Sampling

Background and Motivation

Uncertainty quantification problems often require computing expectations of the form

\[ \bar{h} = \int_\Omega h(x) p(x) dx, \]

where \(h(x)\) is some utility function and \(p(x)\) is the probability density function for the random variable \(x\). Monte Carlo approximations to \(\bar{h}\) use random realizations \(x^{(i)}\sim p(x)\) to approximate \(\bar{h}\) with an estimator \(\hat{h}\) defined by

\[ \hat{h} = \sum_{i=1}^N w_i h(x^{(i)}), \]

where \(w_i\) are appropriately defined weights. Typically, \(w_i = N^{-1}\). MUQ's sampling module provides tools for constructing Monte Carlo estimates like \(\hat{h}\). In particular, MUQ provides a suite of Markov chain Monte Carlo (MCMC) algorithms for generating samples \(x^{(i)}\) that can be used in Monte Carlo.

• The basics of defining a sampling problem in MUQ can be found here.
• An introduction to MCMC algorithms is provided here.
• An introduction to dimension independent MCMC algorithms can also be found here.
• Multi-fidelity and Multi-index MCMC algorithms are described here.

Modules
	Markov chain Monte Carlo