Efficient solution of a three-dimensional inverse heat conduction problem in pool boiling

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In this paper, we consider a three-dimensional transient inverse heat conduction problem arising in pool boiling experiments, i.e., the reconstruction of the surface heat flux from pointwise temperature observations inside a heater. We show that the inverse problem is ill-posed and utilize Tikhonov regularization and conjugate gradient methods together with a discrepancy stopping rule for a stable solution. We investigate the proper choice of regularization terms, which not only affects stability of the reconstructions but also greatly influences the quality of reconstructions in the case of limited observations. For the numerical solution of the governing partial differential equation, a space-time finite element method is used. This allows us to compute exact gradients for the discretized Tikhonov functional, and enables the use of conjugate gradient methods for the solution of the regularized inverse problem. We discuss further aspects of an efficient implementation, including a multilevel optimization strategy, together with an implementable stopping criterion. Finally, the proposed algorithms are applied to the reconstruction of local boiling heat fluxes from experimental data.