Abstract: A non-extensive approach has been proposed for entropy [1]. The Shannon's mathematical theory of information [2] is the basis for a criterion of inference [3]. In inverse problems the entropy has been used as a regularization operator. Recently, the non-extensive entropic form was used as a new regularization operator, using only q=0.5. The parameter "q" plays a central role in the Tsallis' thermostatistics. In the present study several values for q were used: (0.5, 1.5, 2.0, 2.5) (positive and negative values!). This new regularization operator was tested for estimating initial condition in heat conduction problem [5, 6]. Our prelimar results indicate that the Morosov's discrepancy principle is not verified, as a sheme to determine the regularization parameter.
[1] C. Tsallis (1988): Possible generalization of Boltzmann-Gibbs statistics, Journal of Statistical Physics, 52 (1/2), 479-487.
[2] C.E. Shannon, W. Weaver (1949): The Mathematical Theory of Communication, Univ. of Illinois Press.
[3] E.T. Jaynes (1957): Information theory and statistical mechanics, Physical Review, 106 (4), 620-630.
[4] L. Rebollo-Neira, J. Fernandez-Rubio, A. Plastino (1998): A non-extensive maximum entropy based regularization method for bad conditioned inverse problems, Physica A, 261 (3/4), 555-568.
[5] W.B. Muniz, F.M. Ramos, H.F. Campos Velho (2000): "Entropy- and Tikhonov-based regularization techniques applied to the backwards heat equation", Computers & Mathematics with Applications, 40 (8/9), 1071-1084.
[5] W.B. Muniz, F.M. Ramos, H.F. Campos Velho (2000): Entropy- and Tikhonov-based regularization techniques applied to the backwards heat equation, Computers & Mathematics with Applications, 40 (8/9), 1071-1084.
[6] W.B. Muniz, H.F. Campos Velho, F.M. Ramos (1999): A comparison of some inverse methods for estimating the initial condition of the heat equation, Journal of Computational and Applied Mathematics, 103 (1), 145-163.