An application of the method of moments to stochastic equations
A modified form of Galerkin's method is formally applied to an equation involving a stochastic bounded linear operator. The result, in general, is a sequence of stochastic linear algebraic equations. In the case of a statistically homogeneous operator, however, it is possible to obtain a sequence of deterministic linear algebraic equations. The formalism is applied to determining the electric field in a dielectric with a statistically homogeneous random permittivity. Introduction. In formulating a wide variety of continuum problems involving a medium with material properties about which one only has limited information, e.g., the permittivity in a dielectric (e(r)), it is frequently convenient to introduce the concept of a random medium. A random medium may be viewed as a family of media together with a probability distribution defined over the members of the family. The more information one possesses of the relevant material properties, the smaller will be the size of the family encompassed by the random medium and/or the more selective will be the probability distribution. Such formulations result in field equations which are random (stochastic) in the same sense as above. The problem is to invert these equations subject to any auxiliary conditions that may be present, e.g., to determine the family of static electric fields existing in the media (E(r)). The stochastic solution so defined is then a family of solutions together with a probability distribution defined over the members. The carrying out of a solution to the above problem is extremely complicated. In most practical cases it is made impossible by the fact that the probability distribution defined on the family of media involved is not known in detail. Rather, all that is available is some indirect information which is usually expressed in terms of various statistical averages. It is obvious that in such cases, a complete determination of the family of solutions together with a probability distribution to assign to this family is not possible. In the best of situations, it is necessary to be content with obtaining some partial information regarding the solution which likewise is usually in the form of various statistical averages. In most important problems even this is not possible since a desired statistical average of the solution variable is most often not uniquely determined by a finite number of statistical averages of the variable defined by the random medium involved. Complete knowledge of the probability distribution associated with the random medium is required to determine even the simplest statistical average of the solution variable in these cases. Because of this one is led to attempt to look for approximate answers for statistical averages of the solution variable which depend only on a finite number of statistical averages of any parameters which appear in the governing equation. Two approaches to the problem are readily apparent. The first approach is to manip*Keceived August 2, 1967; revised version received August 19, 1968. 522 JOHN J. McCOY [Vol. XXVI, No. 4 ulate and average the stochastic field equations so as to obtain deterministic equations on the statistical averages of the solution variable. The result of such an approach is to obtain an infinite set of deterministic equations. An approximation may then be introduced by truncating the set in some way, usually by invoking some physical argument, and the truncated set of deterministic equations is then solved to obtain the desired statistical averages. This approach is most commonly used and several schemes have been introduced for the systematic truncation of the statistical equations. A second approach would be to attempt to obtain a general solution of the stochastic field equation in the sense that the solution to any member of the family of equations defined by the stochastic equation is a special case of the general solution. Once this has been accomplished the statistical averages of the solution variable that are of interest are directly calculable. The difficulty with this second approach is that the stochastic field equations in the problems of interest are usually of the form of differential or integral equations with coefficients which are random functions of the independent variable (s). There are, in general, no methods available for finding the general solution to such problems exactly, and hence one is required to look for approximate techniques if any success is to be achieved. Toward this end perturbation theory has been used to treat those continuum problems involving a random medium which differs only slightly from a given homogeneous medium. The perturbation theory is of only limited validity, however, and in dealing with stochastic equations it is difficult to ascertain just what are the limits of validity. Another approach which may prove fruitful in obtaining the approximate solution of a differential or integral equation involving stochastic coefficients is to attempt to modify the method first given by Galerkin [1], Galerkin's method has been of great use in approximately solving differential and integral equations with variable, although deterministic, coefficients. Briefly, Galerkin's method consists first of all in introducing a sequence of ever-increasing subspaces of the Hilbert space on which the problem is defined so that the limit of the sequence will be the entire space. Next a projection operation is introduced and used in conjunction with this sequence of subspaces to define a sequence of problems, the elements of which are the projections of the problem to be solved onto each of the subspaces introduced above. Each of the projected problems so obtained requires for its solution the inversion of a set of linear algebraic equations. Intuitively, one might suspect that the solution of the limit of the sequence of projected problems will converge to the solution of the original problem so long as the original problem is selected from some suitably selected class. Convergence proofs have been carried out, particularly in the Russian literature, for several classes of problems [2], [3], [4], [5], [6]. In the absence of such proofs one is forced to rely on intuition strengthened by the usefulness of results obtained by employing the method. Assuming that the sequence of projected solutions converges to the desired solution, then each element of the sequence may be viewed as an approximate solution. How good is such an approximation? Clearly the answer to this depends on the sequence of subspaces which is arbitrarily selected in the first step. Herein lies the art in applying Galerkin's method. It is obvious that a good choice for a sequence of subspaces will depend on the problem to be approximately solved. Herein lies the difficulty in attempting to apply Galerkin's method to a problem involving a stochastic equation. If the stochastic nature of the equation is viewed as an ignorance factor, it is this ignorance which makes a useful choice of base elements difficult. 1969] AN APPLICATION OF THE METHOD OF MOMENTS 523 A modification can be introduced into Galerkin's method which eliminates much of the aspect of art in its successful application. This modification is a formal procedure by which the problem to be solved dictates the sequence of subspaces to be used. Such a modification, termed the method of moments, is presented in a book by Vorobyev [7]. While the method of moments is not as widely applicable as is Galerkin's method, i.e., in its simplest form it is valid only for equations involving linear bounded operators, its loss in generality is counterbalanced by its straightforward application for those problems in which it is applicable. Formal application of the method of moments to a problem involving a stochastic field equation results in a sequence of subspaces, upon which the problem is to be projected, known only in a statistical sense. That is, each member of the family of equations represented by the stochastic equation will result in defining a different sequence of subspace. Similarly, the sequence of projected problems is, in general, also statistical in nature. For a special class of problems, however, it is possible to arrive at a sequence of projected problems which is deterministic. That is, the sequence of problems obtained by projecting a member of the family of equations onto the sequence of subspaces to be associated with this member is independent of the particular member chosen. For this to be true it is necessary that all random functions appearing in the problem be statistically homogeneous, allowing our invoking an ergodic-type hypothesis, i.e., the equating of an ensemble average to an average taken over the independent variable. In the present paper we begin with a short review of Galerkin's method as applied to two types of problems, indicating the calculations to be performed in carrying out a solution. The method of moments procedure is then introduced and it is shown how its formal application to a class of problems involving stochastic operators results in an iteration scheme involving deterministic sets of linear algebraic equations. The method is then applied to the problem of determining the electric field in a medium with statistically homogeneous random variations in permittivity. Details are carried out for a one-term approximation and the result is compared with previously reported work on this problem. 1. General discussion. Let us begin by considering two problems which may be expressed by the equations Ax = / (1.1) and x = ixAx + /. (1.2) In Eqs. (1.1) and (1.2), x is the unknown element and / is the given element of some Hilbert space H, while A is a linear bounded operator defined on II and ^ is a parameter. For clarity it might be well to think of A as a differential or integer operator. For the present we take A and / to be uniquely specified so that we are dealing with a deterministic probl