# SCATTERING PROBLEMS

OVERVIEW

This course introduces students to the theory of wave propagation, both acoustic and electromagnetic, and to the resolution of direct and inverse scattering problems. In the direct scattering problem, by means of information about the source of the wave and the so-called scatterer (that is, 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 of the scatterer are determined.

## AIMS AND CONTENT

LEARNING OUTCOMES

The course describes the physical-mathematical models for the propagation of acoustic and electromagnetic waves, and characterizes the problems of direct and inverse scattering associated with these models. For such problems, some methods of resolution, both analytical and numerical, are analyzed.

AIMS AND LEARNING OUTCOMES

The course allows students to understand the basic mathematical tools for the solution of direct and invers scattering problems, both acoustic and electromagnetic.

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 psychical 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.

Teaching methods

The teaching activity consists of traditional lectures, for a total of 48 hours, in which the subjects are introduced and explained in their classical theoretical setting.

Although attendance is optional, it is strongly recommended.

SYLLABUS/CONTENT

The program focuses on the following main topics:

**Propagation of acoustic waves** in perfect fluids, and subsequent introduction of the Euler equation on the mechanical quantities of pressure, velocity and density of the particles of the medium which the wave propagates in.

D’Alembert (or wave) equation (i.e., differential equation to partial derivatives, PDE), obtained from Euler equations by means of linearization.

Helmholtz equation (PDE): "DELTA u (x) + k ^ 2 (x) u (x) = 0", which characterizes all scattering problems, obtained as a simplification of the D'Alembert wave equation in the hypotheses of time-harmonicity.

Introduction and resolution of the (direct) scattering problem: to compute the (radiative propagating wave) solution u of the Helmholtz equation, for non-homogeneous scatterer, such as an obstacle, represented by the note function k ^ 2 (x), with specific boundary conditions.

Analytical resolution of the Helmholtz equation (explicit in particular cases with simplifying symmetries, and in general by series expansion of appropriate basic functions).

Introduction and resolution of the inverse scattering (ill-posed) problem: to determine information on the scatterer, i.e., k ^ 2 (x), by means of the measurement of the scattered wave u on some domain (or ny means of its effects far from the obstacle, called far field pattern).

Lipmann-Schwinger integral formulation of the Helmholtz differential equation, and its resolution.

**Study of electromagnetic scattering**: differential approach and integral approach, through appropriate characterization of the Helmholtz equation and of the Lipmann-Schwinger equation.

Born approximation of the Lipmann-Schwinger equation, resolution by fixed point techniques, and approximation in the framework of physical optics approximation.

Analysis of the ill-position of the inverse problem and regularization approaches.

Qualitative (Linear Sampling Method) and quantitative methods for the approximate solution of the inverse problem.

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

**Ricevimento:** By appointment via email.

**Ricevimento:** Office hours by appointment via email

Exam Board

CLAUDIO ESTATICO (President)

FEDERICO BENVENUTO

MICHELE PIANA (President Substitute)

## LESSONS

Teaching methods

The teaching activity consists of traditional lectures, for a total of 48 hours, in which the subjects are introduced and explained in their classical theoretical setting.

Although attendance is optional, it is strongly recommended.

LESSONS START

The class will start according to the academic calendar.

## EXAMS

Exam description

The exam is oral.

Assessment methods

The oral exam focuses on the theory, and aims to ascertain its understanding, also through the discussion of the analytical concepts and the examples.