Volume 10, Issue 3

- The problem of reconstruction of surfaces based on a given plane by their shadow contours is considered. The optical projection of the surface point and shadow contours with respect to perpendicular and optical projection is introduced. Necessary and sufficient conditions of unique reconstruction of a plane curve by complete set of optical and perpendicular projections of their points are presented. A uniqueness result on the problem is obtained in the case of smooth hat-shaped surface. A uniqueness theorem for the problem of simultaneous reconstruction of two nonintersecting piecewise-smooth hat-shaped surfaces by complete set of their shadows is proved.

- We consider an integrodifferential system arising in the theory of spread of infection. We formulate the inverse problem, consisting in determine the unknowns u (susceptible population), v (infective population) and the convolution kernel h , (representing the time depending transfer mechanism of infection) in an abstract setting, relating it to a Banach space X . We prove an abstract existence and uniqueness theorem for the abstract inverse problem and we apply it to the concrete case of the epidemic equations.

- Variations in harmonic amplitudes of finite time records of certain recurrent functions f ( t ) ∈ L 1 , in Bochner’s sense, which are encountered e.g. in medical tonometry [1], are analyzed by means of the time-localization principle. The variations are proved to generate a reversed filtration process of the Kallman type. A new sequential “on line” smoothing algorithm is advanced to regularize this ill-posed problem in a weighted Sobolev space with regularizable exponential weight functions.

- In this paper a convergence proof is given for the continuous analog of the Gauss-Newton method for nonlinear ill-posed operator equations and convergence rates are obtained. Convergence for exact data is proved for nonmonotone operators under weaker source conditions than before. Moreover, nonlinear ill-posed problems with noisy data are considered and a priori and a posteriori stopping rules are proposed. These rules yield convergence of the regularized approximations to the exact solution as the noise level tends to zero. The convergence rates are optimal rates under the source conditions considered.

- While applying regularization procedures for obtaining stable solutions of ill-posed problems, one of the crucial step is the choice of the regularization parameter. Among the well considered discrepancy principles in the literature, Morozov’s method and Arcangeli’s method are widely used because of their simplicity for the purpose of applications. Although Morozov’s method and their variations have been considered extensively in the literature for general class of regularization methods, the Arcangeli’s method is known to have applied only for Tikhonov regularization. The reason could be the belief that it can never yield a rate better than Morozov’s procedure, under any smoothness assumption on the solution. However, this belief was misplaced as it has been showed by Nair (1992) that Arcangeli’s method do provide the best rate O ( δ 2 / 3 ) for Tikhonov regularization under sufficient smoothness assumption on the solution, while Morozov’s method gives the rate only up to O ( δ 1 / 2 ). The purpose of this paper is to consider a generalized form of Arcangeli’s method for a general class of regularization methods for the case when there is no error on the modeling, and then extend the procedure which allow error in the modeling as well.

- We investigate the identification of coefficients problem in Elrod- Adams’s model. From a numerical point of view, we relax the control problem; then using penalization methods and a qualification condition we get optimality conditions. Some numerical results are given.