Sampling, integration, and computing volumes

Sampling is a fundamental method for approximating answers that cannot be directly computed. Samples often need to be taken from a non-uniform distribution. In this thesis, I present an algorithm to generate samples from log-concave and nearly log-concave distributions. This sampling algorithm is based on a biased random walk, and the proof of correctness also provides bounds on the mixing time of the Gibbs sampler with the random sweep strategy. To demonstrate the usefulness of sampling from log-concave distributions, I use this algorithm to integrate log-concave functions and to compute the volume of convex sets. This resulting volume algorithm improves on the existing volume algorithms due to Dyer, Frieze, and Kannan, and Lovasz and Simonovits. Samples generated by the sampling algorithm can also be used to estimate marginal densities and as a tool for Bayesian inference.