Determination of the ambient temperature in transient heat conduction.

The restoration of the space- or time-dependent ambient temperature entering a third-kind convective Robin boundary condition in transient heat conduction is investigated. The temperature inside the solution domain together with the ambient temperature are determined from additional boundary measurements. In both cases of the space- or time-dependent unknown ambient temperature the inverse problems are linear and ill-posed. Least-squares penalized variational formulations are proposed and new formulae for the gradients are derived. Numerical results obtained using the conjugate gradient method combined with a boundary element direct solver are presented and discussed

Determination of a source in the heat equation from integral observations

A novel inverse problem which consists of the simultaneous determination of a source together with the temperature in the heat equation from integral observations is investigated. These integral observations are weighted averages of the temperature over the space domain and over the time interval. The heat source is sought in the form of a sum of two space- and time-dependent unknown components in order to ensure the uniqueness of a solution. The local existence and uniqueness of the solution in classical Hölder spaces are proved. The inverse problem is linear, but it is ill-posed because small errors in the input integral observations cause large errors in the output source. For a stable reconstruction a variational least-squares method with or without penalization is employed. The gradient of the functional which is minimized is calculated explicitly and the conjugate gradient method is applied. Numerical results obtained for several benchmark test examples show accurate and stable numerical reconstructions of the heat source