Green s function Monte Carlo algorithms for elliptic problems

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1 Mathematics and Computes in Simulation 63 (2003) Geen s function Monte Calo algoithms fo elliptic poblems I.T. Dimov, R.Y. Papancheva Cental Laboatoy fo Paallel Pocessing, Depatment of Paallel Algoithms, Bulgaian Academy of Sciences, Acad. G. Bonchev St., bl. 25A, 1113 Sofia, Bulgaia Received 2 Septembe 2002; eceived in evised fom 2 July 2003; accepted 28 July 2003 Abstact In many lage-scale poblems, one is inteested to obtain diectly an appoximate value of a functional of the solution. Hee, we conside a special class of gid-fee Monte Calo algoithms fo diect computing of linea functionals of the solution of an elliptic bounday-value poblem. Such kind of poblems appea in envionmental sciences, computational physics and financial mathematics. To ceate the algoithms, we use the Geen s function analysis and define the conditions unde which the integal tansfomation kenel of the integal epesentation fo the bounday-value poblem unde consideation is non-negative. This analysis is done fo a possible set of densities, and it is used to geneate thee diffeent gid-fee Monte Calo algoithms based on diffeent choices of the density of the adius of the balls used in Monte Calo simulation. Only one of the geneated algoithms was known befoe. We shall call it Sipin s algoithm. It was poposed and studied by Sipin. The aim of this wok is to study the two new algoithms poposed hee and based on two othe (than in Sipin s algoithm) possible choices of the densities. The algoithms ae descibed and analyzed. The pefomed numeical tests show that the efficiency of one of the new algoithms, which is based on a constant density is highe than the efficiency of Sipin s algoithm IMACS. Published by Elsevie B.V. All ights eseved. Keywods: Monte Calo method; Selection methods; Density function 1. Intoduction In this pape, new Monte Calo algoithms fo computing functionals of solutions of elliptic boundayvalue poblems ae poposed and studied. Thee ae many lage-scale poblems in computational physics, biology and envionmental sciences, whee it is impotant to be able to compute functionals of the solution without computing the solution itself. It is that because some impotant paametes which can be measued in physics, and some effects on the live matte ae in fact functionals. Fo instance, the damage effect of Coesponding autho. Tel.: ; fax: addesses: ivdimov@bas.bg (I.T. Dimov), umi@canto.bas.bg (R.Y. Papancheva). URL: /$ IMACS. Published by Elsevie B.V. All ights eseved. doi: /s (03)

2 588 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) dange pollution is measued by: f(x)u(x) dx, Ω whee x (x 1,...,x d ) Ω R d, u(x) is the concentation of the pollutant and f(x) is given sensitivity function. Anothe example could be given fom the statistical physics, whee paametes chaacteized a multi-paticle system like the mean-velocity V m and the enegy E ae defined by: V m = f(x)v(x) dx, E = Ω Ω f(x)v 2 (x) dx, whee f(x)(x Ω R d ) is the distibution function. The algoithms poposed in this pape can be efficiently implemented in calculating functionals of the solution. Conside the functional J(u): J(u) (g, u) = g(x)u(x) dx, (1) Ω whee Ω R 3 and x = (x 1,x 2,x 3 ) Ω is a point in the Euclidean space R 3. The functions u(x) and g(x) belong to the Banach space X and to the adjoint space X, espectively, and u(x) is the unique solution of the following Fedholm integal equation in an opeato fom: u = K(u) + f. (2) The algoithms unde consideation use a local integal epesentation of the solution by Geen s function. The Monte Calo appoach with the use of Geen s function is well known (see [8]). The basic poblem is the estimation of the computational complexity and the constuction of Monte Calo algoithms with lowe computational cost. We also conside poblems, in which a linea functional of the solution is seached fo and a ough solution is acceptable. Fo pactical computations, it means that the elative eo is about 5 10%. Fo example, such kind of poblems ae the pollution poblems in envionmental sciences. It is known that the most efficient computational method fo diect computing of functionals is the Monte Calo method (see [2,6]). This pape deals with an analysis of the choice of the density function p() in Lévy s function L p (y, x). We use the Geen s function analysis to ceate the algoithms and define the conditions unde which the integal tansfomation kenel of the integal epesentation fo the consideed bounday-value poblem is non-negative. This analysis is done fo a possible set of densities. The esult is used to geneate diffeent gid-fee Monte Calo algoithms based on diffeent choices of the density of the adius of the balls used in Monte Calo simulation. Only one of the geneated algoithms was known befoe. We call it Sipin s algoithms. It was poposed by Sipin and studied in [9]. Since in the Monte Calo simulation a selection technique is used, it is impotant to obtain an estimation fo the efficiency of the algoithm. The selection algoithm in geneal case woks as follows. Let us assume that we have to calculate a andom vaiable ξ with density function f 1 (x)/f 1 and an easily computed function f 2 (x) exists, with

3 Ω I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) f 1 (x) f 2 (x) fo all x Ω. Hee, F 1 = f 1 (x) dx and F 2 = f 2 (x) dx. Ω Let ξ be a andom vaiable with density function f 2 (x)/f 2, and γ be an unifomly distibuted andom vaiable in the inteval [0, 1]. It has been shown [8] that if: γf 2 (ξ) f 1 (ξ), (3) then ξ is the needed andom vaiable with density function f 1 (x)/f 1. This algoithm allows us to use the function f 2 (x) instead of the oiginal function f 1 (x). Howeve, some vaiables will be ejected, namely when condition (3) will not be fulfilled. We use the following definition fo the efficiency. Definition 1.1. The efficiency of the selection algoithm is defined as the atio: E = F 1 F 2. In 1984, Sipin poved that the efficiency of the selection gid-fee Monte Calo algoithm is geate than 1/2 [9]. An impoved esult fo the efficiency of the Sipin s algoithm was obtained by Dimov in 1988 [3]: E = 1 + α 2 + α, fo some explicit constant α. The last esult was impoved again by Dimov and Guov in 1998 [4]: E = 1 + α 2 + α ε R, fo some ε R, with 0 <ε R < 1. It seems that this impovement is exact, i.e. it could not be impoved again (but the last statement is still not poved). In all these woks [3,4,9], the density function is fixed p() = e k. Hee, we should answe the question if a diffeent function to be taken, how this affects the algoithm efficiency. The aim of this wok is to study the two new algoithms poposed hee and based on two othe (than in Sipin s algoithm) possible choices of the densities. The algoithms ae descibed and analyzed. The pefomed numeical tests show that the efficiency of one of the new algoithms which is based on a constant density is highe than the efficiency of Sipin s algoithm. The pape is oganized as follows. In Section 2, we descibe the fomulation of the poblem and give the conditions fo existing an unique solution u(x) C 2 (Ω) C( Ω).InSection 3, the set of possible densities poviding the conditions, unde which the integal tansfomation kenel is non-negative is defined. Two impotant cases ae consideed: a case of a constant density and a case of an exponential density. We find a set of paametes of the poblem fo which both constant and exponential densities can be applied. In Section 4, we descibe and analyze the algoithms geneated by the chosen densities. Numeical tests ae pesented in Section 5. The esults of the numeical simulation illustate the efficiency of the poposed algoithms. In Section 6, we give some concluding emaks.

4 590 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) Fomulation of the poblem Let X = L 1 (Ω). Then, X = L (Ω) ([8], p. 148). Conside the case when K is an odinay linea integal tansfom: K(u)(x) = k(x, y)u(y) dy. Ω In this case, (2) obtains the fom: u(x) = k(x, y)u(y) dy + f(x). Ω The coesponding Neumann seies conveges when the condition: K(u) L(L1 ) = sup k(x, y) dx q<1, y Ω Ω holds. We have to calculate the functional J(u), whee u(x) is the solution of the following bounday-value poblem: M(u)(x) = Φ(x), x Ω, Ω R 3, (4) u(x) = ψ(x), x Ω, whee the opeato M is defined by: 3 ( 2 M = + b i (x) ) + c(x). x i i=1 x 2 i Hee follows the definition fo the class A (k,λ) of domains Ω. Definition 2.1. The domain Ω belongs to the class A (k,λ) if fo any point x Ω the bounday Ω can be epesented as a function z 3 = σ(z 1,z 2 ) in the neighbohood of x with k σ C (0,λ) (Ω), i.e. k σ(y) k σ(y ) Q y y λ, (5) whee the vectos y (z 1,z 2 ) and y λ (0, 1]. (z 1,z 2 ) ae two-dimensional vectos, Q is a constant, and The definition of the class C (0,λ) (Ω) can be found in [12]. If in the closed domain Ω A (1,λ) the coefficients of the opeato M satisfy the conditions: b i (i = 1, 2, 3), c C (0,λ) ( Ω), c 0 and Φ C (0,λ) (Ω) C( Ω), ψ C( Ω), the poblem (4) and (5) has an unique solution u(x) C 2 (Ω) C( Ω). The conditions fo uniqueness of the solution can be found in [1,11]. The fist step in studying the Monte Calo appoach is obtaining an integal epesentation of the solution in the fom: u(x) = k(x, y)u(y) dy + f(x). B(x)

5 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) Fom an algoithmic point of view, the domain B(x) must be chosen in such a way that the coodinates of the bounday points y B(x) could be easily calculated. So we denote by B(x) the ball: B(x) = B R (x) ={y : = x y R(x)}, whee R(x) is the adius of the ball. We seek a epesentation of the integal kenel k(x, y), using Lévy s function and the adjoint opeato M fo the initial diffeential opeato M. Hee follows a shot definition fo the Lévy s function [12]. Definition 2.2. Let, H(x, y) = 1 4π x y. Evey function L(x, y) continuous in vaiables x and y, fo x and y in Ω, and x y, togethe with its fist and second deivatives with espect to the x i is called a Lévy s function if it satisfies bounds of the following type fo some λ>0 and = x y : L H = O( λ 1 ), [L H] x i = O( λ 2 ), 2 [L H] x i x j = O( λ 3 ), such bounds holding unifomly in evey bounded domain in Ω. The explicit fom of the Lévy s function fo the poblem (4) and (5) can be found in [12]: R ( 1 L p (y, x) = µ p (R) 1 ) p(ρ) dρ, R, (6) ρ whee the following notation is used: p() is a density function: µ p (R) = [4πq p (R)] 1, q p (R) = R 0 p(ρ) dρ. It is known [4] that if the vecto-function b(x) satisfies the conditions b j (x) C(Ω) (j = 1, 2, 3), and div b(x) = 0, then the adjoint opeato M, applied on functions v(x) C 2 (Ω), such that: v(x) x i = v(x) = 0, fo any x Ω, i = 1, 2, 3, has the following fom: M = 3 ( 2 x 2 i=1 i b i (x) ) + c(x). x i

6 592 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) Accoding the esults of Mianda [12], when the Lévy s function is used, the following integal epesentation fo the solution u(x) of the bounday-value poblem is tue: u(x) = (u(y)my L p(y, x) + L p (y, x)φ(y)) dy T(x) + 3 T(x) i=1 [( Lp (y, x) u(y) n i u(y) L ) ] p(y, x) b i (y)u(y)l p (y, x) dy, (7) y i y i whee n = (n 1,n 2,n 3 ) is the exteio nomal to the bounday T(x). Fomula (7) holds fo any domain T(x) A (1,λ) and theefoe it is emaining tue fo evey ball B(x), lying inside the domain Ω. It is shown that it is possible to constuct the function L p (y, x) in such a way that My L p(y, x) is non-negative in B(x), and such that L p (y, x) and its deivatives vanish on B(x). Finally, the integal epesentation of the solution has the following fom: u(x) = My L p(y, x)u(y) dy + L p (y, x)φ(y) dy, (8) whee B(x) M y L p(y, x) = µ p (R) p() + µ p(r) 2 2 B(x) R p(ρ) µ p (R)c(y) ρ [ 3 c(y) + b i (y) y i x i i=1 dρ ] R p(ρ) dρ. Fo solving ou poblem, we use a Monte Calo pocedue that is called Walk on the Balls (WOB). To ensue the convegence of the pocess, we intoduce the ε-stip of the bounday. This pocess is simila to the well-known spheical pocess [7,10,13,14]. The pocess stats at point ξ 0 = x Ω, which is chosen coespondingly with some initial density function π(x). In the paticula case, when π(x) = δ(x x 0 ) the Makov s chain stats at the point x 0 Ω. The next andom point is detemined with tansition density function p(x, y), which is popotional to the kenel of the integal equation (2) inside maximal ball B(x) Ω. This pocess teminates when the point falls into the ε-stip of the bounday. In spheical coodinates, p(x, y) is equal to: p(, w) = p() p (w). We aim at analyzing the oppotunities fo a choice of density function p(). Ou goal is to conside such density functions, in which the choice of bigge adii of the next ball in the Makov chain is moe likely. One may expect that in this case the length of the chains will be shote and the pocess moe effective. 3. Geen s function analysis Hee, we intoduce the following notations: b = sup b(x), x Ω c = sup c(x), x Ω and denote by R the adius of the maximal ball lying inside Ω. b(x) = (b 1 (x), b 2 (x), b 3 (x)),

7 and I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) In [5] was poved that the conditions: My L p(y, x) 0, fo any y B(x), (9) L p (y, x) = L p(y, x) = 0, y i fo any y B(x), i = 1, 2, 3, (10) ae satisfied fo: p() = e k, (11) whee k b + Rc. The density function defined in (11) coesponds to the Sipin s algoithm. Ou aim is to conside the oppotunities and the conditions fo a diffeent choice of the density function p(). The following poblem can be fomulated: find the class of possible densities fo which the integal tansfomation kenel M y L p(y, x) is non-negative and the conditions (10) ae satisfied. The condition: L p (y, x) = 0, fo any y B(x), is obvious and follows diectly fom (6). Since, L p (y, x) y i we obtain: = L p = x i y i y i 3 µ p L p (y, x) = 0, fo any y B(x). y i We shall pove the following theoem. R p(ρ) dρ, Theoem 3.1. If the density function p() satisfies the inequality: p() (b + R4 ) R c p(ρ) dρ, (12) then the integal tansfomation kenel M y L p(y, x) is non-negative fo any y B(x). Poof. My L p(y, x) can be witten as follows [4]: My L p(y, x) = µ p(r) Γ 2 p (y, x), whee ( R R ) p(ρ) 3 Γ p (y, x) = p() + c(y) p(ρ) dρ dρ + b i (y) y i x i ρ i=1 R p(ρ) dρ.

8 594 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) Since ρ 0, it follows that 1 /ρ 0, and theefoe, R R p(ρ) p(ρ) dρ dρ 0. (13) ρ Fo any y: 0 c(y) c. As x y =, we obtain that: 3 i=1 b i (y) y i x i b. Taking in account (13) (15), we obtain: ( R Γ p (y, x) p() c p(ρ) dρ Since 0 ρ R, it follows: 1 ρ 1 R, R p(ρ) ρ and theefoe, ( R R ) p(ρ) ( p(ρ) dρ dρ 1 ) R ρ R Fo 0 R, one has: ( 1 ) R R 4. (14) (15) ) R dρ b p(ρ) dρ. (16) p(ρ) dρ. (17) Finally, using the obtained extemum and the inequalities (16) and (17) we can wite: Γ p (y, x) p() c R R R ( p(ρ) dρ b p(ρ) dρ = p() b + c R ) R p(ρ) dρ. 4 4 Hence the esult. The inequality (12) defines a class of possible densities p(), which can be used in a Monte Calo simulation. We shall conside two possible densities: p() = p = const fo a given set of paametes of the poblem (this case is descibed in Section 3.1) and p() = e m,m>0 (descibed in Section 3.2) Case A: a constant density In this subsection, we descibe the conditions which have to be fulfilled in ode to use a constant density p() = p = const.

9 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) Fo this choice of the function p() is tue that: p() (b + R4 ) R [ c p(ρ) dρ p 1 (b + R4 ) ] c R. Consequently, if (b + (R/4)c )R 1, then M y L p(y, x) 0. Lemma 3.1. If the paametes of the poblem satisfy the inequality: 1 (b Rc )R, and the density function p() is chosen to be: p() = p = const, then the condition (9) is tue. Remak. If the adius of the maximal ball, lying inside the domain Ω satisfies the inequality: R 2 c ( b 2 + c b ), then the condition (9) is tue Case B: an exponential density While in Section 3.1, the adius is unifomly distibuted in [0,R], hee, since the function p() = e m,m>0 is stictly inceasing, the choice of bigge adii will be moe pobable. Let us intoduce a paamete k, defined by the expession: k = b Rc, and conside the density function p() = e k. Fom the inequality (12) one can obtain that: k(r ) ln 2. Consequently if the condition kr ln 2 is fulfilled, then the inequality (12) is tue fo any. Lemma 3.2. If the paametes of the poblem satisfy the inequality: kr ln 2, and the density function p() is chosen to be: p() = e k, k = b Rc, then the condition (9) is fulfilled.

10 596 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) Monte Calo algoithms Conside the poblem of evaluating the functional: J(u) (g, u) = g(x)u(x) dx, Ω whee u(x) is the solution of the integal equation (8). The Monte Calo algoithm fo solving this poblem can be defined as a ball pocess. To ensue the convegence of the pocess, we intoduce the ε-stip of the bounday, i.e. whee Ω ε ={x Ω : B(x) = B ε (x)}, B ε (x) ={y Ω : x y ε}. The andom vaiable, whose mathematical expectation coincides with J(u) is: Θ[g] = g(ξ 0) Q j f(ξ i ), π(ξ 0 ) whee j=0 k(ξ j 1,ξ j ) Q 0 = 1, Q j = Q j 1, j = 1, 2, 3,..., p(ξ j 1,ξ j ) and ξ 0,ξ 1,... is a Makov chain in Ω with initial density function π(x) and tansition densities p(x, y), which ae toleant to g(x) and k(x, y), espectively [2,8,15]. We use the following definition of the toleant density function. Definition 4.1. It is said that the density p(x) is toleant to the function f(x) if p(x) > 0 in all points x Ω, with f(x) 0. In [4] was poved that the integal tansfomation kenel in local integal epesentation can be used as a tansition density function in the Makov pocess. In ou case: { M y L p (y, x), when x Ω\ Ω, p(x, y) = k(x, y) = 0, when x Ω, and { B(x) f(x) = L p(y, x)φ(y) dy, ψ(y), when x Ω\ Ω, when x Ω, The function p(x, y) can be expessed in spheical coodinates as [4]: p(, w) = sin θ 4π p() q p (R) p (w),

11 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) whee [ ] b(x + w) cos(b, w) + c(x + w) R c(x + R w)2 p(ρ) p (w) = 1 + p(ρ) dρ p() p() ρ dρ. Fo simulating andom vaiable with density function p (w) we use the selection algoithm [4]. Since, b R p (w) 1 + p(ρ) dρ = h(), p() the function h() can be used as a majoant. Hee follows the algoithm fo one andom walk [4]: 1. Calculate the adius R(x) of the maximal ball of cente x lying inside Ω. 2. Compute a ealization of the andom vaiable τ with the density: p() q p (R). 3. Calculate the function: b R h() = 1 + p(ρ) dρ. p() 4. Compute independent ealizations w j of an unit isotopic vecto in R Constuct independent ealizations γ j of an unifomly distibuted andom vaiable in the inteval [0, 1]. 6. Repeat steps 4 and 5 until find the paamete j 0 given by: j 0 = min{j : h()γ j p (w j )}. The andom vecto w j0 has the density p (w). 7. Calculate the andom point y, using the following fomula: y = x + w j0. The value = y x is the adius of the ball lying inside Ω and having a cente at x. 8. Stop the algoithm when the andom pocess eaches the ε-stip Ω ε, i.e. y Ω ε.ify/ Ω ε then the algoithm has to be epeated fo x = y Case A: p() = p = const Hee, p() q p (R) = 1 R, and consequently fo sampling the jump in WOB pocess one can use the fomula: = Rγ, whee γ is an unifomly distibuted andom vaiable in the inteval [0, 1].

12 598 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) Fo majoant function h(),wehave: h() = 1 + b (R ), and fo p (w) we obtain: [ 3 ] ( ) R p (w) = 1 + b i (x + w)w i + c(x + w) (R ) c(x + w) 2 ln. i= Case B: p() = e k,k = b + (R/4)c In this case: p() q p (R) = and consequently, kek e kr 1, = 1 k ln(1 + (ekr 1)γ), whee γ is an unifomly distibuted andom vaiable in the inteval [0, 1]. Fo majoant function h(),wehave: h() = 1 + b k (ek(r ) 1) and fo p (w) we obtain: [ 3 p (w) = 1 + b i (x + w)w i + c(x + w) i=1 ] 1 c(x + R w)2 e kρ k (ek(r ) 1) e k ρ dρ. 5. Numeical tests Numeical tests ae pefomed in ode to compae the poposed algoithms on a poblem, which is simila to some eal-life poblems. The examples we conside hee deal with the following bounday-value poblem: 3 ( 2 u(x) i=1 x 2 i + b i (x) u(x) ) + c(x)u(x) = 0 in Ω = [0, 1] 3, x i u(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. In ou tests: 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 ), (18)

13 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) Table 1 CPU time in seconds fo the case when ε = 0.1 and a i = 0.25 (i = 1, 2, 3) fo diffeent values of the numbes of the Makov chains N N p() = const p() = e k p() = e k and c(x) = (a a2 2 + a2 3 ), whee a 1,a 2,a 3 ae paametes of the poblem. We conside thee cases fo the coefficients: the fist case, when a i = 0.25 (i = 1, 2, 3); the second case, when a i = 0.5 (i = 1, 2, 3); the thid case, when a i = 1 (i = 1, 2, 3). Such kind of poblems appea in envionmental mathematics and descibe the pollution tanspot due to advection and diffusion and take into account the deposition of pollution in aeas fee of emission souces (fo moe details, see [16]). We also use diffeent values to define the ε-stips: ε = 0.01; ε = 0.05; ε = 0.1. The numeical expeiments estimate the solution at the point: x 0 = (0.5, 0.5, 0.5), i.e. g(x) = δ(x x 0 ). Some numeical esults ae pesented in Table 1. We compae the computational time in cases when p() = const (algoithm A), p() = e k (algoithm B) and p() = e k. The last case is used as a basic algoithm. One can see, that the algoithm A is much faste than the basic algoithm. It can be also seen, that the algoithm B is even a little bit faste than the basic one. The eason fo such a esult could be that the numbe of opeations of the poposed algoithms is less, o the aveage length of the Makov chains is less. Table 2 shows the aveage length of the Makov chain fo all thee algoithms. One can see, that the length of the Makov chains is pactically the same (fo the both of poposed algoithms the length of the Makov chains is a little bit less). It means that the eason fo deceasing the computational time fo the Table 2 Aveage lengths of the Makov chains in case when ε = 0.1 and a i = 0.25 (i = 1, 2, 3) fo diffeent values of N N p() = const p() = e k p() = e k

14 600 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) Table 3 Aveage lengths of the Makov chains in case when N = 200,000 ε a i (i = 1, 2, 3) p() = const p() = e k p() = e k Table 4 Maximal lengths of the Makov chains in case when N = 200,000 ε a i (i = 1, 2, 3) p() = const p() = e k p() = e k algoithm A is mostly the fact that this algoithm is less time consuming. Ou numeical tests show that the length of the Makov chains (the numbe of Monte Calo iteations) does not depend on the numbe N of the Makov chains. Such a esult can be expected, because the numbe of the Monte Calo iteations depends on the adius of convegence of the iteative opeato and the value of the ε-stip. As smalle is the adius of convegence of the iteative opeato, smalle is the numbe of iteations (that is the length of the Makov chain fo the algoithms unde consideation). As smalle is the ε-stip of the domain, lage p()=const p()=e -k p()=e k CPU N Fig. 1. Dependence of the CPU time fom the numbe of andom tajectoies in case when ε = 0.1, a = 0.25.

15 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) is the numbe of iteation (the aveage length of the Makov chain). Some of ou numeical tests pesented in Table 3 illustate these statements. The fist line pesents the esults of the aveage length of the Makov chain (which coesponds to the aveage numbe of the Monte Calo iteations) in the case when ε = 0.1 and a i = 0.25 (i = 1, 2, 3). The length of the chain is aveaged on a lage numbe of Makov chains (N = 200,000). Tests ae pefomed fo diffeent initial value of the pseudo-andom numbe geneato. The aveage length of the Makov chains pactically does not depend on the initial value of the pseudo-andom numbe geneato. Table 3 contains the same paamete obtained fo the smalle ε-stip (ε = 0.01). In this case, one can obseve a significant inceasing of the aveage numbe of Monte Calo iteations. Some numeical tests wee pefomed in ode to study how the aveage length of the Makov chain depends of the paametes of the poblem. In ou tests we vay the value of the coefficients a i (i = 1, 2, 3). It coesponds to vaiation of the advection pat of the poblem, because these coefficients define the advection coefficients b i (i = 1, 2, 3) (see (18)). The numeical esults show that such kind of changes in the opeato (espectively, in the Monte Calo iteative opeato) does not change a lot the numbe of the Monte Calo iteations. Fom the esults pesented in Table 3, one can see that the aveage length of the Makov chain does not change significantly when the values of a i ae changed fom 0.25 to 0.5. Fo these algoithms it is also impotant to know the maximal length of the Makov chain. Table 4 pesents the esults of length of Makov chains with the maximum states, which coesponds to the maximal numbe of Monte Calo iteations. The Makov chain with the maximal length is chosen among a lage numbe of chains (N = 200,000). The esults ae obtained fo ε = 0.1 and a i = 0.25 (i = 1, 2, 3) Eo p()=const p()=e^{k} p()=e^{-k} N Fig. 2. Convegence of the algoithms fo small values of N, ε = 0.1, a = 0.25, u(x 0 ) =

16 602 I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) Simila esults ae obtained fo ε = One can see, that thee is significant dependence of the maximal length of the Makov chain fom the value of the ε-stip of the domain. The dependence of the CPU time fom the numbe of andom tajectoies (numbe of the Makov chains) is shown in Fig. 1. One can see, that the dependence is linea. It is because the computational wok inceases linealy with the inceasing of the numbe of andom tajectoies N. The diffeence between the exact and the appoximate solutions is pesented in Figs. 2 and 3 as a paamete, called Eo. The convegence of all thee algoithms is pactically the same. One can expect such a esult, because the convegence depends on the adius of convegence of the iteative opeato of the Monte Calo method, which is the same fo all thee algoithms. Some esults of the convegence of the algoithms unde consideation ae pesented in Fig. 3. One can see, that fo values of N lage than 1000 all algoithms convege. The speed of convegence is pactically the same fo all thee algoithms. As the exact solution u(x) is geate than 1 fom Figs. 2 and 3, one can also see that the elative eo is always less than 5%, which satisfies the equiements fo the needed accuacy of the algoithms fo solving this kind of poblems. Unde elative eo, we assume the atio among absolute eo and the exact solution, i.e. Eo = u(x 0) ũ(x 0 ), u(x 0 ) whee ũ(x 0 ) is the appoximate value of the solution u(x) at the point x 0 Ω p()=const p()=e^{k} p()=e^{-k} 0.01 Eo N Fig. 3. Vaiation of the eo fo lage values of N, ε = 0.1, a = 0.25, u(x 0 ) =

17 6. Concluding emaks I.T. Dimov, R.Y. Papancheva / Mathematics and Computes in Simulation 63 (2003) We study two new gid-fee Monte Calo algoithms poposed in this wok. The algoithms unde consideation ae based on two diffeent choices of the density of the adius of the balls used in Monte Calo simulations. The algoithms ae descibed and analyzed. We obtain integal epesentations fo constant and exponential densities and pove that the integal tansfomation kenels of the coesponding epesentations can be used as density functions of the gid-fee Monte Calo algoithms. The pefomed numeical tests show that: The numbe of the Monte Calo iteations (the length of the Makov chain) depends mainly of the value of the ε-stip of the domain. The aveage length of the Makov chains pactically does not depend of the numbe of chains. Fo the both of poposed algoithms this vaiation is smalle than the vaiation fo the Sipin s algoithm consideed as a basic algoithm. The algoithms unde consideation convege; the speed of convegence is pactically the same as the speed of convegence of the Sipin s algoithm. One can also see, that the elative eo is always less than 5%, which satisfies the equiements fo the needed accuacy of the algoithms fo solving this kind of poblems. The efficiency of one of the new algoithms which is based on a constant density is highe than the efficiency of Sipin s algoithm. The main eason fo this is the fact that the algoithm based on the constant density has the lowest computational complexity. One can conclude that algoithm A is pefeable, because of its high algoithmic efficiency and the fact that the speed of convegence of this algoithm is pactically the same as the speed of convegence of algoithm B o the Sipin s algoithm. Acknowledgements Suppoted by BIS Cente of Excellence (Contact No. ICA1-CT ) and by the Ministy of Education and Science of Bulgaia unde Gant #MM 902/99. Refeences [1] A.V. Bitzadze, Equations of the Mathematical Physics, Nauka, Moscow, [2] J.H. Cutiss, Monte Calo methods fo the iteation of the linea opeatos, J. Math. Phys. 32 (4) (1954) [3] I.T. Dimov, Efficiency estimatos the Monte Calo algoithms, in: Bl. Sendov, R. Lazaov, I.T. Dimov (Eds.), Poceedings of the Intenational Confeence on Numeical Methods and Applications, Publishing House of the Bulgaian Academy of Sciences, Sofia, 1989, pp [4] I.T. Dimov, T. Guov, Estimates of the computational complexity of iteative Monte Calo algoithm based on Geen s function appoach, Math. Comput. Simul. 47 (1998) [5] I.T. Dimov, A. Kaaivanova, H. Kuchen, H. Stoltze, Monte Calo algoithms fo elliptic diffeential equations a data paallel functional appoach, J. Paallel Algoithms Appl. 9 (1996) [6] I.T. Dimov, O. Tonev, Monte Calo algoithms: pefomance analysis fo some compute achitectues, J. Comput. Appl. Math. 48 (1993)

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