The problem of creation and investigation of parallel Monte Carlo algorithms for solving LargeScale Air Pollution Problems is studied. We consider a model of Air pollution transport. This model includes diffusion, advection (drift due to the wind), deposition and chemical reactions. For obtaining the initial and boundary conditions of the model it is necessary to deal with Noisy Data. The work consists of three parts. In the first part we obtain a practical efficient Monte Carlo method for estimating the optimum amount of smoothing from the noisy data. In the second part a special Monte Carlo techniques (Monte Carlo with frozen velocities) for calculation of concentrations is developed and studied. An error analysis of this method is realized. The last part of the work deals with parallel realization of Monte Carlo methods on distributed transputer system. A numerical results for estimation of the Speed-Up and Efficiency are obtained. INTRODUCTION Monte Carlo methods are a powerful tool in many fields of mathematics and engineering. It is known that these methods have proved to be very efficient in solving multidimensional problems in composite domains [1,2,3]. Moreover, it is shown that for some problems(including one dimensional problems) in the corresponding functional spaces Monte Carlo methods have better speed of convergence than the optimal deterministic methods in such functional spaces [4,5]. * Supported by the Ministry of Science and Education of Bulgaria, Grand I 210 /1992. Transactions on Information and Communications Technologies vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 480 Applications of Supercomputers in Engineering An important advantage of these algorithms is that Monte Carlo ones allow to find directly the unknown functional of the solution of the problem with a number of operations, necessary to solve the problem in one point of the domain [1,6]. 1 PRELIMINARIES Our approach is based on an integral representation of the solution of the problem. The Fredholm integral equation t u(r, t) = jdtj k(r', f; r, t)u(r', t')dr' + f(r, t), (1) 0 G where re G C R*; t e [0,T]or (x,x')u(x')dx' + f(r,t), (2) D (« = Ku + /) where x = (r, t) £ D = G x [Q,T] and K is an integral operator, is studied. It is assumed that u, / G L, K G [L — •> L], where Lisa Banach space of integrable functions. Monte Carlo method for evaluation of linear functionals of the solution of the following type /t(z)w(z)dz = (12, A) (3) D is considered. The adjoint equation %* = JT w* 4k (4) will be used. In (4) w*, h G L*, K* G [L* — » L*], L* is the adjoint functional space and K* is an adjoint operator. For many applications L = LI and II / lk= / I /(*) I dx; D \\K\\L,<su.p f\k(x,x')\dx' (5) X J D In the case h(x) G Z/oo, sence L\* = L^ and Transactions on Information and Communications Technologies vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 Applications of Supercomputers in Engineering 481 || A||z_=sup|/t(z) |,zED. (6) It is assumed that || K* ||< 1,where n is a natural number, because the studied method uses the Neuman series CO « = £#'/• (7) i=0 But the condition || KTM ||< 1 is not very strong, since as shown by K.Sabelfeld [7] it is possible to construct Monte Carlo method when von Neuman series does not converge. Using method of analytical extension of resolvent through change of spectral parameter [7,p.60] it is possible to create convergent method when von Neuman series does not converge or to accelerate the rate of convergence when it converges slowly. It is easy to show that J = (u,h) = (/,«•). (8) So, consider Monte Carlo method for evaluation of functional (8).(One can see that when fc(x,xi) = 0we have a problem of evaluation of integrals.) Let £ £ D be a random point with a density p(x) and let there be n realizations of the random point £,-(i — 1,2, ...,ra). Let a random variable #(£) be defined in D, such that E6(t) = J Then the computational problem becomes one of calculating repeated realizations of 9 and of combining them into an appropriate statistical estimator of J. Because of the nature of the process every realization of 0 is a Markov chain. So, next we will use the Markov chain as a synonym of the realization of a random variable of this type. As approximate value of the linear functional J is set up