Identification of an unknown time-dependent heat source term from overspecified Dirichlet boundary data by conjugate gradient method

The inverse problem of identifying the unknown time-dependent heat source H(t)H(t) of the variable coefficient heat equation ut=(k(x)ux)x+F(x)H(t)ut=(k(x)ux)x+F(x)H(t), with separable sources of the form F(x)H(t)F(x)H(t), from supplementary temperature measurement h(t)≔u(0,t)h(t)≔u(0,t) at the left end of the rod, is investigated. The Fourier method is employed to illustrate the comparison of spacewise (F(x)F(x)) and time-dependent (H(t)H(t) heat source identification problems. An explicit formula for the Fréchet gradient of the cost functional J(H)=‖u(0,⋅;H)−h‖L2(0,Tf)2 is derived via the unique solution of the appropriate adjoint problem. The Conjugate Gradient Algorithm, based on the gradient formula for the cost functional, is then proposed for numerical solution of the inverse source problem. The algorithm is examined through numerical examples related to reconstruction of continuous and discontinuous heat sources H(t)H(t), when heat is transferred through non-homogeneous as well as composite structures. Numerical analysis of the algorithm applied to the inverse source problem in typical classes of source functions is presented. Computational results, obtained for random noisy output data, show how the iteration number of the Conjugate Gradient Algorithm can be estimated. Based on these results it is shown that this iteration number plays a role of a regularization parameter. Numerical results illustrate bounds of applicability of the proposed algorithm, and also its efficiency and accuracy.