Sequential Correlated Sampling Methods for Some Transport Problems. this volume

0
error
pC ;
W
(2)
where W is the number of random walks used. This means, among other things, that each additional signi cant digit of information requires, on the average, a hundredfold increase in the number of individual random walks simulated. Obviously, this is an unsatisfactory situation when a global solution is sought to great accuracy and when millions of random walks (utilizing hours or even days of the fastest computers) might be needed to achieve errors of perhaps 10%. In 1962, John Halton 4] successfully solved certain matrix problems (which may be thought of as discretized versions of equation (1)) by applying Monte Carlo methods sequentially. Halton's idea was to solve the problem in
0
@ = ? t (x; ) + s R @x ?
1 2
(x; 0 )d 0 ; ?1 1; 0 < x < T; 1; (T; ) = QT ( ); ?1 < 0:
1
(1)
A physical description of this problem and its derivation can be found, for example, in 2] and 3]. In these references one also nds some exploration of the solution, especially in the context of representation through eigenvalues and eigenfunctions. Here we only describe (Fig. 1) the physical phase space that corresponds to problem (1). In Figure 1, we depict a slab of thickness T lled with a homogeneous material. Random walking particles (for example, neutrons or photons), whose source distributions are restricted to x = 0, x = T and are de ned there
successive stages, each of which can be estimated using conventional Monte Carlo methods. With su cient care, the error En after the nth stage can be bounded by the error En? obtained in the previous stage multiplied by a constant (W ) that depends on the number of the random walks used; i.e.,
Sequential Correlated Sampling Methods for Some Transport Problems
Rong Kong & Jerome Spanier
Department of Mathematics Claremont Graduate University 925 N. Dartmouth Ave. Claremont, CA 91711, USA Email: rong.kong@ & jerome.spanier@
1
En
(W )En? :
1
(3)
In fact, often with only a relatively small number of random walks, one can achieve (W ) < 1: Thus, after n stages of such a process, the error can be estimated by En (W )n E ; (4) thus achieving geometric, or exponential, convergence. In this case, additional digits of accuracy can be obtained with only linear, rather than exponential, increases in computer cost, thus o ering enormous potential for improvement over conventional Monte Carlo implementations In this paper, we apply this idea to the continuous transport problem (1), using a generalization of Halton's methods, concentrating here on the algorithms needed and the numerical results obtained. A probabilistic error analysis of this method can be found in 5].
x 0 T
Fig. 1. Phase space corresponding to equation (1)
by Q ( ) and QT ( ); respectively, are injected into the slab from the two boundaries. While moving in the slab, they can be scattered or absorbed, which events are characterized by the total macroscopic cross section, t ; and the scattering cross section, s : For later convenience, we de ne the absorption scattering cross section by a = t ? s The unknown function (x; ) is the particle ux, which is the expected number of particles passing the point x along the direction which makes an angle arccos with the positive x? axis. Our methodology to attack problem (1) will be described in Section 3. Basically, what we do is rst estimate the integral occurring on the right hand side, called the scalar ux, by Monte Carlo methods and then solve the resulting ordinary di erential equation for the full (vector ux) solution. This same idea can also be applied to more general anisotropic transport problems (see 1]). Monte Carlo methods can be useful in solving such problems, but only if the slow convergence associated with conventional applications of the method is overcome. Speci cally, the usual type of probabilistic random walk simulation leads to the estimate of the error, for some constant C;
1 Introduction
In 1], a broad overview is given of Monte Carlo learning algorithms that give rise to geometric convergence for quite general transport problems. In this paper, one of the methods (based on correlated sampling applied sequentially) outlined in 1] is described in much more detail. Similar detail is presented for the method based on importance sampling in 9]. We will consider the following type of transport problem and introduce a new Monte Carlo method to solve it: (0; ) = Q ( ); 0 <