Estimation of spatially varying parameters in partial differential equations†

The estimation of a vector of unknown spatially varying parameters in non-linear partial differential equations from noisy observations is considered. Two algorithms are presented. The first is a method of steepest descent based on consideration of the unknown parameter vector as a control vector. The second is baaed on treating the parameter as an additional state vector and employing least-square filtering. Computational results are presented on the estimation of the diffusivity in the heat equation