Treffer: Regularization of Mellin-type inverse problems with an application to oil engineering

For an ill-posed linear operator equation in a Hilbert space, a regularization technique, in order to guarantee the convergence of the approximate solution, is discussed. The authors propose a regularization method for a class of integral operators, using the Mellin convolution operators. Although the inverse operator is explicitly known for this class of operators, regularization techniques have to be used because of the ill-posedness of the problem. The asymptotic behaviour of the Mellin transform of the kernel, along a certain line in the complex plane, is estimated and a method for Mellin convolution operators that finds the regularization parameter without solving a linear equation is proposed. The regularization parameter is determined by the so called ``Morozov's discrepancy principle''. Only the numerical calculation of a Mellin transform and its inverse is necessary to obtain the regularized solution. The method is applied to derive the Mellin transform of the capillary pressure operator. Some numerical results are performed.