ON THE PROBLEM OF SOLVING THE DIRICHLET PROBLEM

FOR THE LAPLACE EQUATION BY THE MONTE CARLO METHOD

A. V. Pyzhyanov

(Presented by the scientific seminar of the Laboratory of Computer Engineering and Automation, Research Institute of Nuclear Physics, Electronics and Automation)

In studying the motion of charged particles in complex electromagnetic fields, and also in a number of problems of electron optics, it is often necessary to solve the Laplace (Poisson) equations in a three-dimensional domain. For most such problems, the most acceptable approach is the use of the Monte Carlo method [2]. The application of medium-speed digital computers for implementing the Monte Carlo algorithm is limited because of its slow convergence. Recently, considerable attention has been given to the use of hybrid computing equipment, which makes it possible to reduce computation time by 2–3 orders of magnitude [1].

In the present work we consider questions of constructing and using a hybrid computing system (HCS) of a problem-oriented type, consisting of standardized standard elements.

1. The block diagram of an HCS intended for solving three-dimensional Laplace equations is presented in Fig. 1. The diagram consists of three identical channels. Integrators I, together with analog generators of “white” noise, solve, according to the Monte Carlo method algorithm, a system of stochastic differential equations [1] of the form:

\[ \left\{ \begin{aligned} \frac{d}{dt}x(t) &= N_1, \quad x(0)=x_0,\\ \frac{d}{dt}y(t) &= N_2, \quad y(0)=y_0,\\ \frac{d}{dt}z(t) &= N_3, \quad z(0)=z_0, \end{aligned} \right. \]

where \(N_1\), \(N_2\), and \(N_3\) are independent Gaussian processes of the “white noise” type with unit spectral densities.

The sampling-and-hold circuit, implemented on integrators 1 and 2, makes it possible to store voltage values corresponding to the outputs of the wandering point on the boundary of the domain. To form the boundary of the domain under study, a method of one-parameter specification of functions is used, which makes it possible to avoid the need to construct function converters of several variables [3].

The analog part of the system, including digital elements (signature comparator, switching circuits), is built from standardized semi-

conductor elements of the ADC complex [4]. To connect a general-purpose digital computer with the analog part, it is advisable to use series VII conversion devices, manufactured by domestic industry and intended for joint operation with various digital computers in analog-digital computing complexes.

The hybrid computing system considered makes it possible to obtain the solution of the Laplace equation at one point in a time of the order of 120–40 seconds with an error of 1.5–2%.

Fig. 1. Block diagram of the HCS for solving the three-dimensional Laplace equation with Dirichlet boundary conditions.

1. It is known that the computation time by the Monte Carlo method grows in proportion to the number of points of the domain at which the solution must be found. For this reason, the method is used chiefly for estimating potential values at individual points of the domain. For the effective solution of most problems at a large number of points of the domain, we apply an algorithm based on the method of separating elementary subdomains, the essence of which is as follows. The fact that the Monte Carlo method makes it possible to obtain independent solutions at individual points is used to isolate a domain for which an analytic solution can be obtained. Such an algorithm is easily implemented with the aid of a hybrid computing system and makes it possible to reduce the time for computing potentials at a large number of points of the domain by approximately an order of magnitude in comparison with the time required to compute the potentials at the same points by the usual algorithm.

In conclusion, it should be noted that the “hybrid” Monte Carlo method makes it possible simultaneously to solve several problems differing from one another in boundary conditions or in the type of the domain under investigation [5], which is an obvious advantage of the method in modeling various fields. The great flexibility of the HCS considered makes it possible to use it for solving a wide range of problems connected with the calculation of particle motion under the action of complex fields.

REFERENCES

1. G. Korn. Simulation of Random Processes on Analog and Analog-Digital Machines. Mir, 1968.

2. I. M. Bleivas, G. R. Stefanyuk. On the solution of the Dirichlet problem for the Laplace and Poisson equations by the method of statistical trials. Numerical Methods

calculation of electron-optical systems (Proceedings of the 1st All-Union Seminar), Novosibirsk, Nauka, 1967.

1. M. L. Dashevsky, A. N. Kabalevsky. One method for solving the Dirichlet problem for the Laplace equation. Avtomatika i telemekhanika, 1967, No. 9.

2. M. V. Galperin. A high-speed system of elements for control and specialized hybrid computers. In: Analog and Analog-Digital Computing Technology. Issue 4, Moscow, 1971.

3. B. Ya. Kogan, R. Krtolice, I. A. Lapina. On the solution, on a hybrid complex, of nonstationary boundary-value problems with variable boundary conditions. Avtomatika i telemekhanika, 1972, No. 2.