Optimal Filtering for the Backward Heat Equation

Abstract

For the backward heat equation, stabilized by an a priori initial bound, an estimator is determined for intermediate values that is optimal with respect to the bound and the observation accuracy. It is shown how this may be implemented computationally with error estimates for the computed approximation which can be made arbitrarily close to the uncertainty level induced by the ill-posedness of the underlying problem. Thus, the feasibility of this for practical computation, inevitably severely limited by that inherent uncertainty, is as good as possible.