Optimal hybrid parameter selection for stable sequential solution of inverse heat conduction problem

To deal with the ill-posed nature of the inverse heat conduction problem (IHCP), the regularization parameter α can be incorporated into a minimization problem, which is known as Tikhonov regularization method, popular technique to obtain stable sequential solutions. Because α is a penalty term, its excessive use may cause large bias errors. A ridge regression was developed as an estimator of the optimal α to minimize the magnitude of a gain coefficient matrix appropriately. However, the sensitivity coefficient matrix included in the gain coefficient matrix depends on the time integrator; thus, certain parameters of the time integrators should be carefully considered with α to handle instability. Based on this motivation, we propose an effective iterative hybrid parameter selection algorithm to obtain stable inverse solutions. We considered the Euler time integrator to solve IHCP using the finite element method. We then considered β, a parameter to define Forward to Backward Euler time integrators, as a hybrid parameter with α. The error amplified by the inverse algorithm can be controlled by α first by assuming β=1. The total error is then classified into bias and variance errors. The bias error can be computed using the maximum heat flux change, and the variance error can be calculated using the measurement noise error generated by prior information. Therefore, α can initially be efficiently defined by the summation of the bias and variance errors computed in a time-independent manner. Reducing the total error for better stability of the inverse solutions is also available by adjusting β, which is defined to minimize the magnitude of gain coefficient matrix when spectral radius of the amplification matrix is less than one. Consequently, α could be updated with new β in the iteration process. The proposed efficient ridge estimator is essential to implement the iterative hybrid parameter selection algorithm in engineering practice. The possibility and performance of the hybrid parameter selection algorithm were evaluated by well-constructed 1D and 2D numerical examples.