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

2. Grading Policy 60%6 Homework, 10% each60%6 Homework, 10% each 10%Project Proposal10%Project Proposal 30%Project Presentation30%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, 2005Parameter Estimation and Inverse Problems, Aster et al., Elsevier, 2005 Computational Methods for Inverse Problems, Vogel, SIAM, 2002Computational Methods for Inverse Problems, Vogel, SIAM, 2002 Geophysical Inverse Theory, Parker, Princeton University Press, 1994Geophysical Inverse Theory, Parker, Princeton University Press, 1994 ii

4. Outline Introduction to inverse problemsIntroduction to inverse problems Mathematical background: Linear algebra, Functional analysisMathematical background: Linear algebra, Functional analysis Singular value decompositionSingular value decomposition Regularization methodsRegularization methods Iterative optimization methodsIterative optimization methods Methods for choosing regularization parametersMethods for choosing regularization parameters Additional regularization methodsAdditional regularization methods Nonlinear inverse problemsNonlinear inverse problems Bayesian inversionBayesian inversion 1

5. Introduction to Inverse Problems 2 Find tumors or cancers? How can we see internal organs without surgery? Use CT scan. What is CT scan and how does it work?

6. X-Ray Computed Tomography 3

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

8. Forward and Inverse Problems 5 Forward Problem: Given m. Find d. Inverse Problem: Given d. Find m.

9. Well-posedness vs. Ill-posedness 6

10. Classification of Inverse Problems 7

11. Examples 8