Interpolation weights of algebraic multigrid

92 Pages·1.4 MB·English

by  Miranda, Gerald N

NPS ARCHIVE
1997.ce
MIRANDA, G.
NAVAL POSTGRADUATE SCHOOL
Monterey, California
THESIS
INTERPOLATION WEIGHTS OF
ALGEBRAIC MULTIGRID
by
Gerald N. Miranda, Jr.
June 1997
Thesis Advisor: Van Emden Henson
Second Reader: Christopher L. Frenzen
Approved for public release; Distribution is unlimited.
xY KNOXLIBRARY
POSTGRADUATESCHOOL
REPORT DOCUMENTATION PAGE
Form Approved OMB No. 0704-0188
Public reporting burdenfor this collection ofinformation is estimated to average 1 hour per response, including the time for reviewing instruction,
searching existing data sources, gathering and maintaining thedata needed, and completing and reviewing the collection ofinformation. Send
commentsregarding this burden estimateor any otheraspect ofthis collection ofinformation, including suggestions for reducing this burden,
to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204,
Arlington, Va22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188) Washington DC 20503.
1. AGENCY USE ONLY {Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
June, 1997 Master's Thesis
4. TITLE AND SUBTITLE INTERPOLATION WEIGHTS OF 5. FUNDING NUMBERS
ALGEBRAIC MULTIGRID
6. AUTHORS MIRANDA, GERALD N., JR.
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING
ORGANIZATION
Naval Postgraduate School REPORT NUMBER
Monterey CA 93943-5000
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING
AGENCY REPORT NUMBER
11. SUPPLEMENTARY NOTES The views expressed in this thesis are those ofthe author and do not reflect
the official policy or position ofthe Department of Defense or the U.S. Government.
12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE
Approved for public release; distribution is unlimited.
13. ABSTRACT(maximum 200 words)
Algebraic multigrid (AMG) is a numerical method used to solve particular algebraic systems, and interest
in it has risen because ofits multigrid-likeefficiency. Variations in methodology during the interpolation phase
result in differing convergence rates. We have found that regular interpolation weight definitions are inadequate
for solving certain discretized systems so an iterative approach to determine the weights will prove useful. This
iterative weight definition must balance the requirement of keeping the interpolatory set of points "small" in
order to reduce solver complexity while maintainingaccurate interpolation to correctly represent the coarse-grid
function on the fine grid. Furthermore, the weight definition process must be efficient enough to reduce setup
phase costs.
We present systems involvingmatriceswhere thisiterativemethodsignificantlyoutperformsregular AMG weight
definitions. Experimental results show that the iterative weight definition does not improvethe convergence rate
over standard AMG when applied to M-matrices; however, the improvement becomes significant when solving
certain types of complicated, non-standard algebraic equations generated by irregular operators.
14. SUBJECT TERMS 15. NUMBER OF
PAGES
Algebraic Multigrid, Matrix Equations, Interpolation Weights 88
16. PRICE CODE
17. SECURITY CLASSIFI- 18. SECURITY CLASSIFI- 19. SECURITY CLASSIFI- 20. LIMITATION
CATION OF REPORT CATION OF THIS PAGE CATION OF ABSTRACT OF ABSTRACT
Unclassified Unclassified Unclassified UL
NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)
Prescribed by ANSI Std. 239-18 298-102
11
Approved for public release; distribution is unlimited
INTERPOLATION WEIGHTS OF
ALGEBRAIC MULTIGRID
Gerald N. Miranda, Jr.
Lieutenant, Uni11ted States Navy
B.A., University of California, San Diego, 1990
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN APPLIED MATHEMATICS
from the
NAVAL POSTGRADUATE SCHOOL
June 1997

IOXLIBRARY
NAVALPOSTGRADUATESCHOOL
ABSTRACT
Algebraic multigrid (AMG) is a numerical method used to solve particular
algebraic systems, and interest in it has risen because of its multigrid-like efficiency.
Variations in methodology during the interpolation phase result in differing conver-
gence rates. We have found that regular interpolation weight definitions are inade-
quate for solving certain discretized systems so an iterative approach to determine
the weights will prove useful. This iterative weight definition must balance the re-
quirement of keeping the interpolatory set of points "small" in order to reduce solver
complexity while maintaining accurate interpolation to correctly represent the coarse-
grid function on the fine grid. Furthermore, the weight definition process must be
efficient enough to reduce setup phase costs.
We present systems involving matrices where this iterative method signifi-
AMG
cantly outperforms regular weight definitions. Experimental results show that
the iterative weight definition does not improve the convergence rate over standard
AMG
when applied to M-matrices; however, the improvement becomes significant
when solving certain types of complicated, non-standard algebraic equations gener-
ated by irregular operators.
VI
TABLE OF CONTENTS
INTRODUCTION
I. 1
THE FUNDAMENTALS OF MULTIGRID
II. 5
THE PROBLEM STATEMENT
A. 5
NOTATION
B. 6
ITERATIVE METHODS
C. 7
1. The Weighted Jacobi Iteration on the "Model" Problem . 8
2. Frequency Modes of the Error 8
D. THE METHOD OF COARSE-GRID CORRECTION 12
1. The Restriction Operator 12
2. The Interpolation Operator 14
3. The Algorithm of CG 17
THE MULTIGRID V-CYCLE SCHEME
E. 18
THE STRATEGY OF NESTED ITERATION
F. 21
THE FULL MULTIGRID V-CYCLE
G. 22
MULTIGRID ENOUGH?
H. IS 24
ALGEBRAIC MULTIGRID
III. 27
A. WHY ALGEBRAIC MULTIGRID? 28
APPLICATIONS OF AMG
B. 29
C. THE STEPS OF AMG 30
1. The Setup Phase 30
2. The Solution Phase 31
D. DEFINING THE "GRID" 31
CONNECTIONS AND CONVERGENCE
E. 31
CONSTRUCTING THE INTERPOLATION OPERATOR
IV. 35
. .
ALGEBRAIC SMOOTHNESS
A. 35
B. INTERPOLATION ALONG DIRECT CONNECTIONS .... 36
Vll
C. THE ITERATIVE WEIGHT DEFINITION 37
D. INTERPOLATION CONSTRUCTION USING THE ITERA-
TIVE WEIGHT DEFINITION
39
1. Initialization 41
2. Calculation 41
INTERPRETATION OF THE ALGORITHM
E. 43
V. NUMERICAL RESULTS 45
A. ISOTROPIC PROBLEMS 46
B. THE ANISOTROPIC PROBLEM USING THE FINITE DIF-
FERENCE METHOD
48
C. THE ANISOTROPIC PROBLEM USING THE FINITE ELE-
MENT METHOD
50
PROBLEMS WITH COMPLEX DOMAINS
D. 51
1. The Meshes 51
2. Results of Complex Domain Problems 54
COMPLEX DOMAINS AND OPERATORS
E. 55
CONCLUDING REMARKS
F. 58
FUTURE RESEARCH
VI. 61
A. INTERPOLATION USING EIGENVECTORS 61
B. THE LEAST-SQUARES IDEA 62
C. THE COMPOSITE GRID FORMULATION 64
GLOSSARY
67
LIST OF REFERENCES 69
INITIAL DISTRIBUTION LIST 71
vni