Determination of a term in the right-hand side of parabolic equations

The inverse problem of determining a term in the right hand side of parabolic equations from integral observations is investigated. The observations can be regarded as generalized interior point observations which are collected in practice. The problem is then reformulated as a least squares problem in coupling with a Tikhonov regularization term. It is proved that the Tikhonov functional is Fréchet differentiable and a formula for the gradient is derived via an adjoint problem. The variational problem is discretized by the finite element method, the convergence of which is proved. The discretized variational problem is numerically solved by the conjugate gradient method. Some numerical examples are presented for showing the efficiency of the method.