Estimation metrics and optimal regularization in a Tikhonov digital filter for the inverse heat conduction problem

Tikhonov regularization for the inverse heat conduction problem (IHCP) is considered a “whole domain” or “batch” method, meaning that observations are needed over the entire time domain of interest, and that calculations must be performed all-at-once in a batch. This paper examines the structure of the Tikhonov regularization problem and concludes that the method can be interpreted as a sequential filter formulation for continuous processing of data. Several general observations regarding features of the filter formulation are noted. Two error norms are discussed: one regarding temperature and one regarding heat flux. It is shown that these metrics can be split into two parts: one dependent on the heat flux history (bias error) and one dependent on the measurement noise (random error). Two examples demonstrate that the optimal selection of the regularization parameter to minimize the heat flux error yields results similar to the classical Morozov principle defined through temperature error, and that the results are relatively insensitive to the precise selection of the parameter.