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Undergraduate Study

MATH4411 Applied Computational Mathematics

This page contains information on the Honours Mathematics unit Applied Computational Mathematics.

Lecturer for this course: David Ivers.

For general information on honours in the School of Mathematics and Statistics, refer to the relevant honours handbook.

For enrolled students or other authorized people only, here is a link to the Canvas page for MATH4411.

Resources

The information sheet outlines the course, projects for assessment and guidelines for writing up project reports, and their due dates.

Project Lecture notes Matlab codes Material covered
Week 1

Project 1

Rabies in foxes

Lecture 1

Lecture 2

Lecture 3, IVP's

Course outline, projects and project reports; Project 1 problem formulation

Project 1 problem formulation (cont)

IVP's and time-stepping; Adams-Bashforth, Adams-Moulton and Gear methods

Week 2

Project 1: Rabies in foxes

Lecture 4

Lecture 5, FD's

Lecture 6

 

ftcs_dji.m, ftcs_dat, annimate.m

rabies.dat

Adams-Bashforth, Adams-Moulton and Gear methods (cont); Finite-differences

Finite-differences (cont); FTCS scheme for heat equation

Method of lines, numerical scheme for Q1

Week 3

Project 1

Rabies in foxes

Lecture 7

Lecture 8, Interpolation

 

 

Lecture 9

 

 

 

 

ode45_dji.m, rhs45_dji.m

AB2 Matlab implementation; Runge-Kutta methods

Second-order CTCS (leap-frogging scheme); FTCS heat equation stability

CTCS heat equation stability; Q2 steady-state solution stability; Q5 ode45; Interpolation

Week 4

 

Project 2

Spectral Methods for Nonlinear Wave Equations

Project 3

Singular Value Decomposition for Ocean Temperature Data

Lecture 10

 

 

Lecture 11

 

Lecture 12

 

 

 

 

 

svd_image.m, temperature.dat

Interpolation (cont); AB2 local truncation error derivation

Project 2 problem formulation; numerical scheme

Project 3 problem formulation

Weeks 5-13

Lectures will be given online using zoom on the Math4411 Canvas webpage at the scheduled lecture times.

Lecture recordings from Lecture 16 and later will be available on Canvas.

Lecture notes will still be available on this webpage.

Online labs are Fri 1pm on zoom. Email me if you have any difficulties with programming, etc, etc. You should have access to the University Matlab licence on your laptop.

Project 1 is now due noon Wednesday 8 April.

Week 5

Project 3: Singular Value Decomposition for Ocean Temperature Data

Project 4

Magnetic induction equation in 2D, flux expulsion

Lecture 13

 

 

Lecture 14

 

Lecture 15

 

 

Project 3 (cont): problem formulation, SVD; Project 4: problem formulation

Project 1: comments on ode45; Project 4: (cont) background

Project 4 (cont): formulation; numerical scheme, finite-volume method

Week 6

Project 4

Project 3

Lecture 16

 

 

Lecture 17

 

Lecture 18

Lecture 18 (alt)

 

 

Project 4 (cont): spatial discretisation; conservative scheme

Project 4 (cont): time=stepping; DuFort-Frankel leapfrog scheme

Project 3 (cont): SVD derivation

Week 7

Project 3

 

 

 

Project 2

Lecture 19

Lecture 19 (alt)

 

Lecture 20

 

Lecture 21

Lecture 21 (alt)

 

 

Project 3 (cont): SVD derivation

SVD for a linear operator, core surface operator

Project 2 (cont): discrete Fourier series and transform

Week 8

Project 3

 

 

 

Project 2

Lecture 22

 

 

Lecture 23

 

Lecture 24

 

 

Project 3 (cont): svd(A) vs A*A’ and A‘*A

zero singular and eigen values; Project 2 (cont): numerical scheme, fit, iffy and shifts

Project 2 (cont): discrete Fourier series and transform

Weeks 9-13

There are problems with permissions on lecture recordings from Lecture 16 on Canvas. I am trying to fix this.

Two algorithms on Projects 2, 3 or 4 (not the reported second project) is now due noon Wednesday 13 May:

Note correction to Lecture 11 on Project 2.

Second report on Projects 2, 3 or 4 is now due noon Wednesday 13 May.

Week 9

Project 6

Symplectic Integrators

 

Lecture 25

 

Lecture 26

 

Lecture 27

 

 

 

Project 6: Geometric integration, Hamiltonian systems, symplectic maps

symplectic maps (cont), simple harmonic oscillator

simple harmonic oscillator (cont), Euler's method, symplectic numerical integrators

Week 10

Project 7

Numerical Integration of Stochastic Differential Equations

Project 8

Neural Networks

Lecture 28

 

Lecture 29 - note correction

 

Lecture 30

Solar system data

 

 

 

Project 6 (cont): symplectic maps, symplectic integrators

symplectic integrators (cont)

Project 2 (cont) comments; Project 7: background, Wiener processes and numerical simulation

Weeks 11-13

Second report on Projects 2, 3 or 4 is now due and algorithms on Projects 2, 3 or 4 is due noon Wednesday 13 May via turnitin.

There is no need to email the files to me.

 

Note comment on Project 2 Q3: The method solves a periodic extension u of a truncation of the true solution

The periodicity means (i) that boundary fluxes of conserved quantities exactly cancel, which is inherent to the numerical method not the conserved quantity;

and (ii) that conserved quantities give information about the Fourier coefficients of u.

How much information does the "mass" give about the Fourier coefficients of u?

Week 11

Project 7: Numerical Integration of Stochastic Differential Equations

Project 8: Neural Networks

Project 5

Boussinesq Thermal Convection

Lecture 31

 

Lecture 32

 

Lecture 33

horses.dat

 

 

 

Project 7 (cont): Ito's formula; Euler-Maruyama, Milstein stochastic numerical schemes

Milstein scheme (cont); strong and weak convergence; Project 8: neural networks, foundations, modelling

Project 8 (cont) Hebb's rule, Hopfield model; feedforward networks, 2-layer networks

Week 12

Project 8: Neural Networks

Project 5: Boussinesq Thermal Convection

Lecture 34

 

Lecture 35

 

Lecture 36

Project 5 benchmark

 

Project 5 incomplete m-file

 

Project 8 (cont): gradient descent algorithm; Project 5: background

Project 5 (cont): equations for mass, forces and momentum, temperature

Project 5 (cont): Boussinesq approximation; solving equations

Weeks 13 - end

Third report on Projects 5, 6, 7 or 8 is now due noon Wednesday 10 June via turnitin.

Algorithms for the other projects are NOT required.

The third report is now worth 33 marks.

Week 13

Project 5: Boussinesq Thermal Convection

Project 9: Optimisation in a Trade Model

Lecture 37

 

Lecture 38

 

Lecture 39

 

 

 

 

Project 5 (cont): spectral expansions/equations/nonlinear terms in x

Project 5 (cont): linear stabiity problem; solution of Poisson's equation, Matlab code. Project 9: Ricardo's model, absolute & comparative advantage

Project 9 (cont): Gomory's Ricardo model with economies of scale, Cobb-Douglas utility, integer equilibria, upper bound on integer equilibria utilities