MONTE CARLO STUDY OF PARTICLE TRANSPORT PROBLEM IN AIR POLLUTION. R. J. Papancheva, T. V. Gurov, I. T. Dimov

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2 Pliska Stud. Math. Bulga. 14 (23), STUDIA MATHEMATICA BULGARICA MOTE CARLO STUDY OF PARTICLE TRASPORT PROBLEM I AIR POLLUTIO R. J. Papancheva, T. V. Guov, I. T. Dimov Abstact. The actual tanspot of the ai pollutants is due to the wind. This nomally called advection of the ai pollutants. Diffusion and deposition ae othe two majo physical pocesses, which take place duing the tanspot of pollutants in the atmosphee. In this pape we study two classes of gid-fee Monte Calo (MC) algoithms fo solving an elliptic bounday value poblem, whee the patial diffeential equation contains advection, diffusion and deposition pats. The gid-fee MC appoach to solve the above equation uses a local integal epesentation and leads to a stochastic pocess called a andom Walk on balls (WOB). In the fist class of algoithms, the choice of a tansition density function in the Makov chain depends on the adius of the maximal ball, lying inside the domain, in which the poblem is defined, and on the paametes of the diffeential opeato. While the choice of a tansition density function in the second class of algoithms does not depend on the deposition pat of the poblem. The computational complexity of both classes of gid-fee MC algoithms was investigated using vaied numeical tests on a PowePC (G4 w/altivec) 45 MHz unning YDL Mathematics Subject Classification: 65C5 Key wods: Monte Calo algoithms Suppoted by Cente of Excellence BIS-21 gant ICA and by the SF of Bulgaia unde Gants #I 811/98 and #MM 92/99

3 18 R. J. Papancheva, T. V. Guov, I. T. Dimov 1. Fomulation of the poblem Conside the functional (1) J(u) (g, u) = Ω g(x)u(x)dx, whee Ω is a domain in the Euclidean space R 3 and x = (x 1, x 2, x 3 ) Ω. The functions u(x) and g(x) belong to the Banach space X and to the adjoint space X, espectively, and u(x) is a unique solution of the following Fedholm integal equation: (2) u(x) = k(x, y)u(y)dy + f(x). Ω We ae inteested to estimate the functional (1), using MC appoach, whee Eq. (2) is a local integal epesentation of the solution of the following bounday value poblem: (3) Mu = Φ(x), x Ω, Ω R 3, (4) u = ψ(x), The opeato M is defined by: M = 3 ( i=1 2 x 2 i x Ω. + b i (x) x i ) + c(x). Such poblems appea in envionmental sciences. The functional (1) can be consideed as a measue fo the damaging effect of a dange pollution on the life in the natue. In this case g(x) is a given sensitivity function, and u(x) is the concentation of the pollutant, that can be descibed as a solution of the bounday value poblem (3) (4). Hee, we intoduce the following notations: b = max x Ω b(x), c = max x Ω c(x), and denote by R the adius of the maximal ball, lying inside Ω and by R(x) the adius of the maximal ball, lying inside Ω with cente in the point x. We assume that the conditions fo existing of an unique solution of the poblem (3) (4) ae satisfied [1, 11], and divb(x) =. Using the Levy s function L p (y, x) [3] as a Geen s function we obtain following integal fom: (5) u(x) = B(x) M y L p (y, x)u(y)dy + B(x) L p (y, x)φ(y)dy.

4 Monte Calo study of paticle tanspot poblem In the Eq.(5) B(x) is the ball inside Ω with cente in the point x and adius R(x). M = 3 ( ) b i (x) x i + c(x) is the adjoint opeato to M, and (6) i=1 2 x 2 i k(x, y) = M y L p(y, x) = µ p (R) p() 2 + µ p(r) 2 [ c(y) + 3 i=1 R µ p (R)c(y) b i (y) y i x i ] R p(ρ) ρ dρ p(ρ)dρ. The epesentation of u(x) in (5) is the basis fo the poposed MC appoach. Using it a biased estimato fo the solution can be obtained. 2. Monte Calo algoithms The MC estimato, whose mathematical expectation coincides with J(u) is (7) Θ[g] = g(ξ ) π(ξ ) Q j f(ξ j ), j= whee Q = 1, Q j = Q j 1 k(ξ j 1,ξ j ) p(ξ j 1,ξ j ), j = 1, 2, 3,... and ξ, ξ 1,... is a Makov chain in Ω with initial density function π(x) and tansition density p(x, y), which ae toleant to g(x) and k(x, y), espectively [2, 5, 11]. To ensue the convegence of the pocess, we intoduce an ε - stip of the bounday. The pocess stats at point ξ = x Ω, chosen with the initial density function π(x). The next andom point y is detemined by a tansition density function p(x, y). This pocess teminates when the point falls into the ε - stip of the bounday. In the fist class of algoithms (class A), the tansition density is taken to be popotional to the kenel (6), i.e. (8) k(x, y) p(x, y) = k(x, y)dy. B(x) In this case we need to know when the integal tansfomation kenel (6) is non-negative. It is poved in [4] that p(x, y), when the following inequality (9) p() (b + R4 ) R c p(ρ)dρ

5 11 R. J. Papancheva, T. V. Guov, I. T. Dimov is tue. In the second class (class B) the tansition density function in the Makov chain is taken to be the kenel, whee the deposition paamete is equal to zeo, i.e. (1) p(x, y) = k(x, y) = My L p(y, x) with c(y). It is poved in [1] that when the condition: (11) p() b R p(ρ)dρ is fulfilled, p(x, y) can be used as tansition density function in Makov chain. Hee we conside thee cases fo p(): whee case I p() = e k, case II p() = p = const, if kr 1, case III p() = e k, if kr ln 2, b + R 4 c fo class A, k = k = b fo class B. The function p(x, y) can be witten in spheical coodinates as: (12) p(, w) = sin θ 4π p() q p (R) p(w ), whee fo the class A we have: (13) p(w ) = [ ] R b(x + w) cos(b, w) + c(x + w) 1 + p(ρ)dρ p() R c(x + w)2 p(ρ) p() ρ dρ, and fo algoithms fom class B the conditional pobability is: (14) p(w ) = 1 + b(x + w) cos(b, w) p() R p(ρ)dρ.

6 Monte Calo study of paticle tanspot poblem The next andom point y in the Makov chain depends on both the jump into the maximal ball, and on the diection w. We sample the jump with a density function p()/q p (R), using an invese-tansfomation ule. The andom diection w with density function p(w ) = sin θ 4π p(w ) is computed, using an acceptance-ejection (AR) technique [4, 1]. 3. umeical tests As an example, the following bounday value poblem was solved, using two classes of MC algoithms: (15) (16) ( 3 2 u i=1 x 2 i + b i (x) u ) + c(x)u = in Ω = [, 1] 3, x i u = ψ(x), x Ω ε. Two test vaiants ae consideed fo ψ(x), b(x) and c(x): Test 1: ψ(x 1, x 2, x 3 ) = e a 1x 1 +a 2 x 2 +a 3 x 3, (x 1, x 2, x 3 ) Ω ε, b 1 (x) = a 2 a 3 (x 2 x 3 ), b 2 (x) = a 3 a 1 (x 3 x 1 ), b 3 (x) = a 1 a 2 (x 1 x 2 ), Test 2: c(x) = (a a2 2 + a2 3 ); ψ(x 1, x 2, x 3 ) = e a 1(x 1 +x 2 +x 3 ), (x 1, x 2, x 3 ) Ω ε, b 1 (x) = a 2 x 1 (x 2 x 3 ), b 2 (x) = a 2 x 2 (x 3 x 1 ), b 3 (x) = a 2 x 3 (x 1 x 2 ), c(x) = 3a 2 1. Hee a 1, a 2, a 3 ae paametes. It is easy to see that divb(x) =. This condition guaantees the possibility to use the local integal epesentation by Geen s function. Diffeent values fo the coefficients a i, i = 1, 2, 3 ae consideed. In ou tests the following ε-stips ae used: ε =.1, ε =.5, ε =.1. Figues 1 3 show the CPU times fo algoithms of classes A and B, with diffeent values fo coefficients a i. One can see that the algoithm B is faste than the algoithm A, with inceasing of b. This behavio can be explained with the less numbe of the moves in the Makov chains (see Table 1). When the maximum of the advection vecto b(x) inceases the aveage length of Makov

7 112 R. J. Papancheva, T. V. Guov, I. T. Dimov chain (El (ε,b) ) fo the algoithm B is shote than El (ε,a) fo algoithm A. Figue 3 shows that the algoithm B is about 1.75 times faste than the algoithm A in case I when a i = 3, b = The elative eos of the algoithms ae pactically the same fo both classes A and B, and they depend on b (see Figues 4 7). Figue 8 compaes CPU times of the MC algoithms fom the class B with a diffeent choice of the density function p(). One can see that the algoithm in case II is faste than the othes. The less CPU time of this case is due to the computational simplicity. On the othe hand the algoithm in case III is a little bit faste than the algoithm in a case I. This can be explained with esults fo the aveage moves in the WOB that ae pesented in Table 2. We conclude that the numeical esults show that the algoithms of class B ae moe efficient. Also we ecommend to use a case II o a case III, if the paametes of the poblem allow this. a i El (ε,a) El (ε,b) b p() = e k p() = e b Table 1: The aveage lengths of Makov chains with ε =.1. 5 A: p()=e -k B: p()=e -b 4 3 CPU Figue 1: Compaison of the CPU times among algoithms A and B in case when ε =.1, a =.25 fo Test 1.

8 Monte Calo study of paticle tanspot poblem A: p()=e -k B: p()=e-b 4 3 CPU Figue 2: Compaison of the CPU times among algoithms A and B in case when ε =.1, a = 2 fo Test A: p()=e -k B: p()=e -b CPU Figue 3: Compaison of the CPU times among algoithms A and B in case when ε =.1, a = 3 fo Test p()=e -k p()=e -b* Figue 4: Vaiation of the elative eo in case, when ε =.1, a =.25.

9 114 R. J. Papancheva, T. V. Guov, I. T. Dimov.25.2 p()=e -k p()=e -b* Figue 5: Vaiation of the elative eo in case, when ε =.1, a = p()=e -k p()=e -b* Figue 6: Vaiation of the elative eo in case, when ε =.1, a = A: p()=const B: p()=const Figue 7: Vaiation of the elative eo in case, when ε =.1, a =.25 fo Test 2.

10 Monte Calo study of paticle tanspot poblem B: p()=e -b* B: p()=const B: p()=e b* CPU Figue 8: Compaison of the CPU times among algoithms B in case, when ε =.1, a = 1 fo Test 2. Test 1 Test 2 El (ε,a) El (ε,b) El (ε,a) El (ε,b) case I ε = ε = ε = case III ε = ε = ε = Table 2: The aveage lengths of Makov chains in case when a =.25.

11 116 R. J. Papancheva, T. V. Guov, I. T. Dimov R E F E R E C E S [1] A. V. Bitzadze. Equations of the Mathematical Physics, auka, Moscow, [2] J. H. Cutiss, Monte Calo methods fo the iteation of the linea opeatos, J. Math. Phys (1954), [3] I. T. Dimov, T. V. Guov. Estimates of the computational complexity of iteative Monte Calo algoithm based on Geen s function appoach. Mathematics and Computes in Simulation (1998) [4] I. T. Dimov, R. J. Papancheva. Geen s function Monte Calo algoithms fo elliptic poblems, Jounal of Mathematics and Computes in simulations (submitted). [5] S. M. Emakov, G. A. Mikhailov. Statistical Simulation, auka, Moscow, [6] S. M. Emakov, V.V. ekutkin, A.S. Sipin. Random Pocesses fo Solving Classical Equations of the Mathematical Physics, auka, Moscow, [7] T. V. Guov, P. A. Withlock, I. T. Dimov. A gid fee Monte Calo algoithm fo solving elliptic bounday value poblems, Lectue notes in Comp. Sci. (eds. L.Vulkov et al.), Spinge-Velag (21), [8] V. P. Mikhailov. Patial Diffeential Equations, auka, Moscow, [9] C. Mianda, Equasioni alle diivate paziali di tipo ellipttico. Spinge- Velag, Belin, [1] R. J. Papancheva, I. T. Dimov, T. V. Guov. A new class gid-fee Monte Calo algoithms fo elliptic bounday value poblems, In: Poceeding of the MA 2 Confeence , Boovets, Bulgaia (submitted). [11] I. M. Sobol, Monte Calo umeical Methods, auka, Moscow, R. J. Papancheva, T. V. Guov, I. T. Dimov Depatment of Paallel algoithms Cental Laboatoy fo Paallel Pocessing Bulgaian Academy of Sciences Acad. G. Bonchev St., bl. 25A 1113 Sofia, Bulgaia guov@copen.bas.bg, ivdimov@bas.bg, umi@canto.bas.bg