The Living Thing / Notebooks :

Monte Carlo methods

Finding functionals (traditionally integrals) approximately by guessing cleverly. Often, but not always, used for approximate statistical inference, especially certain Bayesian techniques.

Or don’t even guess randomly, just sample cleverly using the new shiny Quasi Monte Carlo.

See also compressed sensing, particle filters, and matrix concentration inequalities, and probably the most important use case, Bayesian statistics.

Samplers

Gibbs, Metropolis, Hamiltonian…

Multi-level Monte Carlo

Hmmm. Also multi scale monte carlo, multi index monte carlo. See also uncertainty quantification.

mimclib is one possible tool here.

Refs

AnHi12
Anderson, D. F., & Higham, D. J.(2012) Multilevel Monte Carlo for Continuous Time Markov Chains, with Applications in Biochemical Kinetics. Multiscale Modeling & Simulation, 10(1), 146–179. DOI.
Gile08
Giles, M. B.(2008) Multilevel Monte Carlo Path Simulation. Operations Research, 56(3), 607–617. DOI.
GiSz14
Giles, M. B., & Szpruch, L. (2014) Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L’{e}vy area simulation. The Annals of Applied Probability, 24(4), 1585–1620. DOI.
GiSz12
Giles, M., & Szpruch, L. (2012) Multilevel Monte Carlo methods for applications in finance. arXiv:1212.1377 [q-Fin].
High15
Higham, D. J.(2015) An Introduction to Multilevel Monte Carlo for Option Valuation. arXiv:1505.00965 [physics, Q-Fin, Stat].
Xia11
Xia, Y. (2011) Multilevel Monte Carlo method for jump-diffusion SDEs. arXiv:1106.4730 [q-Fin].