  Overdetermined systems of first kind integral equations appear in many
applications. When the right-hand side is discretized, the resulting
finite-data problem is ill-posed and admits infinitely many solutions. We
propose a numerical method to compute the minimal-norm solution in the presence
of boundary constraints. The algorithm stems from the Riesz representation
theorem and operates in a reproducing kernel Hilbert space. Since the resulting
linear system is strongly ill-conditioned, we construct a regularization method
depending on a discrete parameter. It is based on the expansion of the
minimal-norm solution in terms of the singular functions of the integral
operator defining the problem. Two estimation techniques are tested for the
automatic determination of the regularization parameter, namely, the
discrepancy principle and the L-curve method. Numerical results concerning two
artificial test problems demonstrate the excellent performance of the proposed
method. Finally, a particular model typical of geophysical applications, which
reproduces the readings of a frequency domain electromagnetic induction device,
is investigated. The results show that the new method is extremely effective
when the sought solution is smooth, but gives significant information on the
solution even for non-smooth solutions.
