A method for identifying a spacewise-dependent heat source under stochastic noise interference

We consider an inverse problem of determining a spacewise-dependent heat source in the time-dependent heat equation using the usual conditions of the direct problem and information from temperature measurements at a given single instant of time. This spacewise-dependent temperature measurement ensures that the inverse problem has a unique solution, but the solution is unstable and hence the problem is ill-posed. Furthermore, it is assumed that information about this temperature measurement is available only at a finite number of points, and moreover, these measurements are blurred by a stochastic noise. The continuous problem is discretized using a finite element method and for the discrete model an estimate of the heat source is obtained by a Tikhonov regularization. It is proved that increasing the number of measurements and decreasing the mesh size produces a sequence of solutions which under appropriate conditions converge to the continuous heat source.