Numerical Methods and Optimisation
Code: 104848 ECTS Credits: 6| Degree | Type | Year | Semester |
|---|---|---|---|
| 2503852 Applied Statistics | FB | 2 | 1 |
Contact
- Name:
- Joan Torregrosa Arus
- Email:
- joan.torregrosa@uab.cat
Teaching groups languages
You can check it through this link. To consult the language you will need to enter the CODE of the subject. Please note that this information is provisional until 30 November 2023.
Prerequisites
It is recommended to have passed the following courses: Àlgebra Lineal, Càlcul 1 and Càlcul 2.
Objectives and Contextualisation
This course will provide students the basic numerical methods to solve real problems which arise from science and mainly from applied statistics.
The purpose of the course is that the students learn the mathematical foundations of the methods, their range of applicability and the type of errors that should be expected. The student should also be able to recognize the problems whose solution requires the use of a numerical method, and to apply a proper method to get an approximate solution in an efficient way.
The student shoud also be able not only to use some programming languages (Maxima, R,...) to implement and test simple algorithms, but to work with the functions provided by the correspondig software.
Competences
- Calculate and reproduce certain mathematical routines and processes with agility.
- Critically and rigorously assess one's own work as well as that of others.
- Make efficient use of the literature and digital resources to obtain information.
- Select and apply the most suitable procedures for statistical modelling and analysis of complex data.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Use quality criteria to critically assess the work done.
- Use software for statistical analysis, numerical and symbolic analysis, graphic visualisation, optimisation or others, to solve problems.
Learning Outcomes
- Calculate and study extrema of functions.
- Compare the respective advantages and disadvantages of analytic methods and numerical methods.
- Critically assess the work done on the basis of quality criteria.
- Make effective use of references and electronic resources to obtain information.
- Master the basic language and tools of linear algebra.
- Reappraise one's own ideas and those of others through rigorous, critical reflection.
- Recognise the usefulness of mathematical methods (calculus, algebra, numerical methods) for optimisation.
- Select and use suitable software to solve problems in algebra, calculus and numerical calculation.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Use numerical methods to solve problems in algebra and calculus.
Content
1. Errors
Floating point arithmetic. Propagation of errors.
Conditioning of a problem.
2. Numerical Linear Algebra
LU decomposition. Perturbation analysis.
QR decomposition. Applications.
Singular value decomposition. Applications.
3. Numerical Solution of Nonlinear Equations
One variable equations: Fixed point methods. Newton-Raphson's method.
Methods for systems of nonlinear equations.
4. Polynomial interpolation
Lagrange polynomial. Divided differences.
Error estimate.
5. Unconstrained Optimitzation
One dimensional minimization.
Line search methods, gradient, Newton.
Methods without derivatives.
6. Constrained Optimitzation
The penalty method.
Augmented Lagrangian method.
7. Numerical Integration
Trapezoidal and Simpson's rules. Monte Carlo method.
Methodology
In the theoretical lectures the teacher will explain the mathematical foundations and basic properties of the numerical methods and will present several illustrative examples.
Different lists of exercises will be proposed so that the student can practice and learn the contents of each topic. In the problem lectures the teacher will work on the lists of exercises, will solve the doubts of the students and will discuss and solve the exercises.
Each computer session will have a script associated. In the computer sessions the student will do the work proposed in the correspondig script under the supervison of the teacher. It is convenient that before the session the student reads carefully the script in order to know the goal of the computer session and the numerical methods to be used. The student must attend the computer sessions.
All the course material will be posted on the Virtual Campus.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Activities
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Problems | 14 | 0.56 | 6, 3, 1, 5, 10, 9, 7, 4, 11 |
| Theory | 26 | 1.04 | 6, 3, 1, 5, 10, 9, 7, 4, 11 |
| Type: Supervised | |||
| Computer sessions | 12 | 0.48 | 6, 3, 1, 2, 8, 10, 9, 11 |
| Type: Autonomous | |||
| Computer work | 21 | 0.84 | 6, 3, 1, 2, 8, 9, 7, 4, 11 |
| Exercises | 35 | 1.4 | 6, 3, 1, 5, 9, 7, 4, 11 |
| Study | 32 | 1.28 | 6, 3, 1, 5, 10, 9, 7, 4, 11 |
Assessment
See the Catalan version.
Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Computer work | 20% | 2 | 0.08 | 3, 1, 2, 8, 10, 7, 4, 11 |
| Final exam | 50% | 3 | 0.12 | 6, 1, 2, 5, 10, 9, 11 |
| Mid-term exam | 30% | 2 | 0.08 | 6, 1, 2, 5, 10, 9, 11 |
| Recovery Exam | 80% | 3 | 0.12 | 6, 1, 2, 5, 10, 9, 11 |
Bibliography
See the Catalan version.
Software
See the Catalan version.