English

KNOWLEDGE AND UNDERSTANDING

At the end of the course the student should be able to prove to know the

fundamental results of numerical analysis, and to know the basics of the

python programming language.

CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING

At the end of the course the student should be able to apply the acquired

knowledge of numerical analysis to solve problems and exercises in

applied mathematics, both at the theoretical level and through the use of

a computer.

JUDGMENT AUTONOMY

At the end of the course the student should be able to) recognize and

apply the most basic techniques of numerical analysis (interpolation,

integration, numerical solution of ODEs, numerical solution of non-linear

equations and non-linear systems, and numerical solution of PDEs and

will also recognize the situations and problems in which these techniques

can be advantageously used (simple models from physics and other

disciplines).

COMMUNICATIVE SKILLS

At the end of the course the student should be able to express

themselves appropriately on the themes of numerical analysis, with clear

language and exposure security.

LEARNING CAPACITY

At the end of the course the student should be able to consult numerical

analysis and computer programming manuals.

Mathematical Analysis, linear algebra, basic knowledge of information

technologies

Introduction to Numerical Analysis and Scientific Computing

Basic concepts (interpolation, integration, solution of linear and nonlinear

systems, numerical solution of ODEs and numerical solution of

PDEs)

For each introduced concept, a corresponding laboratory will attack the

same problem from the practical point of view, presenting

implementational aspects in python

Several examples of mathematical modeling will be used to introduce the

concetps and provide a modeling motivation for the presented topic

Frontal lectures and laboratories, with examples taken from Modern

Mathematical Modeling. In particular we will see the topics of basic

numerical analysis in

Data assimilation in biomechanics, statistics, medicine, electric signals

Model order reduction of matrices

Linear models for hydraulics, networks, logistics

State equations (real gases), applied mechanics systems, grow

population models, financial problems

Applications of ODEs

examples in electric phenomena, signals and dynamics of populations

(Lotke-Volterra)

Models for prey-predator, population dynamics, automatic controls

Applications of PDEs, the poisson problem

- Elastic rope

- Bar under traction

- Heat conductivity

- Maxwell equation

Laboratories will be done in python, using numpy and scipy as the core

libraries.

Frontal Lectures

Basic concepts of Vector spaces and norms

Well posedness, condition numbers, Lax Richtmyer theorem

Polynomial based approximations (Lagrange interpolation, Bernstein

polynomials, Bsplines approximations)

Quadrature rules and orthogonal polynomials

Solution methods for Linear Systems: direct, iterative and least square

methods

Eigenvalues/Eigenvectors

Solution methods for non-Linear systems

Review of ODEs

Review of FEM/Lax Milgram Lemma/Cea’s Lemma/Error estimates

Python Laboratories

Introduction to Python, Numpy, Scipy

Exercises on Condition numbers, interpolation, quadratures

Using numpy for polynomial approximation

Using numpy for numerical integration

Using numpy/scipy for ODEs

Working with numpy arrays, matrices and nd-arrays

Solving non-linear systems of equations

Solving PDEs using Finite Difference

Solving PDEs using Finite Element: 1D case

From one dimensional FEM to N-dimensional exploiting tensor structure

of certain finite elements

Oral examination + Python project

http://www.math.sissa.it/education/2/courses/current

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Any changes to the methods described here, which become necessary to ensure the application of the safety protocols related to the COVID19 emergency, will be communicated on the websites of the Department of Mathematics and Geoscience - DMG and of the Study Program in Mathematics. **********************************************************************

A. Quarteroni, R. Sacco, and F. Saleri. Numerical mathematics, volume 37

of Texts in Applied Mathe- matics. Springer-Verlag, New York, 2000.

[E-Book-ITA] [E-Book-ENG]

A. Quarteroni. Modellistica Numerica per problemi differenziali. Springer,

2008.

[E-Book-ITA]

A. Quarteroni. Numerical Models for Differential Problems. Springer, 2009.

[E-Book-ENG]

A. Quarteroni and A. Valli. Numerical approximation of partial differential

equations. Springer Verlag, 2008.

[E-Book-ENG]

S. Brenner and L. Scott. The mathematical theory of finite element

methods. Springer Verlag, 2008.

[E-Book-ENG]

D. Boffi, F. Brezzi, L. Demkowicz, R. Durán, R. Falk, and M. Fortin. Mixed

finite elements, compatibility conditions, and applications. Lectures given

at the C.I.M.E. Summer School held in Cetraro, Italy June 26–July 1, 2006.

Springer Verlag, 2008.

[E-Book-ENG]

D. Arnold. A concise introduction to numerical analysis. Institute for Mathematics and its Applications, Minneapolis, 2001.

[E-Book-ENG]

A. Quarteroni, F. Saleri, P. Gervasio. Scientific Computing with Matlab and

Octave. Springer Verlag, 2006.

[E-Book-ENG]

B. Gustaffson Fundamentals of Scientific Computing, Springer, 2011

[E-Book-ENG]

Tveito, A., Langtangen, H.P., Nielsen, B.F., Cai, X. Elements of Scientific

Computing, Springer, 2010

[E-Book-ENG]