Simultaneous reconstruction of the spatially-distributed reaction coefficient, initial temperature and heat source from temperature measurements at different times

Abstract

In many practical situations concerned with high temperatures/pressures/loads and/or hostile environments, certain properties of the physical medium, geometry, boundary and/or initial conditions are not known and their direct measurement can be very inaccurate or even inaccessible. In such situations, one can adopt an inverse approach and try to infer the unknowns from some extra accessible measurements of other quantities that may be available. However, the simultaneous identification of several non-constant physical properties along with initial and/or boundary conditions is very challenging, especially when it cannot be decoupled, as it combines both nonlinear as well as ill-posedness features. One such new inverse problem concerning the identification of the space-dependent reaction coefficient, the initial temperature and the source term from measured temperatures at two instants t1,t2 and at the final time T, where 0<t1<t2<T, is investigated in this paper. Insight into the uniqueness of solution is gained by considering various particular cases. Moreover, as in practice the input temperature data are usually noise polluted due to the errors that are inherently present, their influence on the solution of inversion has to be assessed. As such, the least-squares objective functional modelling the gap between the measured and computed data is minimized to obtain the quasi-solution to the inverse problem, and the Fréchet gradients are obtained. The conjugate gradient method (CGM) with the Fletcher–Reeves formula is applied to estimate the three unknown coefficients numerically. Numerical examples are illustrated to show that accurate and stable numerical solutions are obtained using the CGM regularized by the discrepancy principle.