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APPLIED MATHEMATICS (519SM)

A.Y. 2020 / 2021

Period 
First semester
Credits 
6
Duration/Length 
48
Type of Learning Activity 
Programme specific subjects
Study Path 
[PDS0-2018 - Ord. 2018] common
Syllabus 
Teaching language 

English

Learning objectives 

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.

Prerequisites 

Mathematical Analysis, linear algebra, basic knowledge of information
technologies

Contents 

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

Teaching format 

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.

Extended Programme 

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

End-of-course test 

Oral examination + Python project

Other information 

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. **********************************************************************

Texts/Books 

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]


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