Homometric Point Sets and Inverse Problems

The inverse problem of diffraction theory in essence amounts to the reconstruction of the atomic positions of a solid from its diffraction image. From a mathematical perspective, this is a notoriously difficult problem, even in the idealised situation of perfect diffraction from an infinite structure. Here, the problem is analysed via the autocorrelation measure of the underlying point set, where two point sets are called homometric when they share the same autocorrelation. For the class of mathematical quasicrystals within a given cut and project scheme, the homometry problem becomes equivalent to Matheron's covariogram problem, in the sense of determining the window from its covariogram. Although certain uniqueness results are known for convex windows, interesting examples of distinct homometric model sets already emerge in the plane. The uncertainty level increases in the presence of diffuse scattering. Already in one dimension, a mixed spectrum can be compatible with structures of different entropy. We expand on this example by constructing a family of mixed systems with fixed diffraction image but varying entropy. We also outline how this generalises to higher dimension.Comment: 8 page

On A Newton Method For

Iterative methods for inverse eigenvalue problems involve simultaneous approximation of the matrix being sought and its eigenvectors. This paper revisits one such method for the inverse Toeplitz eigenvalue problems by exploring the eigenstructure of centrosymmetric matrices. All iterations are now taking place on a much smaller subspace. One immediate consequence is that the size of the problem is effectively cut in half and hence the cost of computation is substantially reduced. Another advantage is that eigenvalues with multiplicity up to two are necessarily separated into to disjoint blocks and hence division by zero is unmistakably avoided. Numerical experiment seems to indicate that the domain of convergence is also improved. In addition, a new scheme by using the Wielandt-Hoffman theorem is proposed. This new mechanism makes it possible to handle the case when eigenvalues with multiplicity greater than two are present

UP Usual periodogram.

Abstract—The usual nonparametric approach to spectral analysis is revisited within the regularization framework. Both usual and windowed periodograms are obtained as the squared modulus of the minimizer of regularized least squares criteria. Then, particular attention is paid to their interpretation within the Bayesian statistical framework. Finally, the question of unsupervised hyperparameter and window selection is addressed. It is shown that maximum likelihood solution is both formally achievable and practically useful. Index Terms—Hyperparameters, penalized criterion, periodograms, quadratic regularization, spectral analysis, windowing