Optimal combinations of Tikhonov regularization orders for IHCPs

Abstract

In achieving a regularized solution of inverse heat conduction problems (IHCPs), Tikhonov Regularization works based on adding either a zeroth-, a first-, or a second-order term to the sum of squared errors function. In other words, considering combinations of these terms are not often considered. This work investigates employing standard optimization techniques in order to obtain optimal regularization parameters when combinations of these three regularization terms are used. Five different heat pulses are used as test cases: step, triangular, quadratic, quartic, and half-sine. The criterion used to find the optimal value for the regularization parameters is sum of the squares of deviations between the estimated heat flux and the exact heat flux pulse. A hybrid method which utilizes both the Genetic Algorithm and the Pattern Search is used for the optimization through functions defined in MATLAB software. Moreover, a general case containing all five heat flux test cases is considered in finding the optimal combination of three versions of Tikhonov method. All separate and general optimal combined models are used in estimating five pulse heat flux functions, and for each case, the RMS errors are calculated to give an insight toward the combined Tikhonov regularization technique. The proposed approach is also used to recover the surface heat flux using measured temperature data from an experiment. Some discussion about the Morozov discrepancy principle for this application is considered at the end.