## Pictures from 15-859E: Hierarchical Methods for Simulation

Some pictures created in this course.

### N-Body Assignment

assignment handout

Nick Konidaris' 2-D n-body simulator on an orbital "galaxy" of 1000 particles around a central mass. The quadtree used by the Barnes-Hut algorithm for particle clustering is shown with gray lines.

### Multigrid Assignment

assignment handout

Greg Zornetzer's pseudocolor plot of the solution of the 2-D Poisson Equation: -Laplacian(u)=f, where f is positive at one point in the center, negative at one point in the lower left, and zero elsewhere (postscript). You can think of this as the electrostatic potential due to one positive charge and one negative charge.

German Cheung's plot of the solution to Poisson's equation where the right hand side has a constant positive value.

George Bulwinkle's plot of the running time of full multigrid V-cycle (FMV) and Gauss-Seidel relaxation running time versus number of unknowns for the 2-D Poisson Equation, on grids up to 2048^2. (postscript). Multigrid is much faster!

### Wavelet Assignment

assignment handout

Haar wavelet transform on 512x512 grayscale image, using nonstandard decomposition, by Paul Heckbert. Mid-gray represents zero, darker is negative and lighter is positive, following the formula pixel_value = 128+1024*coefficient. Original image.

Haar wavelet image compression with standard decomposition, by George Bulwinkle. Image compressed by keeping only 1% of the largest coefficients of the wavelet transform, and by quantizing those coefficients to 16 bits each before reconstruction. Relative error is 1.5%. The white spots are a small bug. Nonstandard decomposition works a little better, actually. This image can be compressed to 6% of its original size with little visible error using nonstandard Haar.

Haar wavelet image compression with nonstandard decomposition, by German Cheung. Compressed by keeping only .5% of the largest coefficients, quantizing to 8 bits each, and reconstructing. Relative error is 15%. Original image.

15-859E, Hierarchical Methods for Simulation
Paul Heckbert