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The problem of determining a term in the right-hand side of elliptic equations from an observation on a part of the boundary is investigated. The inverse problem is formulated as an operator equation and then stabilized by Tikhonov regularization method. The regularized problem is discretized based on Hinze's variational discretization concept and the regularization parameter is chosen guaranteeing that when noise level and the discretization mesh size tend to zero, the solution of the discretized regularized problem converges to the $ f^* $ f ∗ -minimum norm solution of the continuous inverse problem. Some numerical examples are presented for illustrating the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse problem of finding the source for heat conduction from the integral observation and the final time one. Using least-square method, this problem is reformulated into the optimal one of minimizing the objective functional. The Fréchet differentiability and the derivative of the objective functional are derived via an adjoint problem. Some numerical examples for finding the necessary optimality conditions are presented. [ABSTRACT FROM AUTHOR]