Regularized Optimization Method For Reconstructing The Source Term In Heat Conduction Equations

A lot of works have been done in mathematical and physical inverse problems in the past decades, in which the inverse source identification problems arise in many branches of applied science and engineering and technology, purpose is determining the unknown source from some measurable information related to the source. As we all know, this inverse source problem is ill-posed since small errors inherently presented in the practical measurement can induce enormous and highly oscillatory errors in reconstructing the unknown heat source. The main results of this paper are as follows:Firstly, well-posedness of reconstructing the source term in heat conduction equation is studied by two kinds of the measurement data, i.e, the existence and uniqueness of the solution for two types of inverse problem are obtained respectively, which are the final additional condition u? x,T? and integral additional condition ? ?0,T?u x t dt.Secondly, the inverse problem is turned into a functional optimization problem by regularization technique, and the finite element method and superposition principle of the numerical solution of partial differential equations is used to obtain the discrete form of continuous function. According to the necessary condition of the multivariate function for extreme value, the inverse problem can be turned into the problem of solving linear algebraic equations, then the numerical solution of the unknown source term in the heat conduction equation is obtained. In order to overcome ill-posedness of inverse problems, we use the damped Morozov discrepancy principle to select the regularization parameters.Finally, some numerical results for one- and two-dimensional examples show that the proposed method is efficient and stabilized with respect to data noise.