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School of Mathematics and Statistics

MAST30028 – Numerical Methods & Scientific Computing

2021


Description

Most mathematical problems arising from the physical sciences, engineering, life sciences and finance are suf-ficiently complicated to require computational methods for their solution. This subject introduces students to the process of numerical approximation and computer simulation, applied to simple and commonly encountered stochastic or deterministic models. An emphasis is on the development and implementation of algorithms for the solution of continuous problems including aspects of their efficiency, accuracy and stability. Topics covered will include simple stochastic simulation, direct methods for linear systems, data fitting of linear and nonlinear models, and time-stepping methods for initial value problems.


Learning Outcomes

On completion of this subject, students should:

i. Appreciate the role of computer simulation, as a third method in science, distinct from theory and exper-iment

ii. Understand the distinction between the simulation of stochastic and deterministic models

iii. Be able to use appropriate numerical techniques when undertaking a mathematical or modelling investi-gation.

iv. Be able to write simple numerical programs that utilise a numerical Problem-Solving Environment such as MATLAB or NumPy

v. Understand the significance and role of both roundoff error and truncation error in some standard problems in scientific computing


Classes

The subject has 2 lectures per week and 2 hours’ computer lab per week, taught by

● Dr Hailong Guo, hailong.guo@unimelb.edu.au, Room G19, Peter Hall building.


Prerequisites

One of (MAST20026 Real Analysis or MAST10009 Accelerated Mathematics 2) plus one of (MAST10007 Linear Algebra or MAST10008 Accelerated Mathematics 1) plus one of (COMP10001 Foundations of Computing or COMP20005 Engineering Computation or PHYC20013 Laboratory and Computational Physics 2 or other evidence of competence in computer programming). We assume students to be familiar with: vector spaces, eigenvalues, Taylor series, differential equations and the basics of computer programming.


Class times

● Lectures

– Mon 16:15–17:15, Biosciences 2-124 (Turner Theatre) or Recorded.

– Wed 12:00–13:00, Biosciences 2-124 (Turner Theatre) or Recorded.

● Computer Labs

1 of

– Tue 09:00–11:00, Peter Hall-G69 (Thompson Lab),

∗ Tutor: Rebecca Yuan Yin, Email: yin.r@unimelb.edu.au

– Tue 16:15–18:15, Online Delivery via ZOOM,

∗ Tutor: Hongyi Zhu, Email: hozhu3@unimelb.edu.au

– Wed 14:15–16:15, Peter Hall-G69 (Thompson Lab),

∗ Tutor: Hailong Guo, hailong.guo@unimelb.edu.au

– Thru 14:15–16:15, Online Delivery via ZOOM

∗ Tutor: Shaks Khan, Email: shaks.khan@unimelb.edu.au


Consultation hours

see the Teaching Staff page on the LMS


Textbooks

The recommended textbook is Numerical Computing with MATLAB, SIAM, 2004 by Moler (the developer of MATLAB). You may purchase it or (legally) download the relevant parts, chapters 2, 4, 5 and 7, from the website: http://au.mathworks.com/moler/chapters.html


Weekly schedule

1. Scientific Computing Overview. Array programming in MATLAB.

2. Programming and plotting in MATLAB/NumPy

3. Computer Simulation: probabilistic models, pseudorandom numbers, statistical errors

4. Floating point numbers and roundoff error

5. Root-finding and discretization error

6. Linear systems: GE, LU, pivoting

7. Linear systems: special matrices, conditioning and backward error

8. Data Fitting: linear models, least squares methods

9. Data Fitting: nonlinear model fitting via nonlinear least squares

10. Initial Value problems: first order methods

11. Initial Value problems: higher order and variable-step methods

12. Good software practice: coding style, debugging, testing


Assessment:

Two assignments, worth 20% each (due during semester) and 60% for final assessment (The form of final assessment is yet to be determined).


Assignments

There are two of these, due in roughly Weeks 6 and 10. Submit electronic versions through the LMS.

        It is necessary for you to do programming to complete the assignments, but writing programs is NOT the main aim of the subject. Interpretation of results is a very important part of the assignments.

Late assignments have 100% penalty i.e. are awarded no marks. In answering the assignments, write neatly and explain yourself clearly. A plagiarism form must be submitted for each assignment and assessment.


Programming

In this subject you will be required to write (relatively simple) programs in the software system MATLAB, primarily a numerical computing environment. I will also briefly introduce a programming environment for NumPy, an open source alternative that is becoming increasingly popular.

        We expect many students will have some exposure to MATLAB through MAST10007 Linear Algebra or En-gineering Systems Design subjects but you will need to become more active users. Numerical tasks are to be done in MATLAB. The University has a site license for MATLAB, so it should be available in every University lab. A student version of MATLAB can be purchased online for AUD59, from https://au.mathworks.com, if required.

        Note that the examination will require students to produce answers in a computer laboratory.


Website

Lab sheets, this outline and assignments will also be available from the subject homepage, hosted on the University LMS. We will also provide useful links to various sites relevant to scientific computing throughout the semester.