Almost optimal order approximate inverse based preconditioners for 3-d convection dominated problems on tensor-grids

For a one-dimensional diffusion problem on an refined computational grid we present preconditioners based on the standard approximate inverse technique. Next, we determine its spectral condition number κ2 and perform numerical calculations which corroborate the result. Then we perform numerical calcula-tions which show that the standard approximate inverse preconditioners and our modified versions behave in a similar manner. To finish with we show that a combination of the standard approximate inverse with an additional incomplete factorization leads to an almost optimal order preconditioner in 1, 2 and 3 dimensions, with or without dominant convection.

Local bisection refinement for nn-simplicial grids generated by reflection

A simple local bisection refinement algorithm for the adaptive refinement of $n$-simplicial grids is presented. The algorithm requires that the vertices of each simplex be ordered in a special way relative to those in neighboring simplices. It is proven that certain regular simplicial grids on $[0,1]^n$ have this property, and the more general grids to which this method is applicable are discussed. The edges to be bisected are determined by an ordering of the simplex vertices, without local or global computation or communication. Further, the number of congruency classes in a locally refined grid turns out to be bounded above by $n$, independent of the level of refinement. Simplicial grids of higher dimension are frequently used to approximate solution manifolds of parametrized equations, for instance, as in [W. C. Rheinboldt, Numer. Math., 53 (1988), pp. 165–180] and [E. Allgower and K. Georg, Utilitas Math., 16 (1979), pp. 123–129]. They are also used for the determination of fixed points of functions from ${\bf R}^n$ to ${\bf R}^n$, as described in [M. J. Todd, Lecture Notes in Economic and Mathematical Systems, 124, Springer-Verlag, Berlin, 1976]. In two and three dimensions, such grids of triangles, respectively, tetrahedrons, are used for the computation of finite element solutions of partial differential equations, for example, as in [O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems, Academic Press, Orlando, 1984], [R. E. Bank and B. D. Welfert, SIAM J. Numer. Anal., 28 (1991), pp. 591–623], [W. F. Mitchell, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 146–147], and [M. C. Rivara, J. Comput. Appl. Math., 36 (1991), pp. 79–89]. The new method is applicable to any triangular grid and may possibly be applied to many tetrahedral grids using additional closure refinement to avoid incompatibilities

On error estimation in the fourier modal method for diffractive gratings

The Fourier Modal Method (FMM, also called the Rigorous Coupled Wave Analysis, RCWA) is a numerical discretization method which is often used to calculate a scattered field from a periodic diffraction grating. For 1D periodic gratings in FMM the electromagnetic field is presented by a truncated Fourier series expansion in the direction of the grating periodicity. The grating’s material properties are assumed to be piece-wise constant (called slicing), and next per slice the scattered field is approximated by a truncated Fourier series expansion. The truncation representation of the scattered field and the piece-wise constant approximation of the grating’s material properties cause the error in FMM. This paper presents an analytical estimate/bound for the FMM error caused by slicing

Micro- and macro-block factorizations for regularized saddle point systems

We present unique and existing micro-block and induced macro-block Crout-based factorizations for matrices from regularized saddle-point problems with semi-positive de¿nite regularization block. For the classical case of saddle-point problems we show that the induced macro-block factorizations mostly reduces to the factorization presented in [24]. The presented factorization can be used as a direct solution algorithm for regularized saddle-point problems as well as it can be used a basis for the construction of preconditioners

Orientation identification of the power spectrum

The image Fourier transform is widely used for defocus and astigmatism correction in electron microscopy. The shape of a power spectrum (the square of a modulus of image Fourier transform) is directly related to the three microscope’s controls, namely defocus and two-fold (two-parameter) astigmatism. In this paper the new method for power spectrum orientation identification is proposed. The method is based on the three measures which are related to the microscope’s controls. The measures are derived from the mathematical moments of the power spectrum. The method is tested with the help of a Gaussian benchmark, as well as with the scanning electron microscopy experimental images. The method can be used as an assisting tool for increasing the capabilities of defocus and astigmatism correction a of non-experienced scanning electron microscopy user, as well as a basis for automated application

Derivative-based image quality measure for autofocus in electron microscopy

Automatic focusing methods are based on an image quality measure, which is a realvalued estimation of an image’s sharpness. In this paper we study L_1- or L_2-norm derivative-based image quality measures. For a bench mark case these measures turn out to be quadratic, which implies that after obtaining of at least three images one can find the position of the optimal defocus. The resulting autofocus method is demonstrated for a reference scanning transmission electron microscopy application. Keywords: Electron microscopy · Autofocus · Linear image formation · Image quality measure