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In this paper, a non-linear source term is approximated for an inverse parabolic problem using final observations as additional data. The inverse problem is solved as an optimization problem where objective function is minimized by an iterative algorithm namely the Levenberg-Marquardt Method. Sensitivity analysis is used to find how many terms of the polynomial basis functions are suitable for the source term approximation. Results show that for different boundary conditions, different numbers of the basis functions are appropriate, without considering functional forms of the source term. To examine the accuracy of approximations, some test examples are carried out. The algorithm seems to be very sharp if the corresponding direct problem has a smaller solution's maximum. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose a variation of the soft-thresholding algorithm for finding sparse approximate solutions of the equation , where as the sparsity of the iterate increases the penalty function changes. In this approach, sufficiently large entries in a sparse iterate are left untouched. The advantage of this approach is that a higher regularization constant can be used, leading to a significant reduction of the total number of iterations. Numerical experiments for sparse recovery problems, also with noisy data, are included. [Copyright &y& Elsevier]