Anova for Parameter Dependent Nonlinear PDES and Numerical Methods for the Stochastic ...

90 Pages·2015·1.15 MB·English

by  Zheng Chen

Florida State University Libraries
Electronic Theses, Treatises and Dissertations The Graduate School
2007
Anova for Parameter Dependent Nonlinear
PDEs and Numerical Methods for the
Stochastic Stokes Equations
Zheng Chen
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THE FLORIDA STATE UNIVERSITY
COLLEGE OF ARTS AND SCIENCES
ANOVA FOR PARAMETER DEPENDENT NONLINEAR PDES AND
NUMERICAL METHODS FOR THE STOCHASTIC STOKES EQUATIONS
By
ZHENG CHEN
A Dissertation submitted to the
Department of Mathematics
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Degree Awarded:
Summer Semester, 2007
The members of the Committee approve the Dissertation of Zheng Chen defended on
July 2, 2007.
Max Gunzburger
Professor Directing Dissertation
Fred Huffer
Outside Committee Member
Janet Peterson
Committee Member
Xiaoqiang Wang
Committee Member
The Office of Graduate Studies has verified and approved the above named committee members.
ii
To my parents, my wife and my daughter.
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ACKNOWLEDGEMENTS
My greatest thanks are due to my advisor, Dr. Gunzburger. Over the years of my study
with him, he gave me advice, encouragement and support. His lectures, talks and courses
provided to us were important to me, but I also was privileged to have chats and individual
time with him. I will always admire his skills and extensive knowledge in mathematics,
physics and statistics, and his insight into scientific problems. Only with his help, was I able
to do this work. I hope that I have learned enough from him to be a productive researcher
myself.
I would also like to thank Dr. Yanzhao Cao for giving me great help in preparing my
dissertation including choosing topics and cooperative research throughout these years.
I would also like to thank Dr. Huffer, Dr. Peterson and Dr. Wang for their willingness to
be on my committee. Dr. Huffer allowed me to sit in his class to study stochastic processes;
he is so kind and compassionate and gave me any help I needed. Dr. Peterson is one of my
favorite professors. I learned Finite Element Methods from her in Iowa. I will never forget
her tutoring my spoken English.
Dr. Xiaoming Wang provided the most wonderful advanced PDE courses, from which
I benefited a lot. I also thank him for his concern about my research and my life in these
years.
I also want to take this chance to thank Dr. John Burkardt and Dr. Mark Sussman:
they are always there when I need help in programming. Dr. John Burkardt showed me the
tricks and technology in programming and Unix. He is always passionate and patient. With
him in our group, I do not panic when I have trouble in my work. He also saves me much
time by sharing his codes with me.
I want to thank Dr. Steve Hou and Dr. Scott Hansen at Iowa State University for their
help and concern for my study.
During the pursuit of my PhD, I worked as a teaching assistant for the Department of
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Mathematics and as a research assistant for School of Computational Science. The faculty
and staff of these two departments gave me all kinds of help when I needed, I appreciate it.
Lastly, I am grateful to my wife Min Chen; without her love and continuous support,
it is unimaginable that I could have persisted in my study over these years. My daughter
Rebecca Chen brings me happiness which makes me feel at ease in working hard; for this, I
thank her.
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TABLE OF CONTENTS
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1. General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. The ANOVA expansion and efficient sampling methods for parameter depen-
dent nonlinear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 ANOVA expansions of multivariate functions and effective dimensions . . 8
2.3 The model problem and the approximation property of ANOVA expan-
sions for small perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 TheANOVAexpansionsofandtheireffectivedimensionsforcostfunctionals 16
2.5 Sampling parameter space for building surrogate functionals . . . . . . . 18
2.6 Methods for uniform sampling in hypercubes . . . . . . . . . . . . . . . . 23
3. Review of Results for Stochastic PDEs . . . . . . . . . . . . . . . . . . . . . 25
3.1 Introduction to Numerical Stochastic PDEs . . . . . . . . . . . 25
3.2 Introduction to homogeneous Gaussian noise . . . . . . . . . . . . . . . 29
3.3 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Two concepts in the SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . 39
4. Stochastic Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Discretization of the white noise and error estimates . . . . . . . . . . . 42
4.3 Finite element approximations . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Stochastic Stokes equations driven by homogeneous Gaussian noise . . . 58
5. Conclusions and future research topics . . . . . . . . . . . . . . . . . . . . . . 68
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Future research topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
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BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
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LIST OF TABLES
2.1 The ANOVA expansion terms and values of the effecitve dimensions. . . . . 19
2.2 Absolute error, averaged over 10 realizations, in the location of the minimizing
parameter point for each of the point sampling methods used in the surrogate
construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Absolute error, averaged over 10 realizations, in the minimum value of the
functionals for each of the point sampling methods used in the surrogate
construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
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LIST OF FIGURES
2.1 The data configuration for the model problem used in the computational
examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1 approx integral of mean vs. log(n) (n, partition number in x axis and y axis) 57
4.2 approx integral of variance vs. log(n) (n, partition number in x axis and y axis) 57
4.3 approx mean of 1 norm of pressure vs. log(n) (n, partition number in x axis
−
and y axis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Approx mean of u(x,y) after 5000 realizations in nx = ny = 32 . . . . . . . 59
4.5 Approx mean of u(x,y) after 5000 realizations in nx = ny = 16 . . . . . . . . 59
4.6 Approx variance of v(x,y) after 5000 simulations in nx = ny = 16 . . . . . . 60
4.7 Approx variance of v(x,y) after 5000 simulations in nx = ny = 32 . . . . . . 60
4.8 discretized white noise in nx = ny = 8 . . . . . . . . . . . . . . . . . . . . . 66
4.9 discretized colored noise with alpha = 1.2 and nx = ny = 8 . . . . . . . . . 67
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