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ISSN 10645624, Doklady Mathematics, 2012, Vol. 85, No. 3, pp. 325327. Pleiades Publishing, Ltd., 2012.Original Russian Text G.A. Mikhailov, S.V. Rogazinskii, 2012, published in Doklady Akademii Nauk, 2012, Vol. 444, No. 1, pp. 2830.

325

In this paper, we construct an algorithm for numerically statistically modeling a random variable distributed with probability density of the form

(1)

where

In problems of particle transport theory, the last condition is satisfied by extending the bounded medium toan infinite medium by using an absolute absorber, forwhich capture > 0 (see, e.g., [1]).

In the case under consideration, the realization ofthe standard modeling formula = F1() [1] reducesto the solution (possibly, very burdensome) of theequation

where is a standard random number uniformly distributed in the interval (0, 1).

The distribution with density (1) can be numerically modeled by the method of maximal section [24]involving the recursive calculation of (u), which maybe very burdensome at large n. For this reason, in [5],the randomized modification of this method was suggested (under the name of the majorant frequencymethod), which has made it possible to drasticallyreduce the complexity of the direct statistical modeling of the evolution of interacting particle ensembles

f u( ) u( ) s( ) sd

0

u

, uexp 0,>=

u( ) i u( ), i u( )i 1=

n

0, u( ) ud0

+.= =

s( ) sd

0

,ln=

in approximately solving the Boltzmann nonlinearkinetic equation. The parameter n is the total numberof pairs of particles. In [5], by analogy with [3], therequired algorithm was constructed and substantiatedby transforming the kinetic equation; namely, it wasshown that the introduction of an additional randomized deltascattering preserves the solution of thisequation (a particle flux).

In this paper, we construct and substantiate a modified majorant frequency method as an algorithm forstatistically modeling the distribution with density (1)under the assumption

where the quantities (u) and, therefore, *(u) arefairly easy to evaluate.

Note that, in [5], the case of (u) = was

essentially considered. In what follows, we use thenotation

The modified majorant frequency method is determined by the following assertion, in which the symbol with various indices denotes independent randomvariables uniformly distributed in the interval (0, 1).

Theorem 1. Suppose that

where {k} is a sequence of random variables distributedwith conditional densities

and {l} is a sequence of random numbers such that

i u( ) i* u( ), u( ) * u( ) i* u( ),i 1=

n

=

i*

i** u( )

n

0* u( ) cn.=

0 0, l k, lk 1=

l

1 2 ,, ,= = =

f* u; k 1( ) * k 1 u+( ) * k 1 s+( ) sd

0

u

,exp=

u 0,>

The Modified Majorant Frequency Methodfor Numerical Simulation of the Generalized

Exponential DistributionCorresponding Member of the RAS G. A. Mikhailov and S. V. Rogazinskii

Received December 30, 2011

DOI: 10.1134/S1064562412030064

Ershov Institute of Information Systems, Siberian Branch, Russian Academy of Sciences, pr. Akademika Lavrenteva 6, Novosibirsk, 630090 Russiaemail: gam@osmf.sscc.ru, svr@osmf.sscc.ru

MATHEMATICS

326

DOKLADY MATHEMATICS Vol. 85 No. 3 2012

MIKHAILOV, ROGAZINSKII

(2)

Let N = min l: l . Then

(a) N has generalized exponential distribution withdensity (1);

(b) P(N = i) = , i = 1, 2, , n.

Proof. The sequence of points {l} is a Poisson pointflux with intensity *(u) (see, e.g., [6]). According tothe sampling rule for a Poisson flux [6], we leave pointsl with probability

and obtain a flux { } with intensity (u), for which the

random expectation times with j = 1, 2,

are distributed with conditional densities f(u; )

and = 0. Under the assumptions of the theorem, we

have N = , which implies assertion (a), becausef(u; 0) coincides with (1). The relations

imply (b). This completes the proof of Theorem 1.The modeling algorithm corresponding to Theo

rem 1 is constructed under the assumption that themajorant section *(u) is chosen so that the random variables {k} are sufficiently easy to realize. The algorithm isas follows. After the random variable l = l 1 + l is realized, we choose the value of l in distribution (2) and

the value l (0, 1). If l < , then we set l = N,

and the modeling procedure terminates; the chosenvalues = N and = N are fixed.

Nest, following [4], we state a finiteness criterionfor EN2 and, therefore, EN.

Lemma 1. If (K) < 1, where (K) is the spectralradius of an integral operator K with substochastic kernel

then EN2 < +.

P l i=( )i* l( )* l( ), i 0 1 n., , ,= =

l l( )

l* l( )

i N( ) N( )

i* l( )* l( )

i 1=

n

i l( )

i* l( )

i l( )* l( )

i 1=

n

l( )

* l( )= =

j

j l 1

l 1

0

1

P N i=( )P l i l N l= =( )

P l N l=( ),=

P l N l=( ) l( )

* l( )= ,

P l i l N l= =( )i* l( )* l( )

i l( )

i* l( )

i l( )* l( ),= =

i 1 2 n, , ,=

l l( )

l* l( )

K u' u( ) 1 u'( )* u'( ) f* u u'; u'( ),=

Note that (K) < 1 if either > 0, because

we have ||K|| < 1 in this case, or > 0 for u >

u0 0, because we then have ||K2|| < 1.In problems of transport theory, the mean number

of free paths on the trajectory of a particle is determined by the integral

where R is the corresponding phase space, and (r) isparticle flux. Thus, the performance of the methodunder consideration can be studied by comparing(, *) and (, ).

For the problems with n = considered

in [5], where N is the number of particles in themodel ensemble of interacting particles, we have = , and it is natural to assume that

where vi is the velocity of the ith particle. To apply themodified frequency majorant method, we can set

Instead of the number , we consider the randomnumbers (1) and (2) of two interacting particles. Theorem 1 asserts that the choice of the free path lengthdetermines these numbers.

In [5], the simplest algorithm of this type with

parameters = c1 vi vj| was stated. The point

((1), (2)) is uniformly distributed and can be realizedas [7]

Clearly, the modified method may be substantiallymore economical than that suggested in [5].

This method may also be very efficient in modelingtrajectories of particles in a medium whose densityequals the sum of densities with bounded supports,e.g., of the densities pi defined on the spheres |r ri| i for i = 1, 2, , n. In the case under consideration,we can set

where c is the flux weakening coefficient for p 1. At

large n, the frequency majorant algorithm is substantially more economical than the minimal path algorithm [8], which is realized by the formula

(3)

where li is the random path length for the medium represented by the ith sphere in vacuum. Note that rela

u( )* u( )

u( )* u( )

,( ) r( ) r( ) r,d

R

=

N N 1( )2

i ji j

i j c1 vi vj , i j, 1 2 N,, , ,=

i j* c1 vi vj .=

i j* |i j,

max

1( ) := entier 1n{ } 1, 2( )

:= entier 2 n 1( ){ } 1;+ +

if 2( ) 1( ), then 2( ) := 2( ) 1.+

i* ci,

l li, ii

min 1 2 n,, , ,= =

DOKLADY MATHEMATICS Vol. 85 No. 3 2012

THE MODIFIED MAJORANT FREQUENCY METHOD 327

tion (3) follows directly from the wellknown fact thatthe union of independent Poisson fluxes is a Poissonflux with sum intensity [6].

The frequency majorant method admits a continuous modification in the case where

The continuous modification uses a random parameter l distributed with density

The modeling algorithm considered above, the quantities of the form i(u) are replaced by (u, e) with thecorresponding values of arguments.

The continuous version of Theorem 1 is stated andproved in strict analogy with the discrete, case;Lemma 1 is not changed. The corresponding modification of the majorant frequency method can be veryuseful for solving radiation problems, in which isexpressed by an integral over some spectral interval.

In conclusion, note that the complexity of theabove algorithms can be decreased by reducing thenumber of values generated by a sensor, as in [4].This trick is particularly effective if the discrete distribution (2) is modeled by the standard method, i.e.,l = k, provided that

For the next value of the standard random number wecan take

If ' < or ' , then the current

required standard value can be calculated by the formulas

respectively.By analogy with [4], it is easy to show that the ran

dom variables , ', and '' are independent, and theyare uniformly distributed in the interval (0, 1).

ACKNOWLEDGMENTS

This work was supported by a federal targeted program (State contract no. 07.514.11.4016), by the Russian Foundation for Basic Research (project nos. 120100034 and 120100727), and by interdisciplinaryintegration projects of the Siberian Branch of the Russian Academy of Sciences (project nos. 47 and 126).

REFERENCES

1. S. M. Ermakov and G. A. Mikhailov, Statistical Modeling (Nauka, Moscow, 1982) [in Russian].

2. W. A. Coleman, J. Nucl. Sci. Eng. 32 (1), 7681(1968).

3. G. A. Mikhailov, At. Energ. 28 (2), 175 (1970).4. G. A. Mikhailov and T. A. Averina, Comput. Math.

Math. Phys. 50, 951962 (2010).5. M. S. Ivanov and S. V. Rogazinskii, Dokl. Akad. Nauk

312, 315320 (1990).6. Yu. K. Belyaev, Probability and Mathematical Statistics:

Encyclopaedia (Sovetskaya Entsiklopediya, Moscow,1999) [in Russian].

7. G. A. Mikhailov, S. V. Rogazinskii, and N. M. Ureva,Comput. Math. Math. Phys. 46, 680690 (2006).

8. M. Sh. Nazaraliev and S. A. Ukhinov, in StatisticalModeling in Mathematical Physics (Novosibirsk, 1976),pp. 2937 [in Russian].

u( ) u e,( ) e, u e,( )d

0

E

* u e,( ),

u( ) * u( ) * u e,( ) e.d

0

E

=

* l e,( )

* l e,( ) ed

0

E

, 0 e E.