Direct reconstruction of a multidimensional heat equation

Abstract

We are concerned with a coefficient inverse problem of a multidimensional heat equation. The objective is to reconstruct the sought coefficient from a sequence of observations of the solution taken at a single point. To do so we first obtain an explicit formula for the sought coefficient, and then see how we can approximate it using few observations only. We also show that asymptotics of the solution help reduce the data processing to overcome the curse of dimensionality. This new and direct reconstruction method is fast and gives an alternative to iterative and Newton's type methods. Numerical examples are provided at the end.