OVERVIEW

This course introduces students to the theory of wave propagation (i.e., direct scattering), both acoustic and electromagnetic, and to the resolution of nonlinear inverse scattering problems in biomedical applications. In the direct scattering problem, by means of information about the source of the wave and the so-called scatterer (i.e. the medium), the scattered wave has to be determined. In the inverse problem, from measurements of the scattered wave in an external domain, the sources and characteristics (imaging) of the scatterer are determined.

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to describe the modeling of sound waves in perfect fluids and of direct and inverse scattering problems.

AIMS AND LEARNING OUTCOMES

The course allows students to understand the basic mathematical tools for the solution of direct and inverse scattering problems, both acoustic and electromagnetic. Nonlinear imaging includes nonlinear problems of domain restoration by measurement of the transmitted wave away from the scatterer domain.

At the end of the course the student will have acquired sufficient theoretical knowledge:

• to identify and understand the main mathematical models of acoustic and electromagnetic wave propagation, on the basis of the involved physical models;

• to analyse and mathematically solve direct scattering problems, by computing analytical explicit solutions in the simplest cases, or approximated solution in more general contexts;

• to manage specific mathematical tools for solving wave propagation problems;

• to correctly treat the resolution of inverse problems associated with propagation models;

• to apply numerical-computational tools to direct and inverse scattering problems, useful for a subsequent resolution in real applications, with attention to ecography and tomography.

TEACHING METHODS

The teaching activity consists of traditional lectures, in which the subjects are introduced and explained in their classical theoretical setting. Although attendance is optional, it is strongly recommended.

SYLLABUS/CONTENT

• (Direct) scattering of acoustic waves:

Euler equations, D’Alembert (or wave) equation. Harmonic waves: Helmholtz equation. Sommerfeld condition. Foundamental solution. Helmholtz representation Theorems. Far field pattern.

• Resolution of the Helmholtz equation by single components analysis:

Spherical harmonic functions. Bessel, Neumann e Hankel spherical functions and their properties (outline). Rellich Lemma. Lippmann-Schwinger equation.

• Inverse scattering problem of acoustic waves:

Lippmann-Schwinger as nonlinear fixed point equation. Born approximations. Integro-differential formulation.

• Ultrasound imaging:

Mathematical model for the ecography and its implementation; diagnostic imaging for biological tissues by reflection of ultrasoud acoustic waves.

• Inverse scattering problem of electromagnetic waves:

Lippmann-Schwinger equation for per electrodynamics scattering. Far field and its properties. Eskin Theorem. Ill-posedness and linearization by the Fréchet derivative. Gauss-Newton approach.

• Qualitative methods for inverse scattering:

Linear sampling method and its implementation. Fundamental Theorem. Ill-posedness of the inverse problem. Tikhonov regularization. Tubes of energy flow.​​

RECOMMENDED READING/BIBLIOGRAPHY

In general, the notes taken during class lessons and some downloadable materials from the course web page are sufficient. Course handouts will be also given to the students. Any additional texts will be indicated during the lessons.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam is oral.

Students with DSA certification ("Specific Learning Disorders"), see section FURTHER INFORMATION.

ASSESSMENT METHODS

The oral exam focuses on the theory and applications, as well as on the discussion of applications and their examples.

FURTHER INFORMATION

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it