Bruaset a tveito a (eds) numerical solution of partial differential equations on parallel computers ( 2006)(491s) MNd

Editors Timothy J Barth Michael Griebel David E Keyes Risto M Nieminen Dirk Roose Tamar Schlick Are Magnus Bruaset Aslak Tveito (Eds.) Numerical Solution of Partial Differential Equations on Parallel Computers With 201 Figures and 42 Tables ABC Editors Are Magnus Bruaset Aslak Tveito Simula Research Laboratory P.O Box 134 1325 Lysaker, Fornebu, Norway email: arem@simula.no aslak@simula.no The second editor of this book has received financial support from the NFF – Norsk faglitterær forfatter- og oversetterforening Library of Congress Control Number: 2005934453 Mathematics Subject Classification: Primary: 65M06, 65M50, 65M55, 65M60, 65Y05, 65Y10 Secondary: 65N06, 65N30, 65N50, 65N55, 65F10, 65F50 ISBN-10 3-540-29076-1 Springer Berlin Heidelberg New York ISBN-13 978-3-540-29076-6 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11548843 46/TechBooks 543210 Preface 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 scientific fields of Mathematics 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 efficient solution on a computer, leading to valuable tools for simulation of natural and man-made processes Numerical solution 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 fill 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 Whether being a dedicated multi-processor supercomputer or a loosely coupled cluster of office workstations, the concept of parallelism offers increased data storage and increased computing power In theory, one gets access to the grand total of the resources offered by the individual units that make up the multi-processor environment In practice, things are more complicated, and the need for data communication between the different computational units consumes parts of the theoretical gain of power Summing up the bits and pieces that go into a large-scale parallel computation, there are aspects of hardware, system software, communication protocols, memory management, and solution algorithms that have to be addressed However, over time efficient ways of addressing these issues have emerged, better software tools have become available, and the cost of hardware has fallen considerably Today, computational clusters made from commodity parts can be set up within the budget of a VI Preface typical research department, either as a turn-key solution or as a do-it-yourself project Supercomputing has become affordable and accessible About this book This book addresses the major topics involved in numerical simulations on parallel computers, where the underlying mathematical models are formulated in terms of PDEs Most of the chapters dealing with the technological components of parallel computing are written in a survey style and will provide a comprehensive, but still readable, introduction for students and researchers Other chapters are more specialized, for instance focusing on a specific application that can demonstrate practical problems and solutions associated with parallel computations As editors we are proud to put together a volume of high-quality and useful contributions, written by internationally acknowledged experts on high-performance computing The first part of the book addresses fundamental parts of parallel computing in terms of hardware and system software These issues are vital to all types of parallel computing, not only in the context of numerical solution of PDEs To start with, Ricky Kendall and co-authors discuss the programming models that are most commonly used for parallel applications, in environments ranging from a simple departmental cluster of workstations to some of the most powerful computers available today Their discussion covers models for message passing and shared memory programming, as well as some future programming models In a closely related chapter, Jim Teresco et al look at how data should be partitioned between the processors in a parallel computing environment, such that the computational resources are utilized as efficient as possible In a similar spirit, the contribution by Martin Rumpf and Robert Strzodka also aims at improved utilization of the available computational resources However, their approach is somewhat unconventional, looking at ways to benefit from the considerable power available in graphics processors, not only for visualization purposes but also for numerical PDE solvers Given the low cost and easy access of such commodity processors, one might imagine future cluster solutions with really impressive price-performance ratios Once the computational infrastructure is in place, one should concentrate on how the PDE problems can be solved in an efficient manner This is the topic of the second part of the book, which is dedicated to parallel algorithms that are vital to numerical PDE solution Luca Formaggia and co-authors present parallel domain decomposition methods In particular, they give an overview of algebraic domain decomposition techniques, and introduce sophisticated preconditioners based on a multilevel approximative Schur complement system and a Schwarz-type decomposition, respectively As Schwarz-type methods call for a coarse level correction, the paper also proposes a strategy for constructing coarse operators directly from the algebraic problem formulation, thereby handling unstructured meshes for which a coarse grid can be difficult to define Complementing this multilevel approach, Frank Hăulsemann et al discuss how another important family of very efficient PDE solvers, geometric multigrid, can be implemented on parallel computers Like domain decomposition methods, multigrid algorithms are potentially capable of being order-optimal such Preface VII that the solution time scales linearly with the number of unknowns However, this paper demonstrates that in order to maintain high computational performance the construction of a parallel multigrid solver is certainly problem-dependent In the following chapter, Ulrike Meier Yang addresses parallel algebraic multigrid methods In contrast to the geometric multigrid variants, these algorithms work only on the algebraic system arising from the discretization of the PDE, rather than on a multiresolution discretization of the computational domain Ending the section on parallel algorithms, Nikos Chrisochoides surveys methods for parallel mesh generation Meshing procedures are an important part of the discretization of a PDE, either used as a preprocessing step prior to the solution phase, or in case of a changing geometry, as repeated steps in course of the simulation This contribution concludes that it is possible to develop parallel meshing software using off-the-shelf sequential codes as building blocks without sacrificing the quality of the constructed mesh Making advanced algorithms work in practice calls for development of sophisticated software This is especially important in the context of parallel computing, as the complexity of the software development tends to be significantly higher than for its sequential counterparts For this reason, it is desirable to have access to a wide range of software tools that can help make parallel computing accessible One way of addressing this need is to supply high-quality software libraries that provide parallel computing power to the application developer, straight out of the box The hypre library presented by Robert D Falgout et al does exactly this by offering parallel high-performance preconditioners Their paper concentrates on the conceptual interfaces in this package, how these are implemented for parallel computers, and how they are used in applications As an alternative, or complement, to the library approach, one might look for programming languages that tries to ease the process of parallel coding In general, this is a quite open issue, but Xing Cai and Hans Petter Langtangen contribute to this discussion by considering whether the high-level language Python can be used to develop efficient parallel PDE solvers They address this topic from two different angles, looking at the performance of parallel PDE solvers mainly based on Python code and native data structures, and through the use of Python to parallelize existing sequential PDE solvers written in a compiled language like FORTRAN, C or C++ The latter approach also opens for the possibility of combining different codes in order to address a multi-model or multiphysics problem This is exactly the concern of Lois Curfman McInnes and her co-authors when they discuss the use of the Common Component Architecture (CCA) for parallel PDE-based simulations Their paper gives an introduction to CCA and highlights several parallel applications for which this component technology is used, ranging from climate modeling to simulation of accidental fires and explosions To communicate experiences gained from work on some complete simulators, selected parallel applications are discussed in the latter part of the book Xing Cai and Glenn Terje Lines present work on a full-scale parallel simulation of the electrophysiology of the human heart This is a computationally challenging problem, which due to a multiscale nature requires a large amount of unknowns that have to be resolved for small time steps It can be argued that full-scale simulations of this problem can not be done without parallel computers Another challenging geody- VIII Preface namics problem, modeling the magma genesis in subduction zones, is discussed by Matthew G Knepley et al They have ported an existing geodynamics code to use PETSc, thereby making it parallel and extending its functionality Simulations performed with the resulting application confirms physical observations of the thermal properties in subduction zones, which until recently were not predicted by computations Finally, in the last chapter of the book, Carolin Kăorner et al present parallel Lattice Boltzmann Methods (LBMs) that are applicable to problems in Computational Fluid Dynamics Although not being a PDE-based model, the LBM approach can be an attractive alternative, especially in terms of computational efficiency The power of the method is demonstrated through computation of 3D free surface flow, as in the interaction and growing of gas bubbles in a melt Acknowledgements We wish to thank all the chapter authors, who have written very informative and thorough contributions that we think will serve the computational community well Their enthusiasm has been crucial for the quality of the resulting book Moreover, we wish to express our gratitude to all reviewers, who have put time and energy into this project Their expert advice on the individual papers has been useful to editors and contributors alike We are also indebted to Dr Martin Peters at Springer-Verlag for many interesting and useful discussions, and for encouraging the publication of this volume Fornebu September, 2005 Are Magnus Bruaset Aslak Tveito Contents Part I Parallel Computing Parallel Programming Models Applicable to Cluster Computing and Beyond Ricky A Kendall, Masha Sosonkina, William D Gropp, Robert W Numrich, Thomas Sterling 1.1 Introduction 1.2 Message-Passing Interface 1.3 Shared-Memory Programming with OpenMP 1.4 Distributed Shared-Memory Programming Models 1.5 Future Programming Models 1.6 Final Thoughts References 3 20 36 42 49 50 Partitioning and Dynamic Load Balancing for the Numerical Solution of Partial Differential Equations James D Teresco, Karen D Devine, Joseph E Flaherty 2.1 The Partitioning and Dynamic Load Balancing Problems 2.2 Partitioning and Dynamic Load Balancing Taxonomy 2.3 Algorithm Comparisons 2.4 Software 2.5 Current Challenges References 55 56 60 69 71 74 81 Graphics Processor Units: New Prospects for Parallel Computing Martin Rumpf, Robert Strzodka 89 3.1 Introduction 89 3.2 Theory 97 3.3 Practice 103 3.4 Prospects 118 3.5 Appendix: Graphics Processor Units (GPUs) In-Depth 121 X Contents References 131 Part II Parallel Algorithms Domain Decomposition Techniques Luca Formaggia, Marzio Sala, Fausto Saleri 135 4.1 Introduction 135 4.2 The Schur Complement System 138 4.3 The Schur Complement System Used as a Preconditioner 146 4.4 The Schwarz Preconditioner 147 4.5 Applications 152 4.6 Conclusions 159 References 162 Parallel Geometric Multigrid Frank Hăulsemann, Markus Kowarschik, Marcus Mohr, Ulrich Răude 165 5.1 Overview 165 5.2 Introduction to Multigrid 166 5.3 Elementary Parallel Multigrid 177 5.4 Parallel Multigrid for Unstructured Grid Applications 189 5.5 Single-Node Performance 193 5.6 Advanced Parallel Multigrid 195 5.7 Conclusions 204 References 205 Parallel Algebraic Multigrid Methods – High Performance Preconditioners Ulrike Meier Yang 209 6.1 Introduction 209 6.2 Algebraic Multigrid - Concept and Description 210 6.3 Coarse Grid Selection 212 6.4 Interpolation 220 6.5 Smoothing 223 6.6 Numerical Results 225 6.7 Software Packages 230 6.8 Conclusions and Future Work 232 References 233 Parallel Mesh Generation Nikos Chrisochoides 237 7.1 Introduction 237 7.2 Domain Decomposition Approaches 238 7.3 Parallel Mesh Generation Methods 240 7.4 Taxonomy 255 7.5 Implementation 255 Appendix A Color Figures M6 M∞=0.84, α=3.06 45 M6 M∞=0.84, α=3.06 24 P M6_23k N_p=4 23k N_p=32 23k N_p=4 42k N_p=32 42k N_p=4 94k N_p=32 94k S P ACM,2 N_p=4 M6_23k PS M6_42k 40 473 22 P ACM,2 N_p=4 M6_42k PS M6_94k 20 P ACM,2 N_p=4 M6_94k 35 GMRES iterations GMRES iterations 18 30 25 20 16 14 12 10 15 10 6 time levels 10 time levels 10 Fig A.14 M6 94k Iterations to converge with PS and PACM,2 (left), and iterations to converge with PACM,2 (right) using two different values of Np and 16 processors (This is a color version of Figure 4.12 on page 160) M6 94k, M_infty=0.84, α=3.06, 32 procs M6 94k, M_infty=0.84, α=3.06, 32 procs 45 −3 ACM, N_p=8 ASP−2−ilu0 PS ACM, N_p=8 ASP−2−ilu0 PS 40 −3.5 35 GMRES iterations Time residual −4 −4.5 −5 30 25 20 −5.5 15 −6 −6.5 10 50 100 150 200 250 time iterations CPU time (s) 10 Fig A.15 M6 94k Residual versus CPU-time (right) and iterations to converge at each time level (right), using 32 processors (This is a color version of Figure 4.13 on page 160) M6 316k M∞=0.84 α=3.06 80 M6 316k M =0.84 α=3.06 ∞ 10 P S PACM,2 P S PACM,2 ASP−2−ilu0 70 60 −1 ||r|/||r0|| GMRES iterations 10 50 40 30 −2 10 20 10 −3 time levels 10 12 14 10 10 20 30 40 50 GMRES iterations 60 70 80 Fig A.16 M6 316k Iterations at each time level (left) and converge history at the 14th time step (right) (This is a color version of Figure 4.14 on page 161) 474 Appendix A Color Figures Parallel Mesh Generation N Chrisochoides DD of continuous geometry DD of discrete geometry Fig A.17 Domain decomposition of the continuous geometry [52] and the discrete geometry [17] of a cross section of a rocket pipe (This is a color version of Figure 7.1 on page 239) Appendix A Color Figures Submesh M I 02 Submesh M o Submesh M B Pi Pj t Reques G I 12 Submesh M t remo Expand Cav ABCDE t* C D Expand Cav AFBC H Submesh M o te data Poll Triang ABCDE 475 Service Remote Data Gather Remote data gather Latency Poll A E F Submesh M Poll I 01 Service Remote Completion Service Remote Data Gather Triang AFGHBC (a) (b) Meshing time distribution tee2:16 procs:2M elements:SMGP0 90 Submesh 80 70 Submesh Time (s) 60 Termination Polling Receives Setbacks Encroached W/o active W/active 50 40 P 30 20 Submesh 10 0 10 11 12 13 14 15 Processor Number (c) (d) Fig A.18 a) cavity extension beyond submesh interfaces, b) time diagram with concurrent point insertion, c) a breakdown of execution time for PODM, and finally d) the refinement of a cavity with simultaneous distribution of the newly created elements (This is a color version of Figure 7.4 on page 243) 476 Appendix A Color Figures Parallel PDE-Based Simulation Using the Common Component Architecture L C McInnes et al Fig A.19 State-of-the-art simulation tools are used to help design the next generation of accelerator facilities (Left): Mesh generated for the PEP-II interaction region using the CUBIT mesh generation package Image courtesy of Tim Tautges of Sandia National Laboratories (Right): Excited fields computed using Tau3P Image courtesy of the numerics team at SLAC (This is a color version of Figure 10.2 on page 330) Fig A.20 A 10-cm-high pulsating methane-air jet flame, computed on an adaptive mesh On the left is the temperature field with a black contour showing regions of high heat release rates On the right is the adaptive mesh, in which regions corresponding to the jet shear layer are refined the most (This is a color version of Figure 10.4 on page 333) Appendix A Color Figures 477 Bytes 9.78E+08 4.91504E+08 2.4701E+08 1.24137E+08 6.23865E+07 3.13529E+07 1.57567E+07 7.91871E+06 3.97962E+06 25 Recv proc 20 15 10 0 10 20 Sending proc Average Communication time (secs) Fig A.21 A typical C-SAFE problem involving hydrocarbon fires and explosions of energetic materials This simulation involves fluid dynamics, structural mechanics, and chemical reactions in both the flame and the explosive Accurate simulations of these events can lead to a better understanding of high-energy explosives, can help evaluate the design of shipping and storage containers for these materials, and can help officials determine a response to various accident scenarios The fire image is courtesy of Schonbucher Institut for Technische Chemie I der Universitat Stuttgart, and the images of the container and explosion are courtesy of Eric Eddings of the University of Utah (This is a color version of Figure 10.5 on page 334) 25 np=112 np=7 20 np=14 15 np=56 10 np=28 10 Communication radius Fig A.22 (Left): Communication patterns for 28 processors at timestep 40 (Right): Communication costs as a function of the communication radius at timestep 40 (This is a color version of Figure 10.17 on page 367) 478 Appendix A Color Figures Full-Scale Simulation of Cardiac Electrophysioology on Parallel Computers X Cai and G T Lines t=30ms t=200ms Fig A.23 Snapshots from two time levels of a simulation of the electrical field in the human heart and torso At each time level, the electrical potential distribution on the heart surface is shown at three different angles, while the distribution on the torso surface is shown at two different angles (This is a color version of Figure 11.1 on page 387) Fig A.24 The orientation of the muscle fibers (left) and sheet layers (right) in the heart (This is a color version of Figure 11.2 on page 391) Appendix A Color Figures 479 Fig A.25 An example of partitioning an unstructured heart mesh (left) and an unstructured torso mesh (right) (This is a color version of Figure 11.6 on page 402) x 10 non−computational overlapping points computational overlapping points computational interior points 2.5 1.5 0.5 Subdomain ID Fig A.26 The effect of applying a disjoint re-distribution to the Ω mesh points, where NΩ = 919, 851 and the number of subdomains is (This is a color version of Figure 11.7 on page 406) 480 Appendix A Color Figures Developing a Geodynamics Simulator with PETSc M G Knepley, R F Katz, and B Smith log10η and flow field Depth, km 50 23 100 1400 1200 50 150 1000 110 22 800 100 600 21 200 400 20 250 300 100 200 300 150 19 200 50 100 150 200 24 50 Depth, km Temperature, °C 24 23 1400 1200 50 1000 100 22 150 00 100 11 21 800 600 200 400 250 20 300 19 100 200 Distance, km 300 150 200 50 100 150 200 Distance, km Fig A.27 2D viscosity and potential temperature fields from simulations on processors with 230,112 degrees of freedom Panels in the top row are from a simulation with α=1 in equation (12.1) Panels in bottom row have α=0 The white box in panels (a) and (c) shows the region in which temperature is plotted in panels (b) and (d) (a) Colors show log10 of the viscosity field Note that there are more than five orders of magnitude variation in viscosity Arrows show the flow direction and magnitude (the slab is being subducted at a rate of cm/year) Upwelling is evident in the flow field near the wedge corner (b) Temperature field from the variable viscosity simulation; 1100◦ C isotherm is shown as a dashed line (c) (Constant) viscosity and flow field from the isoviscous simulation Strong flow is predicted at the base of the crust despite the low-temperature rock there No upwelling flow is predicted (d) Temperature field from isoviscous simulation Note that the mantle wedge corner is much colder than in (b) (This is a color version of Figure 12.28 on page 436) Appendix A Color Figures 481 Parallel Lattice Boltzmann Methods for CFD Applications C Kăorner, T Pohl, U Răude, N Thăurey, and T Zeiser 200 Dual Opteron, Myrinet2000, speed-up Dual Opteron, Myrinet2000, scale-up Dual Xeon, GBit, speed-up Dual Xeon, GBit, scale-up SGI Altix, speed-up SGI Altix, scale-up 180 160 140 MLup/s 120 100 80 60 40 20 0 16 32 number of CPUs 48 64 Fig A.28 Scalability tests for modern cluster configurations The domain size is 256 × 129 × 128 for speed-up (fixed domain size) and 1283 per processor for scale-up tests (fixed CPU load by scaling the domain size) For reference the corresponding results of a shared memory system (SGI Altix 3000) are given The common performance unit MLup/s (million lattice site updates per second) has been used (This is a color version of Figure 13.10 on page 455) 482 Appendix A Color Figures Fig A.29 3D foam: The bubbles grow and coalescence occurs The disjoining pressure Π stabilizes the foam and eventually a polygonal structure develops (initial number of bubbles: 1000; system size: 120 × 120 × 140; τ = 0.8; g = 0; σ = 0.01; cΠ = 0.006) (This is a color version of Figure 13.11 on page 456) n set2 F If I E F Iff E F If m If set1 Main Loop arrives at Interface Cell Calculate Mass exchange with Fluid and Interface Cells Stream from adjacent Cells Reconstruct DFs from Empty Cells Reconstruct DFs along Interface Normal n Perform normal Collision 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Springer mathematics books for their personal use, at a discount of 33,3 % directly from Springer-Verlag Addresses: Timothy J Barth NASA Ames Research Center NAS Division Moffett Field, CA 94035, USA e-mail: barth@nas.nasa.gov Michael Griebel Institut für Numerische Simulation der Universität Bonn Wegelerstr 53115 Bonn, Germany e-mail: griebel@ins.uni-bonn.de David E Keyes Department of Applied Physics and Applied Mathematics Columbia University 200 S W Mudd Building 500 W 120th Street New York, NY 10027, USA e-mail: david.keyes@columbia.edu Risto M Nieminen Laboratory of Physics Helsinki University of Technology 02150 Espoo, Finland e-mail: rni@fyslab.hut.fi Dirk Roose Department of Computer Science Katholieke Universiteit Leuven Celestijnenlaan 200A 3001 Leuven-Heverlee, Belgium e-mail: dirk.roose@cs.kuleuven.ac.be Tamar Schlick Department of Chemistry Courant Institute of Mathematical Sciences New York University and Howard Hughes Medical Institute 251 Mercer Street New York, NY 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Timothy J Barth Michael Griebel David E Keyes Risto M Nieminen Dirk Roose Tamar Schlick Are Magnus Bruaset Aslak Tveito (Eds.) Numerical Solution of Partial Differential Equations on Parallel Computers. .. 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,... advantages Creating an additional thread of execution is usually faster than creating another process, and synchronization and context Parallel Programming Models 21 switches among threads are