Application of Monte Carlo methods for solving the regular and degenerate problem of two-phase filtration.pdf Formulation of the problem Let us consider the initially boundary problem for saturation and reduced pressure (5, p) in a given finite domain Qe Rn (n > 2) with the boundary SQ, Q = Q x [0, T], G=dQ.x [0, 7]: c)s - dt m- =drv(K0a'Vs + KlVp + f0), (x,t) e Q, (1) div(KVp + f) = 0, (x, t) e Q, s(x, t) = s0 (x, t), (x, t) e G, (2) (3) p(x, t) = p0 (x, t), (x, t) e G, (4) s(x,0) = su(x,0), xefl, (5) where the coefficients Ко, a, K, f0 , K, and f , as well as the boundary and initial conditions, are given [1]. For the approximate solution of problem (1)-(5), two methods were proposed in [2]: Method 1. 3s - - (6) (7) (8) (9) (10) V,+1 = ~m -p1- + div(X (x= у ) Vs,.+1 ) + B(x, s,.)Vs,.+1 + dt +D( xs )VP,+iVs,+i = 0(xt) e Q, sm (x, t) = s0 (x, t), (x, t) e G, s. j(x,0) = s°(x,0), хеД L2pi+1 = div(AT(x, s,. )Vp(+1 +/(x,s,.)) = 0, (x,r) e Q, Рм 0\\в} = 1, and Px [б) > 0. (28) Proof. From Lemma 2.3.3 (see Lemma 2.3.3, [4] it follows that for the solution u(x) of problem (27) with F = H, ф= 1 the maximum principle is valid: u(x) reaches the smallest value on the boundary of the domain. Hence, u(x) > 0. Let x, be the indicator of an event (the moment of chain termination > i), {An }“=0 be a sequence of o-algebras generated by the chain up to the time moment n. Consider a sequence of random variables n-1 fin = ZH(xi Ух,- +Xnu(xn ). i=0 The sequence {fi }J is a positive martingale with respect to {A }”_0. Indeed, n-1 Mx {fin I An-1 } = Z H (xi )Xi +Mx„_! {Xnu(xn )} = i=0 n-1 = Z H(xi )Xi +xn-1 • J k(xn-1, xn )u (xn )dxn = fin-1. i=0 T ( xn-1) Then, by the martingale convergence theorem [9], there exists a random value such that Mxr\\m q , as и->да with probability 1. Consequently, almost everywhere on the set В H (xB) ->■ 0 as n -> да. Let dCls = {x e Q': dist(x, dCl) < e}, Qs = Q \\ dfls. It is evident that H ( x)> const = c(r.) > 0 on Qs. If /1, is a subset of trajectories from В such that Я(хв) -»0 as n ->■ oo, thenPB [b\\B^ = 0. Hence, if Px |.bJ > 0, then P ldist(xw,aQ)-»o|,B I = Px ldist(xw,aQ)- V о / \\ not Let X = (x0,x1,...,xB,)e51,but dist(xn,3Q)- ->0. Then there also exists £0 and an increasing sequence {nk {nk }l0 such that dist(xB , 5Q) > e0 . Hence, H(xB ) > c(e) > 0, >0, so X g Д. It follows that jx: dist(xn, c< 1) -„ ^ >()j id В . Therefore, if Px (Д) then Px |dist(xB, ci 1) -> 0 |a j = 1. 152 Tastanov M.G., Utemissova A.A., Mayer F.F. Application of Monte Carlo methods Now let us prove that Px (B) > 0. By condition of the lemma, there exists a solution to problem (27) with F = H and ф = 1. Let us denote it as v(x). It can be shown that inf v(x) > 0. x e A martingale r\\n = %nv(xn) is uniformly integrable; therefore, by the martingale convergence theorem, v(x) =MX lim r\\n = 0, if I\\ (В) = Рх{Вг) = 0. And n^>cо this contradiction proves the inequality Px (B1) > 0. The lemma is proved. If we take H(x) = j Л(x, y)dy as H(x), then we can construct estimates for the so- V ( x ) lution u(x) of problem (27) on trajectories of the chain {xm }“=0 with a transition density k(x,y), y eV(x), P(x У) = m j/r \\ [0, y gV (x). The sequence of estimates {"nm}”_0 is determined by the equality m-1 = ^F(x)x, +%■u(x), where x, is the event indicator {the moment of the chain /=0 termination > /'}. Obviously, Mxym = u(x), i.e. estimates ym are unbiased. The se quence {pm }“ forms a martingale with respect to {Am }” - a sequence of o-algebras. Am is generated by the chain up to the time instant m. The last statement is proved in the same way as the Lemma. From this we have Corollary. For a Markovian chain {xn }“=0 determined by the transition density P(x, y) = k(x, y) = NyЛ(у, x), (28) is fulfilled. Let Tj be the moment of chain termination, t2 be the moment when the chain enters the 5-neighborhood of the boundary ts = min(T, t2 ). A sequence {%m}”=0 of unbiased estimates for the solution u(x) of problem (23), (24) is called admissible if there exists a sequence of o-algebras {Ym }m=0 such that Am cYm and Am c Y,^, and % m has the form %m = %m +ymu(xm), where %m Ym are measurable. For an admissible sequence of estimates {%m}”=0, we define a random variable %5 by the equality %п=%тб+Ф( xt6 ), (29) where x denoted a border point closest to x . The definition is correct since T < +“ by virtue of the above Lemma. We finally obtain Theorem. If an admissible sequence of estimates {%m }“_o forms a square integrable martingale with respect to a family of o-algebras, {Ym }“ 0, then the random variable %5 is a e(5)-biased estimate for u(x), its variance is a bounded function of the parameter 5 (e(5) is the modulus of continuity of the function). 153 Механика / Mechanics Proof. Let x denote the indicator of the event {t = t2 } . By the theorem about the free choice transformation [6], Mx?t6 = u(x) therefore; \\u{x)-MJ^\\= \\млч-M£e|<

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