Instructor: Hai Zhang
Office: 3442
Email: haizhang@ust.hk
Office Hours: Tuesday and Thursday 3:30PM—4:30PM
Lectures:
Tuesday and Thursday,
2PM-3: 20PM, G001, CYT Bldg
Main references:
Wave Propagation and Time reversal in Randomly layered Media, J P Fouque,
et.al, Springer, 2007;
Mathematical and Statistical Methods for Multistatic
Imaging, H. Ammari, et.al, Springer, 2013;
Acoustic and Electromagnetic
Equations-Integral Representations for Harmonic Problems, Jean-Claude Nédélec, Springer, 2001;
Prerequisite: College Calculus, Linear algebra,
elementary partial differential equations
Course Description and
Objectives:
Waves are ubiquitous.
They propagate in various media and interact with inhomogeneities therein. The
study of how the presence of inhomogeneity affects the propagation of waves is
called scattering theory, while the study of how to infer useful information
about the inhomogeneities from the perturbed waves is call inverse scattering
theory. The latter lies in the heart of many applications such as bio-medical
imaging, nondestructive testing and geophysics et al. In this course, we aim to
provide an introduction to both scattering and inverse scattering theory with
emphasise on the latter. Other inverse problems and general methods to solve
them will also be briefly introduced. We shall focus on the important ideas and
attempt to avoid technicalities. The following topics are intended to be
covered:
1.
Acoustic waves in homogenous media
1.1
Derivation of accoustic wave equations
1.2
1D wave
1.3
3D waves
1.4
3D wave decomposition
1.5
WelyÕs representation: from spherical waves to plane waves
2.
Wave propagation and scattering in layered media
2.1 Waves in general 1D media
and decomposition
2.2 Wave scattering by a single
interface
2.3 Wave scattering by a single
layer
2.4 Wave scattering by a
multi-layers
3.
Inverse scattering problems for waves in 1D
3.1
Formulation of the problems
3.2
Mathematical analysis and theorems
3.3
Reconstruction algorithm
4.
A general framework to solve inverse problems
4.1
ill-posedness of inverse
problems
4.2
General regularization
methods
4.3 Singular value
decomposition
4.4 Classic
regularization methods
5.
Wave scattering in 2D and 3D
5.1
Representation formulas
5.2
Layer potentials for Laplace equations
5.3
Layer potentials for Helmholtz equations
5.4
Boundary integral formulation of the scattering problems
5.5
The Helmholtz-Kirchhoff theorem
6.
Invese scattering problems in 2D and 3D
6.1
Several inverse scattering problems
6.2
Theoretical results
7 Imaging small inclusions in 2D
and 3D
7.1 Small
volume expansion for conductivity problem
7.2 Small
volume expansion for Helmholtz problems
7.3 Imaging
methods for the conductivity problems
7.4 Imaging
methods for the Helmholtz problems
8 Imaging of extended targets
8.1 Time
reversal imaging method
8.2 Diffraction
tomography by Born approximation
Grading Scheme:
Final Presentation 80%, Attendance 20%, No exams
Final Presentation: at the end of the course, each student will be
assigned to one research paper and are asked to present the results therein.
Tentative schedule
Lecture 01, Sep 01: Course overview and Section
1.1
Lecture 02, Sep 06: Section 1.2
Lecture 03, Sep 08: Section 1.3, 1.4
Lecture 04, Sep 13: Section 1.5, 2.1
Lecture 05, Sep 15: Section 2.2, 2.3
Lecture 06, Sep 20: Section 2.3
Lecture 07, Sep 22: Section 2.4
Lecture 08, Sep 27: Section 2.4, 3.1
Lecture 09, Sep 29: Section 3.2, 3.3
Lecture 10, Oct 04: Section 4.1, 4.2
Lecture 11, Oct 06: Section 4.3, 4.4
Lecture 12, Oct 11: Section 4.4
Lecture 13, Oct 13: Section 5.1
Lecture 14, Oct 18: Section 5.2
Lecture 15, Oct 20: Section 5.3, 5.4
Lecture 16, Oct 25: Section 5.4, 5.5
Lecture 17, Oct 27: Section 6.1
Lecture 18, Nov 01: Section 6.2
Lecture 19, Nov 03: Section 7.1
Lecture 20, Nov 08: Section 7.2
Lecture 21, Nov 10: Section 7.3
Lecture 22, Nov 15: Section 7.3, 7.4
Lecture 23, Nov 17: Section 7.4
Lecture 24, Nov 22: Section 8.1
Lecture 25, Nov 24: Section 8.2
Lecture 26, Nov 29: Final Overview