Computational Projects in Applied Mathematics Lecturer: Sheehan - - PowerPoint PPT Presentation

Computational Projects in Applied Mathematics

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Lecturer: Sheehan Olver Website: www.maths.usyd.edu.au/u/olver/teaching/Computation

COURSE TOPICS

• Topic 1: The fast Fourier transform and function approximation
• Topic 2: Spectral methods for differential equations
• Topic 3: Numerical linear algebra
• Advanced Topics in Fluid Dynamics next semester recommended for

computational topics on time-evolution PDEs

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Background knowledge not covered Please learn on your own (Wikipedia)

• Computer arithmetic

– Integers versus IEEE floating point numbers – Round-off error

• Relative accuracy versus absolute accuracy
• Programming language

– E.g., Julia, Matlab, Mathematica or Python – Fortran or C if you are a masochist

• Big-O, little-O and ~ notation

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Final project

• A final project will be due during exam weeks
• This project is open ended

– The idea is to come up with a computational problem on your own to study

• A proposal will be due 16 April

– For suggestions to ensure it is a good project – Every student’s project must be different, chosen on a first-come basis

• Example projects will be posted on the website

– You obviously can’t use the same idea for a project – Don’t copy projects from other students in previous years

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Intro to function approximation

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• The starting point of this course is that functions can be represented by (finite-

dimensional) vectors

• This corresponds to approximation in a basis
• For example, suppose f(x) is a function (say on [−1, 1]) and ψ0(x), ψ1(x), . . .

are fixed functions, and we can approximate (in some sense) f(x) ≈

n−1

• k=0

fkψk(x)

• Then we can represent the function f by the n-dimensional vector:

f can be represented by

• f0

. . . fn−1

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• A classic and fundamental example is Fourier series
• We take the basis

ψ0(θ) = 1 ψ1(θ) = θ, ψ2(θ) = θ, ψ3(θ) = 2θ, ψ4(θ) = 2θ, . . .

• I.e., we want to approximate on [−π, π) (here n is even):

f(θ) ≈ f0 + f1 θ + f2 θ + · · · + fn−1 n/2θ

Demo

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• We have seen the effectiveness of this approach for computations
• Now we want to understand the mathematics that makes this possible
• We begin with a basic fundamental question:

Given a function f(θ) defined on [−π, π), that we can evaluate pointwise Calculate coefficients f0, . . . , fn−1 so that (n even) f(θ) ≈ f0 + f1 θ + f2 θ + · · · + fn−1 n 2 θ

• To do this, we will need to approximate integrals: quadrature