inverse problems and applications chaiwoot boonyasiriwat last modified on december 6, 2011
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Inverse Problems and Applications
Chaiwoot BoonyasiriwatLast modified on December 6, 2011
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Grading Policy• 60% 6 Homework, 10% each• 10% Project Proposal• 30% Project Presentation
* Homework turned in late will not be graded.
[85%, 100%] = A
[80%, 85%) = B+
[75%, 80%) = B
[70%, 75%) = C+
[65%, 70%) = C
…i
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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
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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
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Seismic tomography
(1980s)
Helioseismology
(1990s)
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Forward and Inverse Problems
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where is data, is a model parameter, and is an operator that maps the model into the data .
Forward Problem: Given m. Find d.
Inverse Problem: Given d. Find m.
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Well-posedness vs. Ill-posedness
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(*) , is a continuous operator,
The above problem is well posed if A has a continuous inverse operator from to .
This means:
1. Existence of solution: there exists , s.t. (*) is satisfied.
2. Uniqueness of solution: there is no more than one satisfying (*).
3. Stability of solution on data: If , .
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Classification of Inverse Problems
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Inverse problem: Finding given
System identification problem: Determining given examples of and .
Parameter identification problem: Finding given data which can be expressed as
where is called the state-to-observation map.
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Examples
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• Linear regression or curve fitting
• 1D steady-state diffusion equation
−𝑑𝑑𝑥 (𝜅 (𝑥 ) 𝑑𝑢
𝑑𝑥 )= 𝑓 (𝑥)