The determination of potential energy surfaces (PES) from values calculated ab initio at a set of points or from spectral data (vibration-rotation energy level information and rotation constants) are important and often difficult problems. The former is a "potential interpolation" problem, the latter a "potential inversion" problem. These are indeterminate problems in which the known data is insufficient to yield a unique solution. We present here a new constrained variational approach to the direct solution of these problems. The constraints are the known exact values of the potential or the exact perturbation corrections desired. The variational functional is chosen to provide control of the magnitude and smoothness of the correction function or potential. The method is very simple, very fast computationally, and yields exact solutions to the perturbation or interpolation equations in a single application. The potential inversion is completed by iteration to converge the perturbation solutions for a given set of assigned levels, and then by repeating with additional levels assigned in sequence to the data set to yield a physically acceptable PES very quickly. This procedure yields a smooth PES from which the energy levels in the dataset are calculated exactly. Information on rotational constants may also be used. Both interpolation and inversion procedures are applied to the the two dimensional (R,theta) PES for ArOH(A (2) Sigma(+)). A combined application of these two procedures is also presented in the paper, where we first interpolate a PES from ab initio points and then correct the ab initio fitted surfaces using spectral data.