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The Bouleuterion at Ephesos. Forschungen in Ephesos|The Bouleuterion at Ephesos Forschungen in Ephesos Band IX/5|
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A simplified L-curve method as error estimator. ETNA - Electronic Transactions on Numerical Analysis
No full textThe L-curve method is a well known heuristic method for choosing the regularization parameter for ill-posed problems by selecting it according to the maximal curvature of the L-curve. In this article, we propose a simplified version that replaces the curvature essentially by the derivative of the parameterization on the y-axis. This method shows a similar behaviour to the original L-curve method, but unlike the latter, it may serve as an error estimator under typical conditions. Thus, we can accordingly prove convergence for the simplified L-curve method
Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case. ETNA - Electronic Transactions on Numerical Analysis
No full textIn computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from thenumerical discretization of a partial differential equation (PDE), knowledge of the spectral distribution of the associated matrix has proved to beuseful information for designing/analyzing appropriate solvers–-especially, preconditioned Krylov and multigrid solvers–-for the considered problem.Actually, this spectral information is of interest also in itself as long as the eigenvalues of the aforementioned matrix represent physical quantities of interest,which is the case for several problems from engineering and applied sciences (e.g., the study of natural vibration frequencies in an elastic material).The theory of multilevel generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices arising from virtually any kind of numerical discretization of PDEs. Indeed, when the mesh-fineness parameter tends to infinity, these matrices give rise to asequence , which often turns out to be a multilevel GLT sequence or one of its “relatives”, i.e., a multilevel block GLT sequence or a (multilevel) reduced GLTsequence. In particular, multilevel block GLT sequences are encountered in the discretization of systems of PDEs as well as in the higher-order finite element or discontinuousGalerkin approximation of scalar/vectorial PDEs.In this work, we systematically develop the theory of multilevel block GLT sequences as an extension of the theories of (unilevel) GLT sequences[Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. I., Springer, Cham, 2017],multilevel GLT sequences[Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. II., Springer, Cham, 2018],and block GLT sequences[Barbarino, Garoni, and Serra-Capizzano, Electron. Trans. Numer. Anal., 53 (2020), pp. 28–112].We also present several emblematic applications of this theory in the context of PDE discretizations
Different Aspects of Mobility and Migration During the Middle Kingdom
No full textFor quite some time it has been assumed that during the Middle Kingdom there was a strong immigration from the Near East to Egypt. The explanatory models are based on many different concepts without being explicitly explained in most cases. This paper examines the question of migration in the early second millennium BCE by addressing existing cultural concepts and associating them with various aspects of mobility. For this consideration it is indispensable to work out and question the meaning of ethnicity and group identity, especially with regard to the period following the Middle Kingdom and the rule of the Hyksos in Egypt
