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A lecture note from a seminar in mathematics class at the university of california, berkeley, taught by thomas shores. The notes cover the topics of tikhonov regularization and iterative methods in the context of inverse theory. Specifically, the notes focus on higher order tikhonov regularization, the tgsvd and gcv methods, and error bounds. The document also includes examples and instructions for recovering the slowness function from a given function.
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Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion
Instructor: Thomas Shores Department of Mathematics
Lecture 22, April 4, 2006 AvH 10
Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion Error Bounds
Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion Error Bounds
Basic Idea We can think of the regularization term α^2 ‖m‖^22 as favoring minimizing the 0-th order derivative of a function m (x) under the hood. Alternatives: Minimize a matrix approximation to m′^ (x). This is a rst order method. Minimize a matrix approximation to m′′^ (x). This is a second order method. These lead to new minimization problems: to minimize
‖G m − d‖^22 + α^2 ‖Lm‖^22.
How do we resolve this problem as we did with L = I?
Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion Error Bounds
Basic Idea We can think of the regularization term α^2 ‖m‖^22 as favoring minimizing the 0-th order derivative of a function m (x) under the hood. Alternatives: Minimize a matrix approximation to m′^ (x). This is a rst order method. Minimize a matrix approximation to m′′^ (x). This is a second order method. These lead to new minimization problems: to minimize
‖G m − d‖^22 + α^2 ‖Lm‖^22.
How do we resolve this problem as we did with L = I?
Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion Error Bounds
Basic Idea We can think of the regularization term α^2 ‖m‖^22 as favoring minimizing the 0-th order derivative of a function m (x) under the hood. Alternatives: Minimize a matrix approximation to m′^ (x). This is a rst order method. Minimize a matrix approximation to m′′^ (x). This is a second order method. These lead to new minimization problems: to minimize
‖G m − d‖^22 + α^2 ‖Lm‖^22.
How do we resolve this problem as we did with L = I?
We will explore approximations to rst and second derivatives at the board.
The minimization problem is equivalent to the problem ( G T^ G + α^2 LT^ L
m = G T^ d
which has solution forms
mα,L =
∑^ p
j= 1
γ j^2 γ^2 j + α^2
UTj d
λj
Xj +
∑^ n
j=p+ 1
UTj d
Xj
Filter factors: fj =
γ j^2 γ^2 j + α^2
, j = 1 ,... , p, fj = 1 , j = p + 1 ,... , n.
Thus
mα,L =
∑^ n
j= 1
fj
UTj d
λj
Xj.
Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion Error Bounds
The Experiment: Place sensors at vertical depths zj , j = 1 ,... , n, in a borehole, then: Generate a seizmic wave at ground level, t = 0. Measure arrival times dj = t (zj ), j = 1 ,... , n. Now try to recover the slowness function s (z), given
t (z) =
∫ (^) z
0
s (ξ) d ξ =
0
s (ξ) H (z − ξ) d ξ
It should be easy: s (z) = t′^ (z). Hmmm.....or is it?
Do Example 4-5 from the CD.
Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion Error Bounds
The Experiment: Place sensors at vertical depths zj , j = 1 ,... , n, in a borehole, then: Generate a seizmic wave at ground level, t = 0. Measure arrival times dj = t (zj ), j = 1 ,... , n. Now try to recover the slowness function s (z), given
t (z) =
∫ (^) z
0
s (ξ) d ξ =
0
s (ξ) H (z − ξ) d ξ
It should be easy: s (z) = t′^ (z). Hmmm.....or is it?
Do Example 4-5 from the CD.
Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion Error Bounds
The Experiment: Place sensors at vertical depths zj , j = 1 ,... , n, in a borehole, then: Generate a seizmic wave at ground level, t = 0. Measure arrival times dj = t (zj ), j = 1 ,... , n. Now try to recover the slowness function s (z), given
t (z) =
∫ (^) z
0
s (ξ) d ξ =
0
s (ξ) H (z − ξ) d ξ
It should be easy: s (z) = t′^ (z). Hmmm.....or is it?
Do Example 4-5 from the CD.
Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion Error Bounds
The Experiment: Place sensors at vertical depths zj , j = 1 ,... , n, in a borehole, then: Generate a seizmic wave at ground level, t = 0. Measure arrival times dj = t (zj ), j = 1 ,... , n. Now try to recover the slowness function s (z), given
t (z) =
∫ (^) z
0
s (ξ) d ξ =
0
s (ξ) H (z − ξ) d ξ
It should be easy: s (z) = t′^ (z). Hmmm.....or is it?
Do Example 4-5 from the CD.
Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion Error Bounds
Model Resolution Matrix: As usual, Rm,α,L = G \G. Comment 1. Comment 2. Comment 3.
Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion Error Bounds
Model Resolution Matrix: As usual, Rm,α,L = G \G. Comment 1. Comment 2. Comment 3.
Chapter 5: Tikhonov Regularization Chapter 6: Iterative Methods A Brief Discussion Error Bounds
Model Resolution Matrix: As usual, Rm,α,L = G \G. Comment 1. Comment 2. Comment 3.
