Inverse Problems and Applications

Chaiwoot BoonyasiriwatLast modified on December 6, 2011

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

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

ii

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?

X-Ray Computed Tomography

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Inverse Problems in Physics

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Seismic tomography

(1980s)

Helioseismology

(1990s)

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.

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 , .

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.

Examples

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• Linear regression or curve fitting

• 1D steady-state diffusion equation

−𝑑𝑑𝑥 (𝜅 (𝑥 ) 𝑑𝑢

𝑑𝑥 )= 𝑓 (𝑥)