## 2. A direct method to solve the variational principleDirect methods in variational problems are well-known especially in
applied mathematics (see, e.g., Gelfand & Fomin 1963). Suppose
that one can find a More precisely, we assume that for any function , there is a choice for the coefficients such that Let us now introduce a sequence of trial mass maps We further require that the function
minimizes the functional Solving this set of The method described here can be easily applied to our problem. In fact, by expanding as in Eq. (8), we find The previous equation, for ,
represents a linear system of However, we note that care must be taken in the choice of a
As our problem involves , the completeness has to be referred to the set of the gradients. In other words, the set is complete if for every implies . 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 can be taken to satisfy a suitable orthonormality condition, so that the gradients of the functions satisfy where for and 0 otherwise. Then Eq. (10) can be rewritten simply as 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
no longer depend on 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
. 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 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 that we want to
calculate. (ii) The method does not require a large amount of memory:
we need to retain only the © European Southern Observatory (ESO) 1999 Online publication: July 16, 1999 |