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## Subject: NUMERICAL METHODS (A.A. 2020/2021)

### master degree course in MATHEMATICS

Course year 1 12 Unit Problemi inversi e applicazioni Models and Application (lesson) TAF: Compulsory subjects, characteristic of the class SSD: MAT/08 CFU: 6 Teachers: Marco PRATO, Federica PORTA Unit Metodi numerici per le equazioni alle derivate parziali Models and Application (lesson) TAF: Compulsory subjects, characteristic of the class SSD: MAT/08 CFU: 6 Teachers: Daniele FUNARO oral final vote Italiano
Contents download ### Overview

The course is splitted in the two modules

1. Numerical methods for partial differential equations
2. Inverse problems and applications

whose aim is to introduce some basic methods for the solution of partial differential equations and linear inverse problems, respectively, to analyze their performances and to implement these methods within the Matlab software environment.

Elements of numerical analysis: numerical integration, numerical methods for solving linear and nonlinear equations, approximation techniques for ordinary differential equations.
Elements of functional analysis: Hilbert spaces, bounded operators.
Basics of Matlab programming.

### Course contents

Module "Numerical methods for partial differential equations"

Numerical approximation of partial differential equations of elliptic, parabolic and hyperbolic type, using different techniques including the finite element method.

Module "Inverse problems and applications"

Linear inverse problems: ill-posedness, ill-conditioning, pseudosolutions, generalized solution and generalized inverse operator.
Linear inverse problems with discrete data.
Regularization algorithms: TSVD, Tikhonov, Landweber. Choice of the regularization parameter: Morozov discrepancy principle, generalized cross validation, L-curve.
Application to the image reconstruction problem: statistical formulation, Maximum Likelihood and Maximum a Posteriori approaches, optimization algorithms, the Computed Tomography case.

### Teaching methods

- Lectures in the classroom, with illustration of the content chapters by means of slides and blackboard. - Laboratory exercises on the solution of the problems described during the lectures, for the practical verification of structured concepts that are the backbone of the course program. Remote attendance will be guaranteed; on the basis of the evolution of the COVID19 health emergency and the specificities of the educational activities, frontal lessons in presence will be taken into account.

### Assessment methods

Examination: oral exam. The candidate must demonstrate a thorough knowledge of: - the course content and teaching training, including both the institutional part, i.e. frontal lessons/classroom activities, and the laboratory practices; Module "Numerical methods for partial differential equations" - the different numerical approximation techniques of elliptic, parabolic and hyperbolic partial differential equations; - the variational formulation of the problems and the Galerkin method; - the finite element method in one or several dimensions; - non symmetrical problems. Module "Inverse problems and applications" - the definition of inverse problems, ill-posedness and ill-conditioning, pseudosolutions, generalized solution, generalized inverse operator; - the dissertation of a linear inverse problem in the presence of discrete data; - the theoretical features of the regularization algorithms presented and of the criteria for the choice of the regularization parameter; - the image reconstruction problem, the algorithms ISRA, EM, SGM and SGP and the Computed Tomography application. The verification is integral with respect to the course's content; it is also verified the student's ability to relate specific subject content with the knowledge listed as pre-requisite. The oral exam consists in the implementation of one of the algorithms analyzed during the class in a Matlab environment and in the in-depth analysis of some topics treated during the lectures. The score of the oral exam, in a scale of thirty, is divided in: 5 points for the communicative skills; 5 points for multidisciplinary skills; 20 points for the knowledge of the contents.

### Learning outcomes

Applying knowledge and understanding:
At the end of the course the student will have adequate knowledge to face some physical and mathematical problems arising from real applications.

Making judgements:
At the end of the course the student must be able to choose the appropriate methods for a specific partial differential equation or inverse problem.

Communicating skills:
At the end of the course the student must be able to explain the analyzed methods in a clear and rigorous way and to discuss their efficiency.

Learning skills:
At the end of the course the student must be able to go deep by himself into the main aspects of the
subjects proposed in the course.