scholarly journals Numerical Approaches for Linear Left-invariant Diffusions onSE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging

2016 ◽  
Vol 9 (1) ◽  
pp. 1-50 ◽  
Author(s):  
Jiong Zhang ◽  
Remco Duits ◽  
Gonzalo Sanguinetti ◽  
Bart M. ter Haar Romeny
Keyword(s):  
Exact Solutions ◽  
Finite Difference Methods ◽  
Retinal Imaging ◽  
Stochastic Methods ◽  
Mathieu Functions ◽  
Resolvent Equations ◽  
Relative Errors ◽  
The Many ◽  
Asymptotical Analysis ◽  
Numerical Approaches

AbstractLeft-invariant PDE-evolutions on the roto-translation groupSE(2)(and their resolvent equations) have been widely studied in the fields of cortical modeling and image analysis. They include hypo-elliptic diffusion (for contour enhancement) proposed by Citti & Sarti, and Petitot, and they include the direction process (for contour completion) proposed by Mumford. This paper presents a thorough study and comparison of the many numerical approaches, which, remarkably, are missing in the literature. Existing numerical approaches can be classified into 3 categories: Finite difference methods, Fourier based methods (equivalent toSE(2)-Fourier methods), and stochastic methods (Monte Carlo simulations). There are also 3 types of exact solutions to the PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in previous works by Duits and van Almsick in 2005. Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches. We compute relative errors of all numerical approaches to the exact solutions, and the Fourier based methods show us the best performance with smallest relative errors. We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions, crucial in implementations of the exact solutions. Furthermore, we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of underlying stochastic processes that overcomes the singular behavior in the origin of time-integrated kernels. Finally, we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.

scholarly journals Transient Pressure-Driven Electroosmotic Flow through Elliptic Cross-Sectional Microchannels with Various Eccentricities

Computation ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 27
Author(s):  
Nattakarn Numpanviwat ◽  
Pearanat Chuchard
Keyword(s):  
Laplace Transform ◽  
Electroosmotic Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Mathieu Functions ◽  
Navier Stokes Equations ◽  
Flow Rates ◽  
Elliptic Cross ◽  
Flow Through ◽  
Relative Errors

The semi-analytical solution for transient electroosmotic flow through elliptic cylindrical microchannels is derived from the Navier-Stokes equations using the Laplace transform. The electroosmotic force expressed by the linearized Poisson-Boltzmann equation is considered the external force in the Navier-Stokes equations. The velocity field solution is obtained in the form of the Mathieu and modified Mathieu functions and it is capable of describing the flow behavior in the system when the boundary condition is either constant or varied. The fluid velocity is calculated numerically using the inverse Laplace transform in order to describe the transient behavior. Moreover, the flow rates and the relative errors on the flow rates are presented to investigate the effect of eccentricity of the elliptic cross-section. The investigation shows that, when the area of the channel cross-sections is fixed, the relative errors are less than 1% if the eccentricity is not greater than 0.5. As a result, an elliptic channel with the eccentricity not greater than 0.5 can be assumed to be circular when the solution is written in the form of trigonometric functions in order to avoid the difficulty in computing the Mathieu and modified Mathieu functions.

scholarly journals NUMERICAL APPROACHES FOR A MATHEMATICAL MODEL OF AN FCCU REGENERATOR

2008 ◽  
Vol 7 (1) ◽  
pp. 71
Author(s):  
J. C. Penteado ◽  
C. O. R. Negrao ◽  
L. F. S. Rossi
Keyword(s):  
Mathematical Model ◽  
Finite Difference ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Finite Difference Methods ◽  
Implicit Finite Difference ◽  
Conservation Principles ◽  
Continuously Stirred Tank Reactor ◽  
Time Changes ◽  
Numerical Approaches

This work discusses a mathematical model of an FCCU (Fluid Catalytic Cracking Unit) regenerator. The model assumes that the regenerator is divided into two regions: the freeboard and the dense bed. The latter is composed of a bubble phase and an emulsion phase. Both phases are modeled as a CSTR (Continuously Stirred Tank Reactor) in which ordinary differential equations are employed to represent the conservation of mass, energy and species. In the freeboard, the flow is considered to be onedimensional, and the conservation principles are represented by partial differential equations to describe space and time changes. The main aim ofthis work is to compare two numerical approaches for solving the set of partial and ordinary differential equations, namely, the fourth-order Runge-Kutta and implicit finite-difference methods. Although both methods give very similar results, the implicit finite-difference method can be much faster. Steady-state results were corroborated by experimental data, and the dynamic results were compared with those in the literature (Han and Chung, 2001b). Finally, an analysis of the model’s sensitivity to the boundary conditions was conducted.

Errors in the numerical solution of the torsion Schr¨odinger equation with Mathieu functions basis set

2019 ◽  
Author(s):  
А.Н. Белов ◽  
В.В. Туровцев ◽  
Ю.Д. Орлов
Keyword(s):  
Fourier Series ◽  
Fourier Expansion ◽  
Continued Fraction ◽  
Basis Set ◽  
Hamiltonian Matrix ◽  
Mathieu Function ◽  
Matrix Elements ◽  
Mathieu Functions ◽  
Expansion Coefficients ◽  
Relative Errors

Рассмотрена погрешность алгоритма аппроксимации функций Матье рядами Фурье, когда коэффициенты ряда Фурье представлены сходящимися цепными дробями. На основании проведенного анализа получены рекуррентные соотношения для абсолютной и относительной погрешностей удерживаемых звеньев цепной дроби и коэффициентов фурье-разложения. Предложен метод оценки точности расчета элементов матрицы гамильтониана торсионного уравнения Шрёдингера в базисе функций Матье. Эффективность предложенного алгоритма подтверждена численными примерами The dependence for the Hamiltonian matrix elements of the Schrodinger torsion equation on the calculation errors of the Mathieu basis set is considered. The Mathieu functions are represented with continued fractions in this study. The analysis of the Mathieu function approximation algorithm using Fourier series expansion is carried out when the coefficients of the Fourier series are represented by convergent continued fractions. It is shown that the major contribution to the errors at the Fourier coefficient calculation is made by the error accumulating in the corresponding elements of the continued fraction. Recurrence relations for the absolute and relative errors of the kept elements of the continued fraction and the Fourier expansion coefficients are obtained. It is shown and illustrated by a numerical example that the absolute and relative errors of the Fourier expansion coefficients in the proposed algorithm are negligible. It is noted that the maximum relative errors of continued fraction are in the highest elements of the kept part. The results of our work are used to estimate the calculation error in the integrals containing Mathieu functions. These integrals constitute the Hamiltonian matrix elements of the Schr¨odinger torsion equation. We developed an algorithm to estimate of the calculation accuracy of the Hamiltonian matrix elements of the Schr¨odinger torsion equation in the basis set of Mathieu functions. We provide the example of this algorithm. The results of the work indicate the adequacy and effectiveness at the application of the Mathieu function basis set to the solution of the Schrdinger torsion equation.¨

Autonomous Systems 2018

2018 ◽  
Keyword(s):  
Sustainable Management ◽  
Autonomous Systems ◽  
Energy Systems ◽  
Stochastic Methods ◽  
Industrial Internet ◽  
Decentralised Systems ◽  
Network Vulnerabilities ◽  
Simulation Results ◽  
The Many ◽  
Intelligent Methods

To meet the expectations raised by the terms Industry 4.0, Industrial Internet and Internet of Things, real innovations are necessary, which can be brought about by information processing systems working autonomously. Owing to their growing complexity and their embedding in ever-changing environments, their design becomes increasingly critical. Thus, the many topics addressed in this book range from data integration on hardware level to methods for security and safety of data and to stochastic methods, data interferences as well as machine learning and search in decentralised systems. Their validity is proven by extensive simulation results. Also, applications for methods from deep learning and neurocomputing are presented. The sustainable management of energy systems using intelligent methods of self-organisation and learning is dealt with in the second major part of this book. As in these particular settings, the assessment of network vulnerabilities plays a crucial role, respective ...

scholarly journals Dynamical Study of Fokker-Planck Equations by Using Optimal Homotopy Asymptotic Method

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 264 ◽  
Author(s):  
H. Younas ◽  
Muhammad Mustahsan ◽  
Tareq Manzoor ◽  
Nadeem Salamat ◽  
S. Iqbal
Keyword(s):  
Convergence Rate ◽  
Exact Solutions ◽  
Fractional Order ◽  
Asymptotic Method ◽  
Dynamical Study ◽  
Optimal Homotopy Asymptotic Method ◽  
Relative Errors ◽  
Fokker Planck Equations ◽  
Fokker Planck

In this article, Optimal Homotopy Asymptotic Method (OHAM) is used to approximate results of time-fractional order Fokker-Planck equations. In this work, 3rd order results obtained through OHAM are compared with the exact solutions. It was observed that results from OHAM have better convergence rate for time-fractional order Fokker-Planck equations. The solutions are plotted and the relative errors are tabulated.

scholarly journals Beta Value Coupled Wave Theory for Nonslanted Reflection Gratings

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Cristian Neipp ◽  
Jorge Francés ◽  
Sergi Gallego ◽  
Sergio Bleda ◽  
Francisco Javier Martínez ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Matrix Method ◽  
Diffraction Gratings ◽  
Wave Theory ◽  
Refractive Indices ◽  
Mathieu Functions ◽  
Reflection Gratings ◽  
Coupled Wave

We present a modified coupled wave theory to describe the properties of nonslanted reflection volume diffraction gratings. The method is based on the beta value coupled wave theory, which will be corrected by using appropriate boundary conditions. The use of this correction allows predicting the efficiency of the reflected order for nonslanted reflection gratings embedded in two media with different refractive indices. The results obtained by using this method will be compared to those obtained using a matrix method, which gives exact solutions in terms of Mathieu functions, and also to Kogelnik’s coupled wave theory. As will be demonstrated, the technique presented in this paper means a significant improvement over Kogelnik’s coupled wave theory.

scholarly journals Comparison between standard and non-standard finite difference methods for solving first and second order ordinary differential equations

2015 ◽  
Vol 4 (2) ◽  
pp. 316
Author(s):  
Abdulrahman Yaghoubi ◽  
Hashem Saberi Najafi
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Finite Difference Methods ◽  
Second Order ◽  
Difference Methods

<p>In this paper, we solve some first and second order ordinary differential equations by the standard and non-standard finite difference methods and compare results of these methods. Illustrative examples have been provided, and the results of two methods compared with the exact solutions.</p>

Modeling of Scrap Dissolution in Molten Iron for the Case of Heat Transfer-Controlled Process by Different Approaches and Comparison of Their Accuracies

2019 ◽  
Vol 142 (1) ◽  
Author(s):  
Ajay Kumar Shukla ◽  
Brahma Deo ◽  
D. G. C. Robertson
Keyword(s):  
Heat Transfer ◽  
Finite Difference Methods ◽  
Molten Iron ◽  
Operating Conditions ◽  
Dissolution Time ◽  
Steel Scrap ◽  
Time Step ◽  
Parabolic Profile ◽  
Static Approach ◽  
Numerical Approaches

Abstract Dissolution of steel scrap in molten iron is studied by using analytical as well as numerical approaches for the case of a heat transfer-controlled process. The approaches used are Green's function approach, quasi-static approach, integral profile method (IPM) (with parabolic profile), and finite difference methods (FDM). Application of FDM explicit and FDM implicit with fixed grid and variable grid is described. Accuracy of all the approaches is compared in terms of total dissolution time evaluated from overall heat balance criterion. The selection rule of time step and deciding the suitable approach for achieving minimum error for different operating conditions, represented by nondimensional Biot number, are described in this paper.

scholarly journals Classical and Multisymplectic Schemes for Linearized KdV Equation: Numerical Results and Dispersion Analysis

Fluids ◽  
2021 ◽  
Vol 6 (6) ◽  
pp. 214
Author(s):  
Adebayo Abiodun Aderogba ◽  
Appanah Rao Appadu
Keyword(s):  
Finite Difference ◽  
Numerical Experiments ◽  
Stability Region ◽  
Kdv Equation ◽  
Finite Difference Methods ◽  
Dispersion Analysis ◽  
Explicit Scheme ◽  
Difference Methods ◽  
Relative Errors ◽  
The Stability

We construct three finite difference methods to solve a linearized Korteweg–de-Vries (KdV) equation with advective and dispersive terms and specified initial and boundary conditions. Two numerical experiments are considered; case 1 is when the coefficient of advection is greater than the coefficient of dispersion, while case 2 is when the coefficient of dispersion is greater than the coefficient of advection. The three finite difference methods constructed include classical, multisymplectic and a modified explicit scheme. We obtain the stability region and study the consistency and dispersion properties of the various finite difference methods for the two cases. This is one of the rare papers that analyse dispersive properties of methods for dispersive partial differential equations. The performance of the schemes are gauged over short and long propagation times. Absolute and relative errors are computed at a given time at the spatial nodes used.

Some exact solutions of the many-dimensional Sine-Gordon equation

1984 ◽  
Vol 41 (14) ◽  
pp. 462-464 ◽  
Author(s):  
W. I. Fushchich ◽  
Yu. N. Seheda
Keyword(s):  
Exact Solutions ◽  
Gordon Equation ◽  
Sine Gordon Equation ◽  
The Many
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