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Overdetermined systems of first kind integral equations appear in manyapplications. When the right-hand side is discretized, the resultingfinite-data problem is ill-posed and admits infinitely many solutions. Wepropose a numerical method to compute the minimal-norm solution in the presenceof boundary constraints. The algorithm stems from the Riesz representationtheorem and operates in a reproducing kernel Hilbert space. Since the resultinglinear system is strongly ill-conditioned, we construct a regularization methoddepending on a discrete parameter. It is based on the expansion of theminimal-norm solution in terms of the singular functions of the integraloperator defining the problem. Two estimation techniques are tested for theautomatic determination of the regularization parameter, namely, thediscrepancy principle and the L-curve method. Numerical results concerning twoartificial test problems demonstrate the excellent performance of the proposedmethod. Finally, a particular model typical of geophysical applications, whichreproduces the readings of a frequency domain electromagnetic induction device,is investigated. The results show that the new method is extremely effectivewhen the sought solution is smooth, but gives significant information on thesolution even for non-smooth solutions.

Regularized minimal-norm solution of an overdetermined system of first kind integral equations