By Peter W. Hawkes (Ed.)
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Sample text
On the other hand, when the number of data points is sufficiently small, an inverse problem with discrete data can be well-posed. Since the number of data points cannot be too small (the information content of the data is then too poor), it follows that there exists an optimum number of data points, which corresponds to a compromise between stability and information content. In other words, there exists an optimum experiment for the determination of the desired physical quantity. To our knowledge, however, such a problem has not yet been solved either from the theoretical or from the practical point of view.
It follows that the computation of the singular values and singular vectors is a standard eigenvalue problem. When this has been solved, the corresponding singular functions uk can be obtained by means of the second of the equations in (126) LINEAR INVERSE AND ILL-POSED PROBLEMS 41 which, using Eq. (127), can be written explicitly as follows where ( v k ) m is the m-th component of the vector v k . Since the singular vectors vk form an orthonormal basis in Y while the singular functions t t k form an orthonormal basis in xN,it is easy to obtain the following representation of the visible component off: As we have already remarked in the Introduction, the determination off ’is a well-posed problem and therefore the propagation of relative errors from the data to the solution is controlled by the condition number.
The interested reader is referred to books on mathematical aspects (Herman and Natterer, 1981; Natterer, 1986a) and applications as well as computational methods (Herman, 1979; Herman, 1980; Sabatier, 1987b). The problem of Radon transform inversion, also called object reconstruction from projections, and the related problem of Abel transform inversion, are two examples of linear inverse scattering problems that arise when the variations of dynamical functions over a given wavelength are so small that diffraction can be neglected.