The modified majorant frequency method for numerical simulation of the generalized exponential distribution

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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.

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