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Publication Type	technical report
School or College	College of Engineering
Department	Computing, School of
Creator	Tsay, Chey-Woei
Title	Fixed Point Computation and Parallel Algorithms for Solving Wave Equations
Date	1994
Description	This dissertation is concerned with two problems: fixed-point computation and parallel algorithms for solving wave equations. Many problems can be formulated as fixed-point problems, which is to solve the equations of the form f(x) - x = 0. We develop efficient algorithms and establish complexity results for such fixed-point computations. For univariate functions, we demonstrate Fixed Point Envelope (FP-EN) algorithms and prove that they minimize the number of function evaluations for computing an approximation to within a prescribed absolute error tolerance. For multivariate contractive functions, we exhibit a Fixed Point Ellipsoid (FP-EL) algorithm. This algorithm is much more efficient than the well-known simple iteration algorithm. We also show an upper bound on the number of function evaluations for this case. For multivariate noncontractive functions, we prove that the complexity is infinite whenever the error tolerance is less than 0.5. As the second problem, we consider the acoustic wave equation and the elastic wave equation that are frequently used in seismology. We develop parallelized finite difference algorithms for solving these equations. This is done by dividing the computational domain into subdomains and then by assigning a processor to each subdomain for computing and storing local grid data. We discuss parallel algorithms on three computing networks: processor arrays, rings, and meshes, with emphasis on the part of communication and synchronization. Finally, we demonstrate that our implementations on transputer linear arrays achieve linear speedup.
Type	Text
Subject	fixed point computation; parallel algorithm; fixed point envelope; fixed point ellipsoid; wave equation; processor array; transputer; linear array
Subject LCSH	Fixed point theory; Parallel processing (Electronic computers); Wave equation
Language	eng
Bibliographic Citation	Tsay, C. (1994). Fixed Point Computation and Parallel Algorithms for Solving Wave Equations.
Series	University of Utah Computer Science Technical Report
Relation is Part of	ARPANET
Format Medium	application/pdf
Format Extent	67,222,126 bytes
File Name	Tsay-Fixed_Point_Computation.pdf
Conversion Specifications	Original scanned with Kirtas 2400 and saved as 400 ppi uncompressed TIFF. PDF generated by Adobe Acrobat Pro X for CONTENTdm display
ARK	ark:/87278/s6mw4jc4
Setname	ir_computersa
ID	102378
Reference URL	https://collections.lib.utah.edu/ark:/87278/s6mw4jc4

Page Metadata

Title	Page 13
Setname	ir_computersa
ID	102262
OCR Text	Show ACKNOWLEDGMENTS My most sincere thanks go to my advisor Dr. Kris Sikorski and my dissertation committee members, Dr. Gerald Schuster and Dr. Frank Stenger. Dr. Sikorski guided me into the field of fixed-point computation. His deep insight and sharp arguments helped me avoid wrong paths during this research. In fact, many results shown in this work are based on his previous work in this field. He also introduced me to the powerful concept of information-based complexity which, I strongly believe, will play an important role in the future development of numerical algorithms. I became acquainted with Dr. Schuster in 1988 when I wrote a parallel program for him to simulate acoustic waves propagating through 3-D acoustic media. He kindly provided me with many geophysics books and papers, and patiently answered many questions. Without him, I could not accomplish developing parallel algorithms for solving PDEs. Dr. Stenger pointed out to me many numerical methods for solving PDEs, in particular, his SINC methods. I benefited from the discussions with my classmates Kwan-Liu Ma, Shyh-Kuang Ueng, and Yun-Hsing Young. Dr. Dennis Shea at IBM Thomas J. Watson Research Center made it possible to use the IBM Victor machine to carry out many experiments. I would also like to thank Colleen Hoopes, who is the most friendly department secretary I have ever known. She helped me to solve many nonacademic problems due to my insensitiveness to the school's regulations. Finally, I dedicate this work to my parents and my wife. They fully backed me during this study. Without their support, the painful dissertation writing would last longer. This work has been funded in part by the University of Utah, the Advanced Combustion Engineering Research Center at the University of Utah, IBM Grant on Parallel Computation with Clusters of Workstations, and IBM Thomas J. Watson Research Center. All opinions, findings, and conclusions expressed in this document are those of the au thor and do not necessarily reflect the views of the sponsoring agencies.
Reference URL	https://collections.lib.utah.edu/ark:/87278/s6mw4jc4/102262