching library for distributed memory, shared memory and mixed
model parallelism. For a full compile time dependence analysis and a
tight distributed memory parallelization, some additional application
knowledge about alignment of arrays or indirect data access can be put
into the application's code data declarations. Results for a multigrid
and a fast multipole benchmark code illustrate the concept.
}
}
@inproceedings{Zumbusch:2007*1,
author = {G. Zumbusch},
title = {Data Dependence Analysis for Parallel Tree Codes},
booktitle = {Applied Parallel Computing. State of the Art in Scientific Computing (PARA 2006)},
editor = {B. K{\aa}gstr{\"o}m and E. Elmroth and J. Dongarra and J. Wasniewski},
volume = {4699},
pages = {890--899},
series = {LNCS},
year = {2007},
publisher = {Springer},
address = {Heidelberg},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/para06.pdf},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/para06.ps.gz},
http = {http://www.springerlink.com/content/},
annote = {refereed},
note = {DOI 10.1007/978-3-540-75755-9_106},
abstract = { Data dependence analysis for automatic parallelization of sequential
tree codes is discussed. Hierarchical numerical algorithms often
use tree data structures for unbalanced, adaptively and dynamically
created trees. Moreover, such codes often do not follow a strict
divide and conquer concept, but introduce some geometric
neighborhood data dependence in addition to parent-children
dependencies. Hence, recognition mechanisms and hierarchical
partition strategies of trees are not sufficient for automatic
parallelization. Generic tree traversal operators are proposed as a
domain specific language. Additional geometric data dependence can
be specified by code annotation. A code transformation system with
data dependence analysis is implemented, which generates several
versions of parallel codes for different programming models.
}
}
@inproceedings{Zumbusch:2008*1,
author = {G. Zumbusch},
title = {A Container-Iterator Parallel Programming Model},
booktitle = {Parallel Processing and Applied Mathematics},
editor = {R. Wyrzykowskii and J. Dongarra and K. Karczewski and J. Wasniewski (PPAM 2007)},
volume = {4967},
pages = {1130--1139},
series = {LNCS},
year = {2008},
publisher = {Springer},
address = {Heidelberg},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/ppam07.pdf},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/ppam07.ps.gz},
http = {http://www.springerlink.com/content/},
annote = {refereed},
note = {},
abstract = {There are several parallel programming models available for numerical
computations at different levels of expressibility and ease of
use. For the development of new domain specific programming models, a
splitting into a distributed data container and parallel data
iterators is proposed. Data distribution is implemented in application
specific libraries. Data iterators are directly analysed and compiled
automatically into parallel code. Target architectures of the
source-to-source translation include shared (pthreads, Cell SPE),
distributed memory (MPI) and hybrid programming styles. A model
applications for grid based hierarchical numerical methods and an
auto-parallelizing compiler are introduced.}
}
@article{Zumbusch:2009*1,
author = {G. Zumbusch},
title = {{F}inite {E}lement, {D}iscontinuous {G}alerkin, and {F}inite {D}ifference evolution schemes in spacetime},
journal = {Class. Quantum Grav.},
year = {2009},
volume = {26},
pages = {175011},
annote = {refereed},
ps = {http://arxiv.org/ps/0901.0851v2.ps},
pdf = {http://arxiv.org/pdf/0901.0851v2.pdf},
abstract = {Numerical schemes for the vacuum Einstein equations are developed. The Einstein equation in harmonic gauge is second order symmetric hyperbolic. It is discretized in four-dimensional spacetime by Finite Differences, Finite Elements, and Interior Penalty Discontinuous Galerkin methods, the latter related to Regge calculus. The schemes are split into space and time and new time-stepping schemes for wave equations are derived. The methods are evaluated for linear and non-linear test problems of the Apples-with-Apples collection.
}
}
@inproceedings{Zumbusch:2009*2,
author = {G. Zumbusch},
title = {Portable Multi-Level Parallel Programming for Cell processor, GPU, and Clusters},
booktitle = {Proc. Para08},
editor = {},
volume = {},
pages = {},
series = {LNCS},
year = {2009},
publisher = {Springer},
address = {Heidelberg},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/para08.pdf},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/para08.ps.gz},
http = {http://www.springerlink.com/content/},
annote = {refereed},
note = {},
abstract = {High performance computers offer lots of parallelism at different
levels of vectorization, thread parallelism, message-passing between
distributed memory architectures and even function off-loading by
hardware accelerators. Large scale numerical simulations often have
lots of parallelism, which may be difficult to express in a high level
programming language. A common abstract parallel programming style is
proposed, which can be translated automatically into parallel code for
one or a combination of common programming styles for different
parallel architectures.}
}
@misc{Zumbusch:2011*1,
author = {G. Zumbusch},
title = {Galerkin Schemes for General Relativity},
howpublished = {Poster, Advances and Challenges in Computational General Relativity (ACCGR), Brown University},
year = {2011},
annote = {unrefereed},
abstract = {Numerical schemes for Einstein's vacuum equation of general relativity are developed. The equation in harmonic gauge is discretized in space-time by Galerkin methods. A split into space and time leads to time-stepping schemes for second order wave equations. Finite Element and Discontinuous Galerkin are covered along with local mesh refinement in space-time.},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/accgr.pdf}
}
@misc{Zumbusch:2012*1,
author = {G. Zumbusch},
title = {Tuning a Finite Difference Stencil},
howpublished = {Poster, GPU Technology Conference 2012 (GTC), San Jose, CA},
year = {2012},
annote = {unrefereed},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/gtc12.pdf}
}
@inproceedings{Zumbusch:2012*2,
author = {G. Zumbusch},
title = {Tuning a Finite Difference Computation for Parallel Vector Processors},
booktitle = {2012 11th International Symposium on Parallel and Distributed Computing},
pages = {63--70},
year = {2012},
editor = {M. Bader and H.-J. Bungartz and D. Grigoras and M. Mehl and
R.-P. Mundani and R. Potolea},
series = {CPS},
publisher = {IEEE Press},
note = {DOI 10.1109/ISPDC.2012.17},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/ispdc12.pdf},
annote = {refereed},
abstract = {Current CPU and GPU architectures heavily use data and instruction
parallelism at different levels. Floating point operations are
organised in vector instructions of increasing vector length. For
reasons of performance it is mandatory to use the vector
instructions efficiently. Several ways of tuning a model problem
finite difference stencil computation are discussed. The combination
of vectorisation and an interleaved data layout, cache aware
algorithms, loop unrolling, parallelisation and parameter tuning
lead to optimised implementations at a level of 90\% peak
performance of the floating point pipelines on recent Intel Sandy
Bridge and AMD Bulldozer CPU cores, both with AVX vector
instructions as well as on Nvidia Fermi/ Kepler GPU architectures.
Furthermore, we present numbers for parallel multi-core/
multi-processor and multi-GPU configurations. They represent
regularly more than an order of speed up compared to a standard
implementation. The analysis may also explain deficiencies of
automatic vectorisation for linear data layout and serve as a
foundation of efficient implementations of more complex expressions.}
}
@inproceedings{Zumbusch:2013*1,
author = {G. Zumbusch},
title = {Vectorized Higher Order Finite Difference Kernels},
booktitle = {PARA 2012, State-of-the-Art in Scientific and Parallel Computing},
year = {2013},
editor = {P. Manninen and P. {\"O}ster},
volume = {7782},
series = {LNCS},
pages = {343--357},
publisher = {Springer},
address = {Heidelberg},
pdf = {http://cse.mathe.uni-jena.de/pub/zumbusch/para12.pdf},
ps = {http://cse.mathe.uni-jena.de/pub/zumbusch/para12.ps.gz},
annote = {refereed},
abstract = {Several highly optimized implementations of Finite Difference
schemes are discussed. The combination of vectorization and an
interleaved data layout, spatial and temporal loop tiling
algorithms, loop unrolling, and parameter tuning lead to efficient
computational kernels in one to three spatial dimensions, truncation
errors of order two to twelve, and isotropic and compact anisotropic
stencils. The kernels are implemented on and tuned for several
processor architectures like recent Intel Sandy Bridge, Ivy Bridge
and AMD Bulldozer CPU cores, all with AVX vector instructions as
well as Nvidia Kepler and Fermi and AMD Southern and Northern
Islands GPU architectures, as well as some older architectures for
comparison. The kernels are either based on a cache aware spatial
loop or on time-slicing to compute several time steps at
once. Furthermore, vector components can either be independent,
grouped in short vectors of SSE, AVX or GPU warp size or in larger
virtual vectors with explicit synchronization. The optimal choice of
the algorithm and its parameters depend both on the Finite
Difference stencil and on the processor architecture.}
}
@mastersthesis{Fleischer:2005,
author = {Stefan Fleischer},
title = {Numerik von {O}ptionsgesch\"{a}ften unter {Z}uhilfenahme der {K}ombinationstechnik},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2005},
type = {Diplomarbeit},
annote = {diplom},
pdf = {http://cse.mathe.uni-jena.de/pub/diplom/fleischer.pdf},
abstract = {Für europäische und amerikanische Optionen sowie
für zwei- und dreidimensionale Basket- Optionen
wird aus den Black-Scholes-Gleichungen der faire
Preis numerisch berechnet. Da in den meisten Fällen
keine geschlossenen Lösungsformeln exisiteren,
müssen die multivariaten parabolischen
Optionspreisaufgaben numerisch gelöst werden. Die
Eindeutigkeit der Gleichungen wird über die
Anfangs- und Randwertprobleme garantiert. Das
Berechnen von Näherungslösungen für diese
Differentialgleichungen geschieht dann durch
Diskretisierung der Gleichungen und durch das Lösen
der daraus resultierenden Gleichungssysteme. Die
Diskretisierung der parabolischen Gleichungen
erfolgt mit Finiten Differenzen auf stark
anisotropen, regulären und rechtwinkligen
Vollgittern. Bei amerikanischen Optionen wird das
Hindernisproblem über das
Projektions-Successive-Overrelaxation-Iterationsverfahren
numerisch gelöst. Die Basket-Optionen beruhen auf
Advektions-Diffusions-Reaktions-Gleichungen. Um den
numerischen Aufwand vertretbar zu halten, wird die
Kombinationstechnik basierend auf der
Dünngitter-Theorie eingesetzt. Der numerische
Aufwand bei steigender Problemdimension kann dabei
substantiell reduziert werden, ohne den
Informationsgehalt der Lösung wesentlich zu
verschlechtern. Damit kann entscheidend dem “Fluch
der Dimensionen“ entgegengewirkt und die
Komplexität der Algorithmen verbessert werden. Die
unterschiedlichen Lösungen auf den stark
anisotropen, regulären, rechtwinkligen und
gröberen Dünngittern werden miteinander geeignet
linear kombiniert und liefern, bis auf eine
logarithmische Dämpfung, denselben numerischen
Fehler wie die numerische Lösung auf einem feineren
isotropen Referenzvollgitter. Anhand von empirischen
Ergebnissen soll die vorgestellte Theorie bestätigt
werden.}
}
@mastersthesis{Langner:2007,
author = {Mathias Langner},
title = {Numerische {B}ehandlung von Contingent Claims},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2007},
type = {Diplomarbeit},
annote = {diplom},
pdf = {http://cse.mathe.uni-jena.de/pub/diplom/langner.pdf},
abstract = {Die vorliegende Arbeit befasst sich mit der
numerischen Evaluierung von zweidimensionalen
Finanzderivaten. Dafür werden zunächst partielle
Differentialgleichungen vorgestellt, die die
Entwicklung dieser Derivate beschreiben. Zur
Diskretisierung der Differentialgleichungen wird
eine finite Volumen Methode verwendet. Im Laufe der
Arbeit werden an dem Verfahren einige Modifikationen
vorgenommen. Zunächst wird auf Konvektionsdominanz
eingegangen. Um stabile Verfahren mit hoher
Konvergenz zu erhalten, werden zwei flux-Limiter
vorgestellt: Der van Leer und van Albada Limiter. Da
es sich bei beiden Limitern um TVD-Schemata handelt
und diese höchstens lineare Konvergenz erreichen,
werden modifizierte Limiter-Schemata entwickelt,
eine Kombination aus zentralen Differenzen und flux-
Limitern. Eine weitere Veränderung des Verfahrens
besteht in der Gittermodifikation. Gradierte Gitter
erlauben eine gezielte lokale Verfeinerung bei
gleich bleibender Anzahl an Diskretisierungspunkten.
Dünne Gitter verfolgen einen Ansatz mit
hierarchischen Basen und erlauben eine Berechnung
mit wesentlich weniger Speicheraufwand, aber einer
nur in geringem Maße schlechteren
Fehlerentwicklung. Ein Nachteil dünner Gitter
besteht in recht komplizierten Strukturen, die durch
die hierarchischen Basen entstehen. Eine Alternative
ist die Kombinationstechnik, die das Gesamtproblem
in mehrere Teilprobleme zerlegt, welche wiederum mit
bekannten Verfahren gelöst werden können. Die
Entwicklung des Fehlers ist äquivalent zu dem auf
dünnen Gittern.}
}
@mastersthesis{Franz:2008,
author = {Thomas Franz},
title = {Schnelle {M}ultipol-{M}ethode},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2008},
type = {Diplomarbeit},
annote = {diplom},
pdf = {http://cse.mathe.uni-jena.de/pub/diplom/franz.pdf},
abstract = {Diese Arbeit befaßt sich mit der Berechnung von
Wechselwirkungen zwischen geladenen Partikeln nach
dem Coulombschen Gesetz. Werden die Kräfte für jedes
Partikel einzeln berechnet, führt dies zu einem
Algorithmus der Ordnung O($N^2$). Dies ist in der
Praxis für große Partikelzahlen nicht praktikabel,
wenn nicht sogar ummöglich, zu berechnen. Deshalb
befassen sich vieleWissenschaftler mit Algorithmen,
die diese Berechnungen vereinfachen und stark
beschleunigen. Im Wesentlichen baut diese Arbeit
auf den Artikeln von L. Greengard, V. Rohklin,
R. Beatson und J. Carrier. Sie entwickelten die
Schnelle Multipol-Methode, einen Algorithmus, der
die Ordnung O($N$ log $N$) besitzt. Dadurch ist es
möglich, auch für eine sehr große Partikelanzahl die
Wechelwirkungen zu berechnen. Ziel der Diplomarbeit
war die Umsetzung des Multipol-Algorithmus in JAVA,
wobei hier besonders großer Wert auf die
Parallelisierung mit Hilfe von Threads gelegt
wurde.}
}
@phdthesis{Peuker:2009,
author = {Frank Peuker},
title = {Simplicial Methods for Solving Selected Problems in General Relativity Numerically. Regge Calculus and the Finite-Element Method},
year = {2009},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
annote = {diplom},
pdf = {http://cse.mathe.uni-jena.de/pub/diss/peuker.pdf}
}
@mastersthesis{Fritzsche:2009,
author = {Marcus Fritzsche},
title = {Parallel Numerical Simulation of {N}avier-{S}tokes-Equations on {GPU}s},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2009},
type = {Diplomarbeit},
annote = {diplom},
pdf = {http://cse.mathe.uni-jena.de/pub/diplom/fritzsche.pdf},
abstract = {This diploma thesis is about solving the Navier-Stokes equations numerically and emphases parallelizing on GPUs. The work has shown that the computation of the Navier-Stokes equations can be accelerated significantly by using GPUs. The speedup factor is at least 20 which has been shown by the simulations. It is the result of parallelizing the SOR-method which is used to solve the discretization of the poisson pressure equation.}
}
@mastersthesis{Riffert:2009,
author = {Till W. Riffert},
title = {Modellierung und {A}pproximation von {Z}ufallsfeldern mit {M}ethoden der hierarchischen {M}atrizen},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2009},
type = {Diplomarbeit},
annote = {diplom}
}
@mastersthesis{Reibiger:2009,
author = {Christian Reibiger},
title = {{L}\"{o}sung elliptischer {R}andwertprobleme mit {H}ilfe der
{CUDA} {T}echnologie},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2009},
type = {Diplomarbeit},
annote = {diplom},
pdf = {http://cse.mathe.uni-jena.de/pub/diplom/reibiger.pdf},
abstract = {In dieser Arbeit wurde zunächst ein
Krylov-Unterraum-Verfahren zur Lösung linearer
algebraischer Gleichungssysteme für spezielle
dünnbesetzte Matrizen (hier als 27-Diagonal-Matrizen
bezeichnet) implementiert. Solche Matrizen erhält
man insbesondere bei der Diskretisierung von
RWA. Durch die Auslagerung großer Teile des
Programms auf die vielen parallel arbeitenden
Prozessoren einer Grafikkarte konnte bereits eine
beachtliche Rechenleistung erzielt werden. Danach
wurde durch das Implementieren und Testen von
einigen Varianten der Gebietszerlegungs- und
Mehrgitterverfahren eine Beschleunigung des Lösers
erreicht. Es ist zu erwarten, dass bei einer
weiterentwickelten Grafikkarte eine noch stärkere
Beschleunigung erreicht werden kann, weil dann
vermutlich größere Teilgebiete in den
Vorkonditionierern verwendt werden können.
Abschließend wurde die theoretische Grundlage
gebildet, um das entwickelte Programm zur Lösung
einer nichtlinearen RWA aus der Astrophysik
anzuwenden.}
}
@mastersthesis{Boos:2009,
author = {Anja Boos},
title = {{S}churkomplement-{V}orkonditionierer f\"{u}r
pseudospektrale {D}iskretisierungen},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2009},
type = {Diplomarbeit},
annote = {diplom},
pdf = {http://cse.mathe.uni-jena.de/pub/diplom/boos.pdf},
abstract = {Ziel der vorliegenden Arbeit ist die Überprüfung
der Vor- und Nachteile von Spektralmethoden. In der
Praxis werden häufig Finite-Elemente-Methoden
genutzt, die stückweise lineare Funktionen
verwenden. Der Vorteil daran liegt auf der Hand.
Die Steifigkeitsmatrix ist dünnbesetzt und kann
somit schnell gelöst werden. In der vorliegenden
Arbeit wird eine partielle Differentialgleichung auf
einem gegebenen Gebiet mit Dirichlet- oder
Cauchy-Randbedingungen numerisch mittels
Spektralmethoden gelöst. Die Umsetzung erfolgt in
der Programmiersprache C/C++. Im Mittelpunkt stehen
die Möglichkeiten den Zeit- und Rechenaufwand zu
optimieren. Dafür werden verschiedene Ansätze
miteinander zu verglichen. Dies können neben
Gebietszerlegungsmethoden auch verschiedene
Vorkonditionierer für die jeweiligen Teilprobleme
sein.}
}
@mastersthesis{schuhmacher:2009,
author = {Kathleen Schuhmacher},
title = {Ausgew\"{a}hlte numerische {P}robleme im gymnasialen {M}athematikuntericht},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2009},
type = {wiss. {H}ausarbeit {S}taatsexamen},
annote = {diplom}
}
@mastersthesis{Buechse:2010,
author = {Katharina B\"{u}chse},
title = {{D}atenstrukturanalyse für die adaptive {F}inite-{E}lemente-{M}ethode},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2010},
type = {Diplomarbeit, {STIFT}-{P}reis 2010},
annote = {diplom},
abstract = {In der vorliegenden Arbeit werden Herausforderungen diskutiert, die sich durch die Verwendung
unterschiedlicher Speichermedien, hier in erster Linie Hauptspeicher und Festplatte,
beim Lösen von adaptiven finiten Elementen ergeben. Es werden Datenstrukturen
auf ihre Eignung für diese konkrete mathematische Problemstellung analysiert und Mittel
bereitgestellt, welche ohne viel Aktualisierungsaufwand die Arbeit mit den benötigten
Daten ermöglichen. Der eigentliche Rechenablauf wird dahingehend angepasst, dass der
Datenaustausch zwischen Hauptspeicher und Festplatte, der einen (Performance-)Engpass
darstellt, so stark wie möglich eingeschränkt wird.}
}
@mastersthesis{Geppert:2010,
author = {Gernot Geppert},
title = {{N}umerische {M}ethoden zur {L}\"{o}sung nichtlinearer, schlecht gestellter
{3D}-{T}omographie-{P}robleme aus der {A}tmosph\"{a}renfernerkundung},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2010},
type = {Diplomarbeit},
annote = {diplom}
}
@mastersthesis{Radszuwill:2011,
author = {Sven Radszuwill},
title = {Effiziente {L}\"{o}sung der {P}oissongleichung mit {M}ulticore-{A}nwendungen},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2011},
type = {Bachelorarbeit},
annote = {diplom}
}
@mastersthesis{Henze:2011,
author = {Richard Henze},
title = {Effizientes {L}\"{o}sen der {W}\"{a}rmeleitungsgleichung auf einer {GPU}},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2011},
type = {Bachelorarbeit},
annote = {diplom}
}
@mastersthesis{Wickles:2011,
author = {Roland Wickles},
title = {Konvergenzuntersuchungen bei {L}\"{o}sungen der {W}ellengleichung mittels finiter {D}ifferenzen},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2011},
type = {Bachelorarbeit},
annote = {diplom}
}
@mastersthesis{Gericke:2013,
author = {Kevin Gericke},
title = {Einwicklung eines numerischen {M}olek\"{u}lnynamikcodes f\"{u}r {K}ettenmolek\"{u}le},
school = {Universit\"{a}t Jena},
year = {2013},
type = {wiss. {H}ausarbeit {S}taatsexamen},
annote = {diplom}
}
@mastersthesis{Feierabend:2013,
author = {J\"{o}rg Feierabend},
title = {Spektralverfahren auf {GPU}s},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2013},
type = {Diplomarbeit},
annote = {diplom}
}
@mastersthesis{Prager:2014,
author = {Ken Prager},
title = {Datenkompression in {CUDA} mittels {H}ierarchischer {B}asen},
school = {Institut f\"{u}r Angewandte Mathematik, Universit\"{a}t Jena},
year = {2014},
type = {Bakkalaureatsarbeit},
annote = {diplom}
}
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