**Important: there will be no class on Thursday September 7. There is an intro video to
replace it.**

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
- Green's functions, geometrical optics
- Scattering series and inversion
- The adjoint-state method: migration and backprojection
- Radar imaging, filtered backprojection, ambiguity and resolution
- Computerized tomography, Radon transform
- Seismic imaging, generalized Radon transforms, velocity analysis
- Microlocal analysis of imaging
- Optimization, regularization, sparsity, dimensionality reduction

Current version of the notes: here.

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 in 2013 at MSRI in Berkeley. Some of the material listed there may 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 3:00 pm to 4:30 pm in room 2-151. Instructor: Laurent Demanet. Contact info. Office hours: W 2-4, or else email.

The first class will be on Tuesday September 12.

The evaluation will consist in problem sets and a project. The project can be computational or theoretical; related to your research or not; or can consist in the 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. Propose a topic for your project to me by early October. Breakdown: 50% hw, 50% project. The presentations will be in class on Th Dec 07 and T Dec 12.

**Homework problems:**

- Pset 1: 5 stars (or more) worth of exercises. Recommended exercises: Chapter 1, exercises 1 or 2; 4; 6; 15 or 16. Due on Tu 09/26.
- Pset 2: 5 stars (or more) worth of exercises. Recommended exercises: Chapter 1, exercises 7 or 8, 12, 14, 16 (version of the notes: Sep 26 or later). Due on Th 10/12.
- Please don't forget to talk to me about your project sometime early to mid October.
- Pset 3: 5 stars (or more) worth of exercises. Recommended exercises: Chapter 3, exercises 2, 6, 8, 9. Due on Th 11/02.
- Pset 4: 5 stars (or more) worth of exercises. Recommended exercises: Chapter 4, exercise 6, and Chapter 6, exercise 2. Due on Th 11/30.

Some advanced papers, list in construction. Some may be adequate for a 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.