The problem of distributing points on a domain, like ball, plays a special
role in the fields like geosciences and medical imaging. Therefore, we present
an equidistribution theory with a focus on obtaining low-discrepancy point
grids on a 3-dimensional ball.
The connection of the discrepancy method and the quadrature points on a
given domain is quite well known. We approximate the integral of a function
given on a bounded domain by the sum of function values at a specific set of
points together with some weights. The idea is to get the best approximation
with the fewest possible function values. The ansatz is logical, if the chosen
data set is well distributed on the whole domain. This perspective, with the
ball as a domain, enables us to get nice configurations as well as suitable
approximations to the integrals of functions on the ball.
It is, for instance, important for choosing the centres of the radial basis
functions as they are needed for regularization methods such as the RFMP
algorithm and the ROFMP algorithm, developed by the Geomathematics
Group at the University of Siegen for ill-posed inverse problems with particular
focus on the sphere and the ball as domains of the unknown functions.
Additionally, it is also important for computational purposes. For instance,
for the wavelet methods with data given on the ball, where one needs to have
an appropriate quadrature rule.