NUMERICAL VALIDATION OF PROBABILISTIC LAWS TO EVALUATE FINITE ELEMENT ERROR ESTIMATES

We propose a numerical validation of a probabilistic approach applied to estimate the relative accuracy between two Lagrange finite elements Pk and Pm,(k < m). In particular, we show practical cases where finite element Pk gives more accurate results than finite element Pm. This illustrates the theoretical probabilistic framework we recently derived in order to evaluate the actual accuracy. This also highlights the importance of the extra caution required when comparing two numerical methods, since the classical results of error estimates concerns only the asymptotic convergence rate.

ALGORITHMIC PROVISION OF ERROR-CORRECTING MULTI-COLORED BARCODE PATTERNS BASED ON GF(p)

Background. In recent years, there has been a steady trend towards the using of multi-colored barcodes. This increases the information density of the data compared to black and white barcodes. However, this complicates the processes of recognition and decoding of bar code images. Therefore, in order to reliably read multi-colored barcodes from an object, it is necessary to ensure noise immunity of bar code patterns – the minimum structural units of the bar code image. Objective. The purpose of the paper is development of a method for the synthesis of symbolics of multi-colored bar codes, which have the property of noise immunity at the level of bar code patterns. Methods. This goal is achieved through the using of multivalued incomplete Hemming codes as the basis for constructing bar code patterns. The numerical equivalent of the bar code pattern is a code word of the multi-valued Hemming code, in which the encoding-decoding operations are performed according to the rules of a finite field GF(p). Results. A number of error-correcting multi-colored barcodes have been proposed, in which one-time distorted element (error) is corrected at the level of barcode pattern and a significant number of multiple distortions are detected. The error-correcting capability of multi-colored barcode patterns has been evaluated. Conclusions. The proposed method of the synthesis of symbolics of multi-colored barcodes allows creating barcodes with improved error-correcting characteristics, which provides the appropriate level of reliability of the process of reading multi-colored barcodes in automatic identification systems.

Pressure-robust error estimate of optimal order for the Stokes equations: domains with re-entrant edges and anisotropic mesh grading

The velocity solution of the incompressible Stokes equations is not affected by changes of the right hand side data in form of gradient fields. Most mixed methods do not replicate this property in the discrete formulation due to a relaxation of the divergence constraint which means that they are not pressure-robust. A recent reconstruction approach for classical methods recovers this invariance property for the discrete solution, by mapping discretely divergence-free test functions to exactly divergence-free functions in the sense of $${\varvec{H}}({\text {div}})$$ H ( div ) . Moreover, the Stokes solution has locally singular behavior in three-dimensional domains near concave edges, which degrades the convergence rates on quasi-uniform meshes and makes anisotropic mesh grading reasonable in order to regain optimal convergence characteristics. Finite element error estimates of optimal order on meshes of tensor-product type with appropriate anisotropic grading are shown for the pressure-robust modified Crouzeix–Raviart method using the reconstruction approach. Numerical examples support the theoretical results.

Fast convolution-based performance estimation method for diffraction-limited source with imperfect X-ray optics

Although optical element error analysis is always an important part of beamline design for highly coherent synchrotron radiation or free-electron laser sources, the usual wave optics simulation can be very time-consuming, which limits its application at the early stage of the beamline design. In this work, a new theoretical approach has been proposed for quick evaluations of the optical performance degradation due to optical element error. In this way, time-consuming detailed simulations can be applied only when truly necessary. This approach treats the imperfections as perturbations that convolve with the ideal performance. For simplicity, but not by necessity, the Gaussian Schell-model has been used to show the application of this theoretical approach. The influences of the finite aperture size and height error of a focusing mirror are analysed using the proposed theory. The physical explanation of the performance degradation acquired from the presented approach helps to give a better definition of the critical range of error spatial frequencies that most affect the performance of a mirror. An example comparing two mirror surface errors with identical power spectral density functions is given. These two types of mirror surface errors result in very different intensity profiles. The approach presented in this work could help beamline designers specify the error tolerances on general optical elements more accurately.

New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints

We study first-order necessary optimality conditions and finite element error estimates for a class of distributed parabolic optimal control problems with pointwise state constraints. It is demonstrated that, if the bound in the state constraint and the differential operator in the governing PDE fulfill a certain compatibility assumption, then locally optimal controls satisfy a stationarity system that allows to significantly improve known regularity results for adjoint states and Lagrange multipliers in the parabolic setting. In contrast to classical approaches to first-order necessary optimality conditions for state-constrained problems, the main arguments of our analysis require neither a Slater point, nor uniform control constraints, nor differentiability of the objective function, nor a restriction of the spatial dimension. As an application of the established improved regularity properties, we derive new finite element error estimates for the dG(0)-cG(1)-discretization of a purely state-constrained linear-quadratic optimal control problem governed by the heat equation. The paper concludes with numerical experiments that confirm our theoretical findings.