Numerical Analysis and Simulation

Code:

M4076

Acronym:

M4076

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 2S

Active?	Yes
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:

- Know and apply the fundamental methods of numerical linear algebra for linear systems, eigenvalues and least squares. Master the issues concerning convergence, conditioning and errors control, algorithms and computational implementation.

- 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).

- To understand and use important techniques in Computational Statistics.
To study computational methods used in Statistics with emphasis on large-scale computations and develop understanding and skills how to use these appropriately in research and applications

- 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), iterative methods (Jacobi and Gauss-Seidel). Eigenvalues: localization, power method and deflation, Householder triangularization, tridiagonal matrices, SVD. QR factorization and least squares. Conditioning.

Introduction to statistical simulation and computation. Regression: numerical aspects. 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

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)
Lars Elden; Matrix Methods in Data Mining and Pattern Recognition, SIAM, 2007. ISBN: 0898718864, 9780898718867
Monahan John F.; Numerical methods of statistics. ISBN: 978-0-521-13951-9 (Numerical methods of statistics / John F. Monahan)

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.

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	30,00
Teste	20,00
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 the evaluation components (exam, oral presentation, report)

Calculation formula of final grade

Final classification : 0.2 T + 0.30 O + 0.50 R,

T – Computational test

O – oral presentation + discussion

R- Report ( including the computacional part)

At ER the final exam (E) replaces T in the formula.

Minimum mark in each component T, E, O, or  R  is 40%. 

Any component not  concluded in the schedule  and/or established conditions is considered as not performed.

Examinations or Special Assignments

n.a.

Special assessment (TE, DA, ...)

n.a.

Classification improvement