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.