Numerical Analysis and Simulation

Code:

M4076

Acronym:

M4076

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2019/2020 - 2S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=299
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master in Mathematical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:A_ASTR	6	Plano de Estudos oficial desde_2013/14	1	-	6	56	162
			2
M:ENM	16	Official Study Plan since 2013-2014	1	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

It is intended that the students learn the paradigm of computational simulation based on Monte Carlo methods, namely MCMC, as well as the principles of numerical linear algebra, in a framework of critical application as well as their application in interdisciplinary areas involving the social, life or computational sciences.

Learning outcomes and competences

The student should be able to:

- To Know and apply the fundamental methods of numerical linear algebra related to systems of equations, eigenvalues and least squares. Numerical Optimization Topics with application to Deep Learning.

- Know and apply the principles of generation of random variables and integration of Monte Carlo, with results analysis and control of the variance. Understand and apply Monte Carlo methods via Markov Chain (MCMC).

- Apply critically the studied methods to selected case studies of interdisciplinary areas involving the social, life or computational sciences.

Working method

Presencial

Program

Systems of Linear Equations: direct methods (LU factorization and Cholesky decomposition). Eigenvalues: symmetric or self-adjunct operators. Singular Value Decompostion (SVD). Application to the analysis of major componentse (PCA). Computational implementation in Octave and Python. Gradient descent, Stochastic gradient descent and application to Feedforward neural networks. Network backpropagation and optimization. Other neural network architectures.

Introduction to statistical simulation and computation. Comprehensive hands-on excursion of Monte Carlo methods: from random number generation algorithms and Monte Carlo integration, to Markov Chain Monte Carlo. Metropolis-Hastings and Gibbs algorithms, including convergence monitoring.

Mandatory literature

Golub Gene H.; Matrix computations. ISBN: 0-8018-5414-8
Trefethen Lloyd N.; Numerical linear algebra. ISBN: 0-89871-361-7
Kroese Dirk P.; Handbook of monte carlo methods. ISBN: 978-0-470-17793-8
Robert Christian P.; Introducing monte carlo methods with R. ISBN: 978-14419-1575-7

Complementary Bibliography

Brezinski Claude; Méthodes numériques directes de l.algèbre matricielle. ISBN: 2-7298-2246-1
Lange Kenneth; Numerical analisys for statisticians. ISBN: 0-387-94979-8 (2nd Edition 2010 available under Springer Link at FCUP)
Higham Desmond J.; Matlab guide. ISBN: 0-89871-469-9 (Matlab guide / Desmond J. Higham, Nicholas J. Higham, SIAM 2000)

Comments from the literature

Other Bibliography under Springer Link available at FCUP

Teaching methods and learning activities

Lectures TP organized in accordance with the syllabus and the intended outcomes to present and illustrate the topics. Problems / Projects with strong laboratorial computation component using (Matlab, R). The curricular unit has a strong practical component and classes with computers are essential. The computational projects allow the consolidation and critical application of the syllabus topics.

Note: 

e-learning due to restrictive measures caused by the COVID pandemic19.-

Software

Matlab
R

keywords

Physical sciences > Mathematics > Applied mathematics
Physical sciences > Mathematics > Applied mathematics > Numerical analysis
Physical sciences > Mathematics > Statistics

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Prova oral	40,00
Teste	10,00
Trabalho escrito	50,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	103,00
Frequência das aulas	56,00
Trabalho escrito	3,00
Total:	162,00

Eligibility for exams

40% in all evaluation components (exam, oral and written) in the simulation module.

At least 10 values in the numerical analysis module, obtained by submitting a report and its evaluation.

Calculation formula of final grade

 

Arithmetic mean of the classification in the 2 modules: numerical linear algebra (AN) and simulation (S). Final classification in the simulation module:  0.2T + 0.30 O + 0.50 R T - computational test O - oral exam (presentation and discussion) R- Report (Written work, including the computational component) At the time of appeal (ER) the exam (E) replaces the tests in the calculation formula. The classification of components O or R should not be less than 40%. Any component not carried out within the term and / or conditions established on the discipline's pages will be considered as not carried out. Final classification in the numerical linear algebra module: Individual report (Written work, including the computational component) quoted for 20 values. The presentation of the report is mandatory to obtain frequency.

Examinations or Special Assignments

n.a.

Special assessment (TE, DA, ...)

n.a.

Classification improvement

Only component E in the can be improved (ER) in the Simulation Module.

The improvement in the numerical linear algebra module will be done through the submission of an individual report.

Observations

Juri: Ana Paula Rocha, Zelia Rocha e João Nuno Tavares Teaching and on line synchronous assessment since 18 March Simulation - Classes finish  2  April Test  Simulation- 2 April  ER _____Calendar to be fixed by  CPFCUP ON-LINE and syncrhonous Zoom assessement, with possible on line  oral exam/discussion   Numerical Analysis Module - Distance learning with recorded sessions and synchronous sessions, via ZOOM. Beginning on April 15th and ending on May 28th. The last classes will be for presentation and oral discussion (by videoconference) of the individual works.