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].