Assignment 'Volume of d-dimensional unit sphere'

You will use Monte Carlo experiments to numerically compute the volume of the d-dimensional unit sphere: This is fully described in Exercise 3.8 in the course book. This page gives you some help on the C++ code that is readily available.

1D Monte Carlo integration

We start with a short guideline how to get the code up and running for the one-dimensional case. You will run these first few Monte Carlo experiments as suggested in Exercise 3.4 in the course book. Follow these steps:

• Open the mc1d.cc file.
• Start programming the hit-or-miss and sample-mean functions.
• Run some experiments to see how the error decreases as you increase the number of points. Does the predicted order O(1/N^1/2) show?

d-Dimensional integration

For the multi-dimensional MC, you will use three files:

• hitormiss.cc: The main program. It calls the hitormiss procedure with two different point distributions (initially just one, you will add the discrepancy distribution yourself).
  It produces four output files, e.g. when calling it as follows (100.000 points in 5d):

  [labsc@oeleboeter:labsci-1b]$ ./hitormiss 5 100000
  

  • points_random_5d_10000.txt: The randomly generated points by your RNG. Interesting to plot in MATLAB (2d/3d only), remember plotpoints.
  • points_discrep_5d_10000.txt: The discrepancy-sampled points (See Sec. 3.3). Also plot these in MATLAB.
  • hom_random_5d_10000.txt: The computed volume for various numbers of points using random sampling.
  • hom_discrep_5d_10000.txt: The computed volume for various numbers of points using discrepancy sampling.
• sphere.cc: Function that tests whether a point lies inside the unit sphere (needed for hitormiss).
• discrep.cc: DISCREP is a subclass (special form) of RVG. It samples points in [0,1]^d by discrepancy sampling, instead of random sampling.

Additionally, there is the plotsphere.m MATLAB function that reads and plots the files that hitormiss produces. Have a look at the file to find out how it works.

Discussions

There is nothing to hand in now, but remember that the results and conclusions following from Ex. 3.4 & 3.8 should appear in your final report. Investigate the order of the error, what do you expect? How tight is the upper bound, and is discrepancy sampling really noticeably better? How is the efficiency? Be critical of your results.