Real Time Computing with the Parareal Algorithm

Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2008 Realtime Computing with the Parareal Algorithm Christopherr.Harden Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES REALTIME COMPUTING WITH THE PARAREAL ALGORITHM By CHRISTOPHERR.HARDEN A Thesis submitted to the School of Computational Science in partial fulfillment of the requirements for the degree of Master of Science Degree Awarded: Spring Semester, 2008 The members of the Committee approve the Masters Thesis of Christopher R. Harden defended on April 8, 2008. Janet Peterson Professor Directing Masters Thesis Max Gunzburger Committee Member Robert Van Engelen Committee Member Approved: Max Gunzburger, Director Department of School of Computational Science The Office of Graduate Studies has verified and approved the above named committee members. ii This thesis is dedicated to all of the people who have helped and guided me throughout my research including but not limited to Janet Peterson, Max Gunzburger, John Burkardt, Robert Van Engelen, and the many other professors who have provided me with an excellent level of instruction throughout all of my course work here at FSU. Also, I would like to dedicate this work to my wife, Jennifer Alligood, and my son, Youth, for their infinite depth of understanding of my situation as a graduate student and for their unending support throughout this endeavor. iii ACKNOWLEDGEMENTS I would like to acknowledge my gratitude to my committee members, who have taken the time to review my work and to pose the difficult questions which have kept me honest and thus have allowed me to grow as an academic throughout this process. A special acknowledgment is in order for my advisor who took a chance in bringing me under her wing and teaching me how to become a researcher and to John Burkardt, who took a lot of time out of his schedule to teach me many of the tools that were necessary for me to be able to complete this work. Also, some thanks are owed to Clayton Webster for all of his TEX support. iv TABLE OF CONTENTS List of Tables ...................................... vii List of Figures ..................................... viii Abstract ........................................ x 1. INTRODUCTION ................................. 1 2. The Parareal Algorithm .............................. 4 2.1 The Basic Algorithm ............................. 4 2.2 A Simple Example .............................. 5 2.3 Comments on Some Mathematical Properties of the Parareal Algorithm . 8 3. The Finite Element Method and The Parareal Algorithm ............ 14 3.1 A Finite Element Method .......................... 14 3.2 The Finite Element Method and the Parareal Algorithm for Nonlinear PDE’s ..................................... 22 4. Combining the Parareal Algorithm and Reduced Order Modeling ....... 25 4.1 Reduced Order Modeling with Proper Orthogonal Decompositions .... 25 4.2 Reduced Order Modeling and The Parareal Algorithm .......... 29 4.3 Implementation ................................ 33 5. Computational Experiments and Results ..................... 35 5.1 FEM and The Parareal Algorithm Results ................. 35 5.2 ROM and The Parareal Algorithm Results ................. 38 6. Performance Analysis and Scalability ....................... 44 6.1 Introduction to performance analysis concepts and terminology ..... 44 6.2 Problem Parameters in our FEM and ROM Parareal Implementations .. 50 6.3 Strong Scaling Trends of the Parareal Algorithm .............. 57 7. Conclusions and Future Work ........................... 69 7.1 Conclusions .................................. 69 7.2 Future Work ................................. 70 REFERENCES ..................................... 72 v BIOGRAPHICAL SKETCH ............................. 74 vi LIST OF TABLES 5.1 Comparison of errors using standard finite element approach and the parareal/FEM approach ...................................... 36 5.2 Speedup results for the parareal/FEM approach compared to the serial FEM approach. ..................................... 37 5.3 (Comparison of errors for 4-parameter problem using standard ROM approach and the parareal/ROM algorithm. Errors are calculated by comparing to the full finite element solution.) ........................... 40 6.1 Results of FEM Test Case 1 ........................... 57 6.2 Results of FEM Test Case 2 ........................... 57 6.3 Results of FEM Test Case 3 ........................... 58 6.4 Results of FEM Test Case 4 ........................... 58 6.5 Results of FEM Test Case 5 ........................... 59 6.6 Results of FEM Test Case 6 ........................... 59 6.7 Results of ROM Test Case 1 ........................... 60 6.8 Results of ROM Test Case 2 ........................... 60 vii LIST OF FIGURES 2.1 Illustration of the coarse and fine grids ..................... 6 2.2 B2, Exact Solution ................................ 10 2.3 B2, Parareal Solution on the Coarse Grid After 2 iterations of the Correction Scheme ....................................... 11 2.4 B1, Phase Portrait ................................ 12 2.5 B1, Coarse Grid Solution and Divergent Refined Solution. 13 4.1 The H-cell domain of the building ventilation problem ............. 32 5.1 speedup of Parareal/FEM Implementation : Blue-∆t = 0.01 and Red-∆t = 0.005 ........................................ 38 5.2 speedup of Parareal/ROM Implementation : Blue-∆t = 0.01 and Red-∆t = 0.005 ........................................ 41 5.3 The H-cell domain of the building ventilation problem, with boundary parameters illustrated. .................................. 42 5.4 speedup of Parareal/ROM Implementation of the Navier-Stokes Problem : Blue-∆t = 0.01 and Red-∆t = 0.005 ...................... 43 6.1 Suite A, Speedup vs. Processors ........................ 53 6.2 Speedup of Parareal/FEM with h = 0.1, ∆t = 0.005, and T =1 . 61 6.3 Speedup of Parareal/FEM with h = 0.1, ∆t = 0.001, and T = 1 . 62 6.4 Speedup of Parareal/FEM with h = 0.05, ∆t = 0.005, and T = 1 ...... 63 6.5 Speedup of Parareal/FEM with h = 0.05, ∆t = 0.001, and T = 1 ...... 64 6.6 Speedup of Parareal/FEM with h = 0.1, ∆t = 0.001, and T = 10 ...... 65 6.7 Speedup of Parareal/FEM with h = 0.05, ∆t = 0.001, and T = 10 ...... 66 viii 6.8 Speedup of Parareal/ROM with h = 0.1, ∆t = 0.005, and T = 1 ....... 67 6.9 Speedup of Parareal/ROM with h = 0.1, ∆t = 0.001, and T = 10 ...... 68 ix ABSTRACT This thesis presents and evaluates a particular algorithm used for the real time computations of time dependent ordinary and partial differential equations which employs a parallelization strategy over the temporal domain. We also discuss the coupling of this method with another popular technique used for real time computations, model reduction, which will be shown to provide more gains than either method alone. In particular, we look at reduced order modeling based on proper orthogonal decompositions. We present some applications in terms of solving time dependent nonlinear partial differential equations and solving these equations with a coupled approach of combining model reduction and the parareal algorithm . The performance of this method, both numerically and computationally, is discussed in terms of the gains in speedup and efficiency, and in terms of the scalability of the parallelization of the temporal domain on a larger and larger set of compute nodes or processors. x CHAPTER 1 INTRODUCTION Many of the equations and systems governing the various mechanisms of nature implicitly involve time evolution and time dependencies in general. Many successful attempts have been made to construct computational schemes for integration in time. Until recently, many of these schemes have shared the common theme of being purely serial in implementation. Traditional schemes seem to have taken the view that time, itself, is sequential and many schemes have been designed in ways such that computations at a current time step rely on values computed at previous time steps. In contrast to this traditionally sequential view of the temporal domain there are all sorts of varieties of ways in which researchers have dealt with the spatial domain: serial, parallel, or a combination of both. In the context of time dependent ordinary and partial differential equations many parallel schemes have been proposed in which discretizations of the spatial domain are implemented in parallel. One class of methods in which such implementations are prevalent is in domain decomposition methods. These methods have been shown to be useful in both serial and parallel implementations. The scheme presented here shares much of the flavor of these domain decomposition methods. There are two significant points of departure though. The first is the focus on decomposing the temporal domain and not the spatial domain. The second lies in the fact that this scheme has absolutely no value in a serial computation and is, therefore, a purely parallel algorithm. Also, many people will notice its similarity to some of the various flavors of methods for iterative improvement for linear systems. The Parareal algorithm was introduced by Lions, Maday, and Turinici in 2001 [13] as a numerical method to solve time-evolution problems in parallel. The name of the algorithm already indicates the intention of its design. The purpose is for parallel, real time 1 computations involving time evolution equations whose solutions

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