This class covers the mathematics of inverse problems involving waves, with examples taken from reflection seismology, synthetic aperture radar, and computerized tomography. The course is suitable for graduate students from all departments who have affinities with applied mathematics.

**Topics:**

- Acoustic, elastic, electromagnetic wave equations (09/11 to 09/20)
- Scattering series and inversion (09/25 and 09/27)
- Migration and backprojection: adjoint-state methods (10/02 to 10/11)
- Radar imaging, filtered backprojection, ambiguity and resolution (10/16 to 10/25)
- Computerized tomography, Radon transform (10/30 and 1/01)
- Seismic imaging, geometrical optics, generalized Radon transform (11/06 to 11/22)
- Optimization, regularization, velocity estimation, autofocus (11/27 to 12/04)

Current version of the notes: here.
Old versions of the notes: Sep 11,Sep 13, Sep 18, Oct 1, Oct 3, Oct 10, Oct 16,
Oct 20,Nov 11,
Nov 24, Nov 25, Dec 4.

** For future reference, the most current version of the notes will be at the top of the resources page.**

There is not one textbook. The material will be inspired from various sources. Here is a list of references that sometimes go way beyond what we'll do in class.

- For acoustics, see the book "Fundamental of acoustics" by Kinsler, Frey, Coppens, and Sanders.
- For continuum mechanics, a decent (cheap!) book is "Continuum mechanics" by A. J. M. Spencer, in the Dover collection, but if you know of a better reference I'll be happy to list it here.
- For optics, Born and Wolf, Principles of Optics, is the bible.
- For radar imaging, see the review article by Margaret Cheney as well as her book with Brett Borden on Fundamentals of radar imaging (published by SIAM, 2009).
- For computerized tomography, see the book "Mathematical Methods in Image Reconstruction" by Natterer and Wubbeling
- For seismology, see the review notes "Mathematics of reflection seismology" by Bill Symes (find it on google scholar).
- For the introductory treatment of wave equations, I've used "Partial differential equations" by L.C. Evans, "Introduction to partial differential equations" by G. Folland, "Partial differential equations" by F. John, "Linear and nonlinear waves" by G. Whitham, and also the "Notes on the algebraic structure of wave equations" by Steven Johnson.
- For geometrical optics, I like "Lectures on geometrical optics" by J. Rauch.
- We gave a summer school on Waves and Imaging last year in Vancouver. Some of the material listed there will be useful as well.

**Prerequisites:** Some undergraduate familiarity with partial
differential equations, Fourier transforms, distributions (the Dirac
delta), linear algebra and least squares, as well as some basic
physics. Basic computer programming.

We will meet T-Th from 2:30 pm to 4 pm in room 2-136. Instructor: Laurent Demanet. Contact info. Office hours: W 2-4, or else email.

The first class will be on Thursday September 6.

The evaluation will consist in occasional problem sets and an oral presentation of a good (landmark, foundational) paper from the literature. Talk to your advisor or to me if you'd like a recommendation of a good paper. Breakdown: 70% hw, 30% presentation. The lowest hw score will be dropped. The presentations will be on December 6 and 11, in class.

**Homework problems:**

- Pset 1: solve exercises 2, 3, 7, 8 in the exercises section at the end of chapter 1. Due Th 10/04 in class or in the box outside my office.
- Pset 2: exercises 2, 5, 7 in the exercises section at the end of chapter 3. Due Th 10/18.
- Pset 3: wording, dataset. Due Tu 11/27.
- Pset 4: exercises 4 in chapter 4, 1 in chapter 5, and 2 in chapter 2. Due Tu Dec 4.

Some advanced papers, list in construction. Some may be adequate for the term paper presentation: consult with me first.

- R. Lewis and W. Symes, On the relation between the velocity coefficient and boundary value for solutions of the one-dimensional wave equation. Inverse Problems 7 (1991) 597-631.
- Gregory Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform, J. Math. Phys. 26, 99 (1985); doi:10.1063/1.526755
- A. P. E. ten Kroode, , D. -J. Smit and A. R. Verdel, A microlocal analysis of migration, Wave Motion, Volume 28, Issue 2, September 1998, Pages 149-172
- Alfred M. Bruckstein, Bernard C. Levy and Thomas Kailath, Differential Methods in Inverse Scattering, SIAM Journal on Applied Mathematics, Vol. 45, No. 2 (Apr., 1985), pp. 312-335
- Gelfand, I. M. and Levitan, B. M. (1951). On the determination of a differential equation from its spectral function, Amer. Math. Transl. 1(2), 239253.