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The theory of hybrid stochastic algorithms. (English) Zbl 0732.65005
Probabilistic methods in quantum field theory and quantum gravity, Proc. NATO Adv. Res. Workshop, Cargèse/Fr. 1989, NATO ASI Ser., Ser. B 224, 209-223 (1990).

[For the entire collection see Zbl 0722.00046.]

In this paper a family of stochastic algorithms for performing Monte Carlo calculations in quantum field theory is considered. The author deals with the problem of evaluating functional integrals numerically in the presence of fermion fields. It is necessary to evaluate the expectation value of some operator ${\Omega }$ which depends on some fields which is denoted by ${\Phi }$ : $<{\Omega }>={z}^{-1}\int |d{\Phi }|exp\left(-S|{\Phi }|\right){\Omega }\left({\Phi }\right),$ where S is the action and $|d{\Phi }|$ is the measure. The Metropolis algorithm of the Monte Carlo integration is discussed.

Hybrid and Langevin algorithms are studied from the viewpoint that they are approximate versions of the hybrid Monte Carlo method. The paper comprises results of recent progress in this area explaining higher order integration schemes, the asymptotic large-volume behaviour of the various algorithms, and some simple exact results obtained by applying them to free field theory.

Reviewer: I.Dimov (Sofia)
##### MSC:
 65C05 Monte Carlo methods 65D32 Quadrature and cubature formulas (numerical methods) 81T80 Simulation and numerical modelling (quantum field theory)