Diffusion tensor regularization with metric double integrals

In this paper, we propose a variational regularization method for denoising and inpainting of diffusion tensor magnetic resonance images. We consider these images as manifold-valued Sobolev functions, i.e. in an infinite dimensional setting, which are defined appropriately. The regularization functionals are defined as double integrals, which are equivalent to Sobolev semi-norms in the Euclidean setting. We extend the analysis of [14] concerning stability and convergence of the variational regularization methods by a uniqueness result, apply them to diffusion tensor processing, and validate our model in numerical examples with synthetic and real data.

A Numerical Method for Computing Double Integrals with Variable Upper Limits

We propose and justify a numerical method for computing the double integral with variable upper limits that leads to the variableness of the region of integration. Imposition of simple variables as functions for upper limits provides the form of triangles of integration region and variable in the external limit of integral leads to a continuous set of similar triangles. A variable grid is overlaid on the integration region. We consider three cases of changes of the grid for the division of the integration region into elementary volumes. The first is only the size of the imposed grid changes with the change of variable of the external upper limit. The second case is the number of division elements changes with the change of the external upper limit variable. In the third case, the grid size and the number of division elements change after fixing their multiplication. In these cases, the formulas for computing double integrals are obtained based on the application of cubatures in the internal region of integration and performing triangulation division along the variable boundary. The error of the method is determined by expanding the double integral into the Taylor series using Barrow’s theorem. Test of efficiency and reliability of the obtained formulas of the numerical method for three cases of ways of the division of integration region is carried out on examples of the double integration of sufficiently simple functions. Analysis of the obtained results shows that the smallest absolute and relative errors are obtained in the case of an increase of the number of division elements changes when the increase of variable of the external upper limit and the grid size is fixed.

Integration method of non-elementary exponential functions using iterated Fubinni integrals

The present work presents a new method of integration of non-elementary exponential functions where Fubinni's iterated integrals were used. In this research, some approximations were used in order to generalize the results obtained through mathematical series, in addition to integration methods and double integrals. In addition to the integration methods, the Taylor series was used, where the value found and compatible with the values ​​of the power series that are used to calculate the value of the exponential function demonstrated in the work was verified. In addition to the methods described, a comparison of the values ​​obtained by the series and the values ​​described in the method was improvised, where it was noticed that the higher the value of the variable, the closer the results show a stability for the variable greater than the value 4, described in table 01. The conclusions point to a great improvement, mainly for solving elliptic differential equations and statistical functions.

Post-quantum Ostrowski type integral inequalities for functions of two variables

In this study, we give the notions about some new post-quantum partial derivatives and then use these derivatives to prove an integral equality via post-quantum double integrals. We establish some new post-quantum Ostrowski type inequalities for differentiable coordinated functions using the newly established equality. We also show that the results presented in this paper are the extensions of some existing results.

A cookbook for approximating Euclidean balls and for quadrature rules in finite element methods for nonlocal problems

The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms involving double integrals which lead to discrete systems having higher assembly and solving costs due to possibly much lower sparsity compared to that of FEMs for PDEs. In addition, one may encounter nonsmooth integrands. In many nonlocal models, nonlocal interactions are limited to bounded neighborhoods that are ubiquitously chosen to be Euclidean balls, resulting in the challenge of dealing with intersections of such balls with the finite elements. We focus on developing recipes for the efficient assembly of FEM stiffness matrices and on the choice of quadrature rules for the double integrals that contribute to the assembly efficiency and also posses sufficient accuracy. A major feature of our recipes is the use of approximate balls, e.g. several polygonal approximations of Euclidean balls, that, among other advantages, mitigate the challenge of dealing with ball-element intersections. We provide numerical illustrations of the relative accuracy and efficiency of the several approaches we develop.

Some New Results Concerning the Classical Bernstein Cubature Formula

In this article, we present a solution to the approximation problem of the volume obtained by the integration of a bivariate function on any finite interval [a,b]×[c,d], as well as on any symmetrical finite interval [−a,a]×[−a,a] when a double integral cannot be computed exactly. The approximation of various double integrals is done by cubature formulas. We propose a cubature formula constructed on the base of the classical bivariate Bernstein operator. As a valuable tool to approximate any volume resulted by integration of a bivariate function, we use the classical Bernstein cubature formula. Numerical examples are given to increase the validity of the theoretical aspects.

Implementation of Preisach-Krasnoselskii Hysteresis Model with the Use of Artificial Neural Networks

Accurate modeling of hysteresis is essential for both the design and performance evaluation of electromagnetic devices. This project proposes the use of feedforward meural networks to implement an accurate magnetic hysteresis model based on the mathematical difinition provided by the Preisach-Krasnoselskii (P-K) model. Feedforward neural networks are a linear association networks that relate the ouput patterns to input patterns. By introducing the multi-layer feedforward neural networks make the hysteresis modeling accurate without estimation of double integrals. Simulation results provide the detailed illustrations. The comparisons with the experiments show that the proposed approach is able to satisfactorily reproduce many features of obsereved hysteresis phenomena an in turn can be used for many applications of interest.