Transcript Extended Diffraction-Slice Theorem for Wavepath Traveltime

Inverse Problems and
Applications
Chaiwoot Boonyasiriwat
Last modified on December 6, 2011
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Textbooks
• Parameter Estimation and Inverse Problems, Aster
et al., Elsevier, 2005
• Computational Methods for Inverse Problems,
Vogel, SIAM, 2002
• Geophysical Inverse Theory, Parker, Princeton
University Press, 1994
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Outline
• Introduction to inverse problems
• Mathematical background: Linear algebra,
Functional analysis
• Singular value decomposition
• Regularization methods
• Iterative optimization methods
• Methods for choosing regularization parameters
• Additional regularization methods
• Nonlinear inverse problems
• Bayesian inversion
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Introduction to Inverse Problems
Find tumors or cancers?
How can we see internal
organs without surgery?
Use CT scan.
What is CT scan and how
does it work?
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X-Ray Computed Tomography
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Inverse Problems in Physics
Seismic tomography
(1980s)
Helioseismology
(1990s)
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Forward and Inverse Problems
𝑑 = 𝐴(𝑚)
where 𝑑 is data, 𝑚 is a model parameter, and 𝐴 is an
operator that maps the model 𝑚 into the data 𝑑.
Forward Problem: Given m. Find d.
Inverse Problem: Given d. Find m.
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Well-posedness vs. Ill-posedness
(*) 𝐴𝑥 = 𝑦, 𝐴 is a continuous operator, 𝐴: 𝑋 → 𝑌
The above problem is well posed if A has a
continuous inverse operator 𝐴−1 from 𝑌 to 𝑋.
This means:
1. Existence of solution: ∀𝑦 ∈ 𝑌 there exists ∀𝑥 ∈
𝑋, s.t. (*) is satisfied.
2. Uniqueness of solution: ∀𝑦 ∈ 𝑌 there is no more
than one ∀𝑥 ∈ 𝑋 satisfying (*).
3. Stability of solution on data: If 𝑦 − 𝑦0 → 0,
𝑥 − 𝑥0 → 0.
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Classification of Inverse Problems
𝐴𝑚 = 𝑑
Inverse problem: Finding 𝑚 given 𝑑
System identification problem: Determining 𝐴 given
examples of 𝑚 and 𝑑.
𝐴 𝑞 𝑢=𝑓
Parameter identification problem: Finding 𝑞 given
data 𝑑 which can be expressed as
𝑑 = 𝐶𝑢
where 𝐶 is called the state-to-observation map.
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Examples
• Linear regression or curve fitting
𝑦 = 𝑚𝑥 + 𝑐
• 1D steady-state diffusion equation
𝑑
𝑑𝑢
−
𝜅 𝑥
= 𝑓(𝑥)
𝑑𝑥
𝑑𝑥
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