  We consider a time-dependent linear diffusion equation together with a
related inverse boundary value problem. The aim of the inverse problem is to
determine, based on observations on the boundary, the non-homogeneous diffusion
coefficient in the interior of an object. The method in this paper relies on
solving the forward problem for a whole family of diffusivities by using a
spectral Galerkin method in the high-dimensional parameter domain. The
evaluation of the parametric solution and its derivatives is then completely
independent of spatial and temporal discretizations. In case of a quadratic
approximation for the parameter dependence and a direct solver for linear least
squares problems, we show that the evaluation of the parametric solution does
not increase the complexity of any linearized subproblem arising from a
Gauss-Newtonian method that is used to minimize a Tikhonov functional. The
feasibility of the proposed algorithm is demonstrated by diffusivity
reconstructions in two and three spatial dimensions.
