Statistical modelling

A method in applied and computational mathematics, consisting of the computer realization of special stochastic models of phenomena or objects under consideration. The use of statistical modelling has been extended as a result of the rapid development of technology and, particularly, of multi-processor computing systems, which allow many independent statistical experiments to be simulated simultaneously. Moreover, classical computing methods are in many cases unsatisfactory for research into the increasingly complex mathematical models of the phenomena in question. This also serves to increase the role of statistical modelling, whose efficiency depends weakly on the dimensions and geometric details of the problem. Other positive properties of this method include the simplicity and the natural nature of its algorithms, as well as the fact that modifications can be made in the light of new information regarding the solution (see Monte-Carlo method; Statistical experiments, method of).

Problems that are solved by the method of statistical modelling can be conditionally divided into two classes. The first includes problems of a stochastic nature, in which a direct simulation of a natural probability model is used. The second class covers deterministic problems, in which a probability process is constructed artificially and a formal solution of the problem is thereby obtained. This process is then modelled on a computer and a numerical solution is constructed in the form of statistical estimators. An intermediate class between these two classes also exists. The problems in this class, which are described by deterministic equations, have either coefficients or boundary conditions or right-hand sides that are random. Particularly effective here is "dual randomizationdual randomization" (see [1]), which means that for a given realization of random parameters, only a small number of trajectories of the process that solves the equation is constructed.

The areas in which statistical modelling is used are examined below.

The solution of the problems of radiation transfer can be achieved by modelling the trajectories of the particles: neutrons, photons, gamma-quanta, electrons, etc. The algorithms of statistical modelling for solving problems of atmospheric optics (see [2]) and neutron physics (see [3]) are well developed. Statistical modelling is also useful in research into impure diffusion (see [4]), especially in stochastic velocity fields (see [5]).

Statistical modelling is used in solving a number of problems in statistical physics (see [7]) by means of averaging "over time" a certain model with stochastic kinetics (often artificially). Statistical modelling has been used to obtain new results in the theory of phase transitions, solid bodies in disorder (especially in magnetics), surface phenomena, etc. (see [7]). In solving complex problems in the theory of rarefied gases, an effective technique is to modify the method of direct statistical modelling related to the decomposition of the non-linear kinetic Boltzmann equation (see [6]). This equation can also be related to certain branching Markov processes (see [1]).

Statistical modelling can be used to solve boundary value problems of mathematical physics on the basis of special probability models (see [5]). Statistical modelling is useful in solving stochastic problems: diffraction of waves on random surfaces, the theory of elasticity with random loads, etc.

Statistical modelling is widely used in solving problems of queueing theory and other complex stochastic systems (see, for example, [1], [8]).

References

[1]	S.M. Ermakov, G.A. Mikhailov, "Statistical modelling" , Moscow (1982) (In Russian)
[2]	G.I. Marchuk, et al., "The Monte-Carlo method in atmospheric optics" , Novosibirsk (1976) (In Russian)
[3]	A.D. Frank-Kamenetskii, "Simulation of neutron trajectories in reactor calculus by the Monte-Carlo method" , Moscow (1978) (In Russian)
[4]	L.M. Galkin, "Solution of diffusion problems by the Monte-Carlo method" , Moscow (1975) (In Russian)
[5]	B.S. Elepov, et al., "Solving boundary value problems by the Monte-Carlo method" , Novosibirsk (1980) (In Russian)
[6]	O.M. Belotserkovskii, A.I.Erofeev, V.E. Yanitskii, "A non-stationary method of direct statistical modelling of rarefied gas flows" USSR Comp. Math. Math. Phys. , 20 : 5 (1980) pp. 82–112 Zh. Vych. Mat. i Mat. Fiz. , 20 : 5 (1980) pp. 1174–1204
[7]	K. Binder, D.W. Heermann, "Monte-Carlo simulation in statistical physics" , Springer (1988)
[8]	N.P. Buslenko, et al., "The method of statistical trials (the Monte-Carlo method)" , Moscow (1962) (In Russian)

References

[a1]	G.A. Bird, "Molecular gas dynamics" , Clarendon Press (1976)
[a2]	C. Cercignani, "The Boltzmann equation and its applications" , Springer (1988)
[a3]	C. Cercignani, "Mathematical methods in kinetic theory" , Plenum (1990)
[a4]	G.A. Bird, "Direct simulation and the Boltzmann equation" Phys. Fluids , 13 (1970) pp. 2676–2687
[a5]	K. Koura, "Transient Couette flow of rarefied binary gas mixtures" Phys. Fluids , 13 (1970) pp. 1457–1466
[a6]	K. Nanbu, "Direct simulation scheme derived from the Boltzmann equation" J. Phys. Soc. Japan , 49 (1980) pp. 2042–2049
[a7]	H. Babovsky, "A convergence proof for Nanbu's Boltzmann simulation scheme" European J. Mech. B/Fluids , 8 (1989) pp. 41–55
[a8]	H. Babovsky, R. Illner, "A convergence proof for Nanbu's simulation method for the full Boltzmann equation" SIAM J. Numer. Anal. , 26 (1989) pp. 45–65
[a9]	D.N. Chorafas, "Systems and simulation" , Acad. Press (1965)
[a10]	S.M. Ermakov, V.V. Nekrutkin, A.S. Sipin, "Random processes for the classical equations for mathematical physics" , Kluwer (1989) (Translated from Russian)