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Astron. Astrophys. 348, 38-42 (1999)

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2. A direct method to solve the variational principle

Direct methods in variational problems are well-known especially in applied mathematics (see, e.g., Gelfand & Fomin 1963). Suppose that one can find a complete set of functions [FORMULA] on the domain [FORMULA] (the full definition of "complete" will be given below), so that any function on [FORMULA] can be represented as a linear combination of the form

[EQUATION]

More precisely, we assume that for any function [FORMULA], there is a choice for the coefficients [FORMULA] such that

[EQUATION]

Let us now introduce a sequence of trial mass maps

[EQUATION]

We further require that the function [FORMULA] minimizes the functional S: in other words, [FORMULA] are chosen so that the functional S has minimum value. This obviously happens when

[EQUATION]

Solving this set of n equations, we obtain the n coefficients [FORMULA], and thus the function [FORMULA]. By repeating this operation for a sequence of values of n, we find a sequence of functions [FORMULA]. These functions, under suitable assumptions (verified in our problem), have the following properties (see Gelfand & Fomin 1963 for a detailed discussion): (i) Let us call [FORMULA] the value of S when [FORMULA] is replaced by the function [FORMULA]. Then, obviously, the sequence [FORMULA] is not increasing. (ii) If the set [FORMULA] is complete, then the functions [FORMULA] converge to the solution [FORMULA] of the problem. This method thus provides a way to obtain the function [FORMULA] with desired accuracy.

The method described here can be easily applied to our problem. In fact, by expanding [FORMULA] as in Eq. (8), we find

[EQUATION]

The previous equation, for [FORMULA], represents a linear system of n equations for the n variables [FORMULA]. Its solution is thus the set of coefficients to be used in Eq. (8).

However, we note that care must be taken in the choice of a complete set of functions. Let us define, for the purpose, the product [FORMULA] between two generic vector fields [FORMULA] and [FORMULA] as

[EQUATION]

As our problem involves [FORMULA], the completeness has to be referred to the set of the gradients. In other words, the set [FORMULA] is complete if

[EQUATION]

for every [FORMULA] implies [FORMULA]. It is easy to show that this condition is equivalent to Eq. (7).

The direct method can be further simplified if a set of functions [FORMULA] can be taken to satisfy a suitable orthonormality condition, so that the gradients of the functions [FORMULA] satisfy

[EQUATION]

where [FORMULA] for [FORMULA] and 0 otherwise. Then Eq. (10) can be rewritten simply as

[EQUATION]

Thus, with the use of an orthonormal set of functions we have secured two important advantages: (i) The linear system (10) has been diagonalized, so that its solution is trivial. (ii) The coefficients [FORMULA] no longer depend on n: that is, the coefficients of the exact solution are given by [FORMULA].

Because of these advantages, whenever possible an orthonormal set of functions should be used. We note, however, that the orthonormality condition (13) depends on the field of observation [FORMULA]. Even if the existence of an orthonormal set of functions is always guaranteed by the spectral theory for the Laplace operator (see Brezis 1987), for "strange" geometries, it may be non trivial to find a complete orthonormal set of functions. In such cases, we need to solve the linear system (10).

The direct method described above has several advantages with respect to the "kernel" method and to the over-relaxation method: (i) The method is fast in the case where an orthonormal set of functions can be found. In fact, we need only to evaluate one integral for each coefficient [FORMULA] that we want to calculate. (ii) The method does not require a large amount of memory: we need to retain only the n values of the coefficients [FORMULA]. (iii) The precision of the inversion is driven in a natural way by the value of n. Typically, the larger [FORMULA] is, the smaller the length scale of [FORMULA] (see below). (iv) In some cases, the decomposition of the mass density [FORMULA] in terms of the functions [FORMULA] can be useful.

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© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999
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