Inverse problems basis functions, choice

FIGURE 6.1.9 Cumulative distribution for the size of breakage fragements from the solution of the inverse problem for various choices of the number of basis functions (From Sathyagal et al, 1995). (Reprinted with permission from Elsevier Science.) Note that x here represents the ratio of the breakage rate of broken fragment to that of the parent particle. 

The choice of the basis functions, 0 , is important in both forward and inverse electromagnetic problems. It depends on the specific geophysical problem and the required accuracy of discretization. 

The choice of the basis functions depends crucially on the behavior of the self-similar distribution (see footnote 5). For example, suppose that the self-similar distribution z6 (z) has the asymptotic behavior (ju < 1) in the region of z close to zero. Then it is possible to show (see Appendix of Sathyagal et al, 1995) that the function g u) is approximated by for u close to zero. In other words, g(u) is of order 0(m ). Consequently, g u) is not analytic at w = 0, and a very large number of basis functions in the expansion (6.1.9) are required to describe adequately the behavior near the origin. This problem can be overcome by choosing basis functions that have the same dependence on u near w = 0, as g u) does. Incorporating as much known analytical information as possible about the nature of the solution is an important aspect of the solution of inverse problems. Let us see how... 

We now return to the issue of the choice of basis functions for solution of the inverse problem (6.2.8). The behavior of the aggregation frequency that relates to the small- / behavior of the function (j)(rj) is the issue of specific interest. We choose to fit with y-distributions that can accommodate either a singular or nonsingular nature of the self-similar distribution near the origin and accordingly set... 

In this example, there are nine uncoupled states and, of course, nine coupled states. The coefficients that have been worked out are linear expansion coefficients, and so they can be arranged in a 9 x 9 matrix. This is the transformation (matrix) from the uncoupled basis to the coupled basis. This matrix and its inverse contain the coefficients for making up one kind of basis function from the other set. These coefficients are used in many problems in quantum physics, and they are often called vector coupling coefficients. Since they can be worked out once and for all for specific choices of /j and there are many tabulations of the values, and many alternate procedures have been devised for finding them. They are sometimes scaled by different factors for convenience in different circumstances. Names given to vector coupling coefficients under various prescriptions include Clebsch-Gordon coefficients, Racah coefficients, 3 -j symbols, and 6- j symbols. 

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