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Inverse Problems and Applications

Inverse Problems and Applications. Chaiwoot Boonyasiriwat Last 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

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Inverse Problems and Applications

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  1. Inverse Problems and Applications Chaiwoot Boonyasiriwat Last modified on December 6, 2011

  2. 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

  3. 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

  4. 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 1

  5. 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? 2

  6. X-Ray Computed Tomography 3

  7. Inverse Problems in Physics Seismic tomography (1980s) Helioseismology (1990s) 4

  8. Forward and Inverse Problems where is data, is a model parameter, and is an operatorthat maps the model into the data . Forward Problem: Given m. Find d. Inverse Problem: Given d. Find m. 5

  9. Well-posednessvs. Ill-posedness (*) , is a continuous operator, The above problem is well posed if A has a continuous inverse operator from to . This means: Existence of solution: there exists , s.t. (*) is satisfied. Uniqueness of solution: there is no more than one satisfying (*). Stability of solution on data: If , . 6

  10. Classification of Inverse Problems 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. 7

  11. Examples • Linear regression or curve fitting • 1D steady-state diffusion equation 8

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