AMATH 483 / 583 (Roche) - Homework Set 8
Due Friday June 6, 5pm PT
May 30, 2025
Homework 8 (80 points)
1. (+20) Fourier transforms. Evaluate the Fourier transform. of the following functions by hand. Use the definitions I provided (includes this is common in physics but also now the default used in WolframAlpha - a powerful math AI tool) as well as the definition for Dirac delta I used if needed.
(a) f(x) =
(b) f(t) = sin(ω0t) , ω0 constant
(c) f(x) = e−a|x| and a > 0
(d) (distribution) f(t) = δ(t)
2. (+10) Correlation. By definition, correlation is and measures how similar one signal or data function is to another. Let p() = hpi + δp() and q() = hqi + δq(), where <> and δ() denote the mean values and fluctuation functions (deviations about the mean). Two functions are defined to be uncorrelated when Evaluate of the following functions:
3. (+10) Autocorrelation. Aside, periodic functions exhibit pronounced autocorrelations as shifting such functions by their period puts the function directly on itself. Alternatively, random functions or noise is characterized as being uncorrelated. Evaluate the autocorrelation of the following function:
4. (+20) Fourier transform. di↵usion equation solve. Consider the di↵usion equation where T(x, t) describes the temperature profile of a long metal rod.
(a) Assume you know T(x, 0) and define the Fourier transform. of T(x, t) to be (k, t). Transform. the original equation and initial conditions into k-space. Solve the resulting equation. Inverse transform. the result to obtain the solution in terms of the original variables.
(b) Find the temperature in the rod given initial conditions and
5. (+20) Compare FFTW to CUFFT on HYAK. Measure and plot the performance of calculating the gradient of a 3D double complex plane wave defined on cubic lattices of dimension n3 from 163 to n = 2563, stride n⇤ = 2 for both the FFTW and CUDA FFT (CUFFT) implementations on HYAK. Let each n be measured ntrial times and plot the average performance for each case versus n, ntrial ≥ 3. Submit your performance plot and C++ test code. Your plot will have ’flops’ on the y-axis (or some appropriate unit of FLOPs) and the dimension of the cubic lattices (n) on the x-axis. You will need to estimate the operation count of computing the derivative using FFT on a lattice.