Reconstruction of an additive space- and time-dependent heat source

In this paper, we consider the inverse problem of simultaneous determination of an additive space- and time-dependent heat source together with the temperature in the heat equation, with Dirichlet boundary conditions and two over-determination conditions. These latter ones consist of a specified temperature measurement at an internal point and a time-average temperature condition. The mathematical problem is linear but ill-posed since the continuous dependence on the input data is violated. In discretised form, the problem reduces to solving an ill-conditioned system of linear equations. We investigate the performances of several regularisation methods and examine their stability with respect to noise in the input data. The boundary element method combined with either the truncated singular value decomposition, or the Tikhonov regularisation, using various methods for choosing regularisation parameters, e.g. The L-curve method, the generalised cross-validation criterion, the discrepancy principle and the L-surface method, are utilised in order to obtain accurate and stable numerical solutions.