Simultaneous reconstruction of the perfusion coefficient and initial temperature from time-average integral temperature measurements

Inverse coefficient identification formulations give rise to some of the most important mathematical problems because they tell us how to determine the unknown physical properties of a given medium under inspection from appropriate extra measurements. Such an example occurs in bioheat transfer where the knowledge of the blood perfusion is of critical importance for calculating the temperature of the blood flowing through the tissue. Furthermore, in many related applications the initial temperature of the diffusion process is also unknown. Therefore, in this framework the simultaneous reconstruction of the space-dependent perfusion coefficient and initial temperature from two linearly independent weighted time-integral observations of temperature is investigated. The quasi-solution of the inverse problem is obtained by minimizing the least-squares objective functional, and the Fréchet gradients with respect to both of the two unknown space-dependent quantities are derived. The stabilisation of the conjugate gradient method (CGM) is established by regularising the algorithm with the discrepancy principle. Three numerical tests for one- and two-dimensional examples are illustrated to reveal the accuracy and stability of the numerical results