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 | |
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| Week 1 |
Rabies in foxes |
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 |
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| Week 2 |
Project 1: Rabies in foxes |
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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 |
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| Week 3 |
Project 1 Rabies in foxes |
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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 |
Spectral Methods for Nonlinear Wave Equations Singular Value Decomposition for Ocean Temperature Data |
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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. |
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| Week 5 |
Project 3: Singular Value Decomposition for Ocean Temperature Data Magnetic induction equation in 2D, flux expulsion |
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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 |
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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 |
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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 |
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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. |
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| Week 9 |
Symplectic Integrators
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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 |
Numerical Integration of Stochastic Differential Equations Neural Networks |
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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? |
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| Week 11 |
Project 7: Numerical Integration of Stochastic Differential Equations Project 8: Neural Networks Boussinesq Thermal Convection |
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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 |
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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. |
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| Week 13 |
Project 5: Boussinesq Thermal Convection Project 9: Optimisation in a Trade Model |
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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 |