We have developed an analytical technique that predicts both in-plane and out-of-plane distortion arising from an arbitrary stress distribution on a membrane surface. Our technique can also solve the inverse problem; i.e., we can predict the stress distribution which, when applied to any existing distortion, eliminates it. These techniques, based on the variational method, require one to formulate the total energy that results from the membrane distortion, which is a straightforward task even for complicated stress distributions. One then iteratively searches for the distortion map that minimizes this total energy, which, according to minimum energy principle, should be the true membrane distortion for a given stress distribution. The inverse problem is solved analytically by performing a functional minimization on the total energy and solving for the stress distribution. This calculational technique should enable us to correct for arbitrary distortions by, for example, applying another film or differential heating.