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2006 | Buch

Parallel Programming Models Applicable to Cluster Computing and Beyond Kapitel lesen Erstes Kapitel lesen

Numerical Solution of Partial Differential Equations on Parallel Computers

herausgegeben von: Are Magnus Bruaset, Aslak Tveito

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Computational Science and Engineering

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Since the dawn of computing, the quest for a better understanding of Nature has been a driving force for technological development. Groundbreaking achievements by great scientists have paved the way from the abacus to the supercomputing power of today. When trying to replicate Nature in the computer’s silicon test tube, there is need for precise and computable process descriptions. The scienti?c ?elds of Ma- ematics and Physics provide a powerful vehicle for such descriptions in terms of Partial Differential Equations (PDEs). Formulated as such equations, physical laws can become subject to computational and analytical studies. In the computational setting, the equations can be discreti ed for ef?cient solution on a computer, leading to valuable tools for simulation of natural and man-made processes. Numerical so- tion of PDE-based mathematical models has been an important research topic over centuries, and will remain so for centuries to come. In the context of computer-based simulations, the quality of the computed results is directly connected to the model’s complexity and the number of data points used for the computations. Therefore, computational scientists tend to ?ll even the largest and most powerful computers they can get access to, either by increasing the si e of the data sets, or by introducing new model terms that make the simulations more realistic, or a combination of both. Today, many important simulation problems can not be solved by one single computer, but calls for parallel computing.

Inhaltsverzeichnis

Frontmatter

Parallel Computing

Frontmatter
1. Parallel Programming Models Applicable to Cluster Computing and Beyond
Summary
This chapter centers mainly on successful programming models that map algorithms and simulations to computational resources used in high-performance computing. These resources range from group-based or departmental clusters to high-end resources available at the handful of supercomputer centers around the world. Also covered are newer programming models that may change the way we program high-performance parallel computers.
Ricky A. Kendall, Masha Sosonkina, William D. Gropp, Robert W. Numrich, Thomas Sterling
2. Partitioning and Dynamic Load Balancing for the Numerical Solution of Partial Differential Equations
Summary
In parallel simulations, partitioning and load-balancing algorithms compute the distribution of application data and work to processors. The effectiveness of this distribution greatly influences the performance of a parallel simulation. Decompositions that balance processor loads while keeping the application’s communication costs low are preferred. Although a wide variety of partitioning and load-balancing algorithms have been developed, their effectiveness depends on the characteristics of the application using them. In this chapter, we review several partitioning algorithms, along with their strengths and weaknesses for various PDE applications. We also discuss current efforts toward improving partitioning algorithms for future applications and architectures.
James D. Teresco, Karen D. Devine, Joseph E. Flaherty
3. Graphics Processor Units: New Prospects for Parallel Computing
Summary
This chapter provides an introduction to the use of Graphics Processor Units (GPUs) as parallel computing devices. It describes the architecture, the available functionality and the programming model. Simple examples and references to freely available tools and resources motivate the reader to explore these new possibilities. An overview of the different applications of GPUs demonstrates their wide applicability, yet also highlights limitations of their use. Finally, a glimpse into the future of GPUs sketches the growing prospects of these inexpensive parallel computing devices.
Martin Rumpf, Robert Strzodka

Parallel Algorithms

Frontmatter
4. Domain Decomposition Techniques
Summary
We introduce some parallel domain decomposition preconditioners for iterative solution of sparse linear systems like those arising from the approximation of partial differential equations by finite elements or finite volumes. We first give an overview of algebraic domain decomposition techniques. We then introduce a preconditioner based on a multilevel approximate Schur complement system. Then we present a Schwarz-based preconditioner augmented by an algebraic coarse correction operator. Being the definition of a coarse grid a difficult task on unstructured meshes, we propose a general framework to build a coarse operator by using an agglomeration procedure that operates directly on the matrix entries. Numerical results are presented aimed at assessing and comparing the effectiveness of the two methodologies. The main application will concern computational fluid dynamics (CFD), and in particular the simulation of compressible flow around aeronautical configurations.
Luca Formaggia, Marzio Sala, Fausto Saleri
5. Parallel Geometric Multigrid
Summary
Multigrid methods are among the fastest numerical algorithms for the solution of large sparse systems of linear equations. While these algorithms exhibit asymptotically optimal computational complexity, their efficient parallelisation is hampered by the poor computation-to-communication ratio on the coarse grids. Our contribution discusses parallelisation techniques for geometric multigrid methods. It covers both theoretical approaches as well as practical implementation issues that may guide code development.
Frank Hülsemann, Markus Kowarschik, Marcus Mohr, Ulrich Rüde
6. Parallel Algebraic Multigrid Methods — High Performance Preconditioners
Summary
The development of high performance, massively parallel computers and the increasing demands of computationally challenging applications have necessitated the development of scalable solvers and preconditioners. One of the most effective ways to achieve scalability is the use of multigrid or multilevel techniques. Algebraic multigrid (AMG) is a very efficient algorithm for solving large problems on unstructured grids. While much of it can be parallelized in a straightforward way, some components of the classical algorithm, particularly the coarsening process and some of the most efficient smoothers, are highly sequential, and require new parallel approaches. This chapter presents the basic principles of AMG and gives an overview of various parallel implementations of AMG, including descriptions of parallel coarsening schemes and smoothers, some numerical results as well as references to existing software packages.
Ulrike Meier Yang
7. Parallel Mesh Generation
Summary
Parallel mesh generation is a relatively new research area between the boundaries of two scientific computing disciplines: computational geometry and parallel computing. In this chapter we present a survey of parallel unstructured mesh generation methods. Parallel mesh generation methods decompose the original mesh generation problem into smaller sub-problems which are meshed in parallel. We organize the parallel mesh generation methods in terms of two basic attributes: (1) the sequential technique used for meshing the individual subproblems and (2) the degree of coupling between the subproblems. This survey shows that without compromising in the stability of parallel mesh generation methods it is possible to develop parallel meshing software using off-the-shelf sequential meshing codes. However, more research is required for the efficient use of the state-of-the-art codes which can scale from emerging chip multiprocessors (CMPs) to clusters built from CMPs.
Nikos Chrisochoides

Parallel Software Tools

Frontmatter
8. The Design and Implementation of hypre, a Library of Parallel High Performance Preconditioners
Summary
The hypre software library provides high performance preconditioners and solvers for the solution of large, sparse linear systems on massively parallel computers. One of its attractive features is the provision of conceptual interfaces. These interfaces give application users a more natural means for describing their linear systems, and provide access to methods such as geometric multigrid which require additional information beyond just the matrix. This chapter discusses the design of the conceptual interfaces in hypre and illustrates their use with various examples. We discuss the data structures and parallel implementation of these interfaces. A brief overview of the solvers and preconditioners available through the interfaces is also given.
Robert D. Falgout, Jim E. Jones, Ulrike Meier Yang
9. Parallelizing PDE Solvers Using the Python Programming Language
Summary
This chapter aims to answer the following question: Can the high-level programming language Python be used to develop sufficiently efficient parallel solvers for partial differential equations (PDEs)? We divide our investigation into two aspects, namely (1) the achievable performance of a parallel program that extensively uses Python programming and its associated data structures, and (2) the Python implementation of generic software modules for parallelizing existing serial PDE solvers. First of all, numerical computations need to be based on the special array data structure of the Numerical Python package, either in pure Python or in mixed-language Python-C/C++ or Python/Fortran setting. To enable high-performance message passing in parallel Python software, we use the small add-on package pypar, which provides efficient Python wrappers to a subset of MPI routines. Using concrete numerical examples of solving wave-type equations, we will show that a mixed Python-C/Fortran implementation is able to provide fully comparable computational speed in comparison with a pure C or Fortran implementation. In particular, a serial legacy Fortran 77 code has been parallelized in a relatively straightforward manner and the resulting parallel Python program has a clean and simple structure.
Xing Cai, Hans Petter Langtangen
10. Parallel PDE-Based Simulations Using the Common Component Architecture
Summary
The complexity of parallel PDE-based simulations continues to increase as multimodel, multiphysics, and multi-institutional projects become widespread. A goal of component- based software engineering in such large-scale simulations is to help manage this complexity by enabling better interoperability among various codes that have been independently developed by different groups. The Common Component Architecture (CCA) Forum is defining a component architecture specification to address the challenges of high-performance scientific computing. In addition, several execution frameworks, supporting infrastructure, and general-purpose components are being developed. Furthermore, this group is collaborating with others in the high-performance computing community to design suites of domain-specific component interface specifications and underlying implementations.
This chapter discusses recent work on leveraging these CCA efforts in parallel PDE-based simulations involving accelerator design, climate modeling, combustion, and accidental fires and explosions. We explain how component technology helps to address the different challenges posed by each of these applications, and we highlight how component interfaces built on existing parallel toolkits facilitate the reuse of software for parallel mesh manipulation, discretization, linear algebra, integration, optimization, and parallel data redistribution. We also present performance data to demonstrate the suitability of this approach, and we discuss strategies for applying component technologies to both new and existing applications.
Lois Curfman McInnes, Benjamin A. Allan, Robert Armstrong, Steven J. Benson, David E. Bernholdt, Tamara L. Dahlgren, Lori Freitag Diachin, Manojkumar Krishnan, James A. Kohl, J. Walter Larson, Sophia Lefantzi, Jarek Nieplocha, Boyana Norris, Steven G. Parker, Jaideep Ray, Shujia Zhou

Parallel Applications

Frontmatter
11. Full-Scale Simulation of Cardiac Electrophysiology on Parallel Computers
Summary
In this chapter, we will present an advanced parallel electro-cardiac simulator, which employs anisotropic and inhomogeneous conductivities in realistic three-dimensional geometries for modeling both the heart and the torso. Since partial differential equations (PDEs) constitute the main part of the mathematical model, this chapter thus demonstrates a concrete example of solving PDEs on parallel computers. It will be shown that good overall parallel performance relies on at least two factors. First, the serial numerical strategy must find a parallel substitute that is scalable with respect to both convergence and work amount. Second, care must be taken to avoid unnecessary duplicated local computations while maintaining an acceptable level of load balance.
Xing Cai, Glenn Terje Lines
12. Developing a Geodynamics Simulator with PETSc
Summary
Most high-performance simulation codes are not written from scratch but begin as desktop experiments and are subsequently migrated to a scalable, parallel paradigm. This transition can be painful, however, because the restructuring required in conversion forces most authors to abandon their serial code and begin an entirely new parallel code. Starting a parallel code from scratch has many disadvantages, such as the loss of the original test suite and the introduction of new bugs. We present a disciplined, incremental approach to parallelization of existing scientific code using the PETSc framework. In addition to the parallelization, it allows the addition of more physics (in this case strong nonlinearities) without the user having to program anything beyond the new pieces of discretization code. Our approach permits users to easily develop and experiment on the desktop with the same code that scales efficiently to large clusters with excellent parallel performance. As a motivating example, we present work integrating PETSc into an existing plate tectonic subduction code.
Matthew G. Knepley, Richard F. Katz, Barry Smith
13. Parallel Lattice Boltzmann Methods for CFD Applications
Summary
The lattice Boltzmann method (LBM) has evolved to a promising alternative to the well-established methods based on finite elements/volumes for computational fluid dynamics simulations. Ease of implementation, extensibility, and computational efficiency are the major reasons for LBM’s growing field of application and increasing popularity. In this paper we give a brief introduction to the involved theory and equations for LBM, present various techniques to increase the single-CPU performance, outline the parallelization of a standard LBM implementation, and show performance results. In order to demonstrate the straightforward extensibility of LBM, we then focus on an application in material science involving fluid flows with free surfaces. We discuss the required extensions to handle this complex scenario, and the impact on the parallelization technique.
Carolin Körner, Thomas Pohl, Ulrich Rüde, Nils Thürey, Thomas Zeiser
Backmatter
Metadaten
Titel
Numerical Solution of Partial Differential Equations on Parallel Computers
herausgegeben von
Are Magnus Bruaset
Aslak Tveito
Copyright-Jahr
2006
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-540-31619-0
Print ISBN
978-3-540-29076-6
DOI
https://doi.org/10.1007/3-540-31619-1

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