Scientific Computing · Cluster
Simulation
Monte Carlo and discrete event models.
- 01 Inverse-Transform Sampling and the Generalized Inverse of the CDF Why mapping a uniform variate through the quantile function produces the right distribution, what the discrete case changes, and where floating-point resolution destroys the tails.
- 02 Hyperplane Structure in Linear Congruential Generators Why successive outputs of a congruential generator lie on a small number of lattice planes, how the spectral test scores that defect, and what state size buys in a modern generator.
- 03 Rejection Sampling and the Price of the Envelope The acceptance rate of a rejection sampler is exactly the reciprocal of its envelope constant, which is why a proposal that looks cheap can make the loop run thousands of trials per variate.
- 04 Why Monte Carlo Error Falls as the Square Root of the Sample Count The convergence rate follows from the variance of a sum, and the practical consequences are a fixed digit budget per factor of a hundred in work and an error bar that is itself estimated.
- 05 Weight Variance and Effective Sample Size in Importance Sampling Reweighting a proposal sample is exact only in the limit, and the estimator variance is governed by the second moment of the weights, which the effective sample size summarizes.
- 06 Antithetic Variates and the Synchronization of Paired Runs Pairing each draw with its mirror cancels the leading fluctuation of an estimator when the integrand is monotone, and the price is a stream whose consumption must stay aligned.
- 07 Detailed Balance and the Stationary Distribution of a Markov Chain A single equation on the transition kernel guarantees that a distribution is stationary, which is what turns a chain of biased moves into a sampler for an arbitrary target.
- 08 Birth-Death Processes and the Queue Length Equilibrium The transition diagram of a single-server queue is a birth-death chain whose stationary probabilities satisfy a flow-balance equation that yields the M/M/1 geometric distribution.
- 09 Next-Event Time Advance and the Simplicity of the Priority Queue A discrete-event simulator advances time by jumping to the next scheduled event, and the correctness of that jump rests on a heap whose ordering and tie-breaking are both specified.
- 10 Rare-Event Estimation with Splitting and the Cross-Entropy Shift When the event of interest has probability a million to one, plain sampling sees nothing, and the standard tools bias the proposal or multiply the paths to reach the tail in bounded steps.