Monte Carlo method
The Monte Carlo method is a numerical procedure, devised by S. Ulam (on his hospital bed, playing cards) to solve a problem by randomly sampling it. It is a very powerful method that we use quite a lot. It can apply even to quite unlikely cases (see for instance Knuth's 3:16 study of the Bible).
It is also a favorite for teaching, as it allows one to do nontrivial stuff very quickly. Rejection sampling is both explained and implemented very easily.
A quite advanced but still nice work is by L. Devroye.[1]
Multivariate sampling
Strangely enough, multivariate sampling—which seems to be a very common need—is little documented on the web, the best thing one can find being people trying to guess the procedure. It is, if taken at face value, a complicated problem, which requires dedicated methods for what are likely highly tricky particular cases that require the method. Even so:
Computational sampling from univariate distributions is effectively a solved problem.— Dolgov et al.[2]
The simplest method goes as follows:
This page is still in progress.
Links
- See [1]
Quantum Monte Carlo
The method has been developed for quantum mechanics in the early 1990s by roughly three schools:
- Dalibard, Castin & Molmer's MCWF (Monte Carlo Wave Function).
- Carmichael's trajectories.
- Hegerfeldt's quantum jumps.
- With Camilo, we implemented it for Time and frequency resolved correlations.[3]
References
- ↑ 🕮Handbooks in Operations Research and Management Science, chapter "4. Nonuniform Random Variate Generation", p. 83. L. Devroye. North Holland, 2006. [DOI: 10.1016/s0927-0507(06)13004-2]
- ↑ Dolgov, S., Anaya-Izquierdo, K., Fox, C. et al. Approximation and sampling of multivariate probability distributions in the tensor train decomposition. Stat Comput 30, 603–625 (2020). They give references to more direct discussions: cf. Devroye, Johnson & Hörmann.
- ↑ Frequency-resolved Monte Carlo. J. C. López Carreño, E. del Valle and F. P. Laussy in Sci. Rep. 8:6975 (2018).
- Multivariate Statistical Simulation: A Guide to Selecting and Generating Continuous Multivariate Distributions, By Mark E. Johnson