Integral Equation Algorithm Research Articles

News media coverage and market efficiency research based on Fredholm integral equation algorithm

News media coverage and market efficiency research based on Fredholm integral equation algorithm

Response to “Comment on ‘Simultaneous density and thermal conductivity depth profile reconstructions from noised thermal-wave amplitude and phase data using a combined integral equation and imperialist competitive algorithm method’” [J. Appl. Phys. 134, 056101 (2023)

Response to “Comment on ‘Simultaneous density and thermal conductivity depth profile reconstructions from noised thermal-wave amplitude and phase data using a combined integral equation and imperialist competitive algorithm method’” [J. Appl. Phys. 134, 056101 (2023)

A high–order spectral algorithm for the numerical simulation of layered media with uniaxial hyperbolic materials

Electromagnetic metamaterials are artificial media assembled from components which have dimensions much smaller than the wavelength of the illuminating radiation. It has been demonstrated that these media can have properties beyond those found in conventional materials with important applications to many areas of science and engineering. Among the collection of such materials currently garnering significant attention are the Hyperbolic Metamaterials. These are highly anisotropic structures which have a hyperbolic dispersion relation due to the fact that one principal component of the relative permittivity or permeability tensor has the opposite sign of the other two. For such materials scattered wave information at all scales is transmitted far away which suggests a number of important applications, the most immediate of which is imaging objects below the diffraction limit (superlensing). Clearly it is of great current interest to have numerical simulation capabilities for these fascinating materials. As the hyperbolic response is quite strong and its nature is very sensitive, numerical simulations of these configurations should be robust and highly accurate. For this reason we focus on High–Order Spectral algorithms which efficiently produce high fidelity solutions. More specifically, we describe a High–Order Perturbation of Surfaces approach which enjoys the greatly reduced operation counts and memory savings of interfacial methods while avoiding the complexities and indefinite linear systems faced by Integral Equation algorithms. We give a full discussion of our formulation in terms of Impedance–Impedance Operators (which avoids spurious singularities of other formulations) and implementation details, followed by numerical validation and simulation of results which appear in the engineering literature.

Algorithms for Non-Relativistic Quantum Integral Equation

In low energy scattering in Non-Relativistic Quantum Mechanics, the Schödinger equation in integral form is used. In quantum scattering theory the wave self-function is divided into two parts, one for the free wave associated with the particle incident to a scattering center, and the emerging wave that comes out after the particle collides with the scattering center. Assuming that the scattering center contains a position-dependent potential, the usual solution of the integral equation for the scattered wave is obtained via the Born approximation. Assuming that the scattering center contains a position-dependent potential, the usual solution of the integral equation for the scattered wave is obtained via the Born approximation. The methods used here are arbitrary kernels and the Neumann-Born series. The result, with the help of computational codes, shows that both techniques are good compared to the traditional method. The advantage is that they are finite solutions, which does not require Podolsky-type regularization.

Numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types)

Purpose This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types). Design/methodology/approach The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations. Findings Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods. Originality/value The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.

Contraction Integral Equation for Modeling of EM Scattering by 3-D Bodies Buried in Multilayered Anisotropic Medium

Integral equation (IE) is regarded as an efficient method for 3-D modeling of electromagnetic (EM) responses. The main contribution of this article is to develop a contraction integral equation (CIE) algorithm with a high performance for modeling of EM scattering by 3-D bodies buried in multilayered anisotropic medium. The introducing of contraction operator can ensure that the discretized IE will be iteratively convergent in large conductivity contrast between abnormal bodies and background medium. Moreover, the contraction operator can also accelerate the iteration convergence of the IE. During the discretization process of CIE, direct waves are extracted from dyadic Green's functions (DGFs) of multilayered medium, and the singular integrals of direct waves are eliminated using equivalent sphere and ellipsoid methods. By splitting DGF into convolution terms and correlation terms, 3-D FFT can be used to accelerate the product evaluation of DGFs integral and scattering current encountered in each iteration. Validity of the proposed CIE is verified by comparing with the existing finite volume method. Modeling results show that the conventional IE will be iteratively divergent when the conductivity contrast between bodies and background is large, but the CIE is always iteratively convergent even under very large conductivity contrast.

Fast algorithms using orthogonal polynomials

We review recent advances in algorithms for quadrature, transforms, differential equations and singular integral equations using orthogonal polynomials. Quadrature based on asymptotics has facilitated optimal complexity quadrature rules, allowing for efficient computation of quadrature rules with millions of nodes. Transforms based on rank structures in change-of-basis operators allow for quasi-optimal complexity, including in multivariate settings such as on triangles and for spherical harmonics. Ordinary and partial differential equations can be solved via sparse linear algebra when set up using orthogonal polynomials as a basis, provided that care is taken with the weights of orthogonality. A similar idea, together with low-rank approximation, gives an efficient method for solving singular integral equations. These techniques can be combined to produce high-performance codes for a wide range of problems that appear in applications.

A new strategy for the numerical solution of nonlinear Volterra integral equations with vanishing delays

The aim of this paper is to develop a high-order algorithm for nonlinear Volterra integral equations with vanishing variable delays. The algorithm is a non-trivial extension of the single-step methods. This method is based on the variational form, so the algorithm is relatively simple. This scheme enjoys high order accuracy and can be implemented in a stable and efficient manner. We prove the existence and uniqueness of the numerical solution, and derive error estimates of the algorithm. Numerical results show a good agreement with the theoretical analysis.

Implementation of the Equivalence Principle Algorithm for Potential Integral Equations

A novel implementation of the equivalence principle algorithm (EPA) employing potential integral equations (PIEs) is presented. EPA is generalized to be compatible with PIEs that are used to formulate inner problems inside equivalence surfaces. Based on the stability of PIEs, the resulting EPA-PIE implementation is suitable for low-frequency problems involving dense discretizations with respect to wavelength. Along with the formulation and a clear demonstration of the EPA-PIE mechanism, high accuracy, stability, and efficiency of the implementation are presented on canonical problems.

Some convergence results of OS-EM algorithm for ill-posed integral equations

ABSTRACT In this paper, we study the properties of iteration OS-EM reconstruction method. We propose a speed-enhanced tomographic reconstruction algorithm that is accelerated to ordered-subset expectation-maximization by overrelaxation parameters. In the case of exact data, we concentrate on showing the monotonicity properties of the methods and convergence to the solution of the equation. Furthermore, in the case of perturbed data, we obtain some similar qualitative conclusions of the EM reconstruction method.

A Computerized Boundary Element Algorithm for Modeling and Optimization of Complex Magneto-Thermoelastic Problems in MFGA Structures

Aims: The aim of this article is to propose a boundary integral equation algorithm for modeling and optimization of magneto-thermoelastic problems in multilayered functionally graded anisotropic (MFGA) structures.
 Study Design: Original research paper.
 Place and Duration of Study: Jamoum laboratory, January 2018.
 Methodology: a new dual reciprocity boundary element algorithm was implemented for solving the governing equations of magneto-thermoelastic problems in MFGA structures.
 Results: A numerical results demonstrate validity, accuracy, and efficiency of the presented technique.
 Conclusion: Our results thus confirm the validity, accuracy, and efficiency of the proposed technique. It is noted that the obtained dual reciprocity boundary element method (DRBEM) results are more accurate than the FEM results, the DRBEM is more efficient and easy to use than FEM because it only needs the boundary of the domain needs to be discretized.

Fast Algorithms for Boundary Integral Equations on Elliptic Domains and Related Inverse Problems

Fast Algorithms for Boundary Integral Equations on Elliptic Domains and Related Inverse Problems

Domain decomposition based integral equation modeling of 3-dimensional topography in frequency domain for well electromagnetic field

As an efficient geophysical exploration technology, well electromagnetic method is particularly applicable to oil and gas exploration in China's complex terrain areas (deserts, mountains, etc.). A serious influence of topographic relief area on the electromagnetic response of well is inevitable but challenging. To the best of our knowledge, there is no literature on modeling the electromagnetic response of three-dimensional (3D) topography with well electromagnetic method. Based on the domain decomposition, an integral equation method is presented to simulate the electromagnetic response of 3D topography in frequency domain via the well electromagnetic method. Compared with the finite difference and finite element method based on partial differential equation, this method is very efficient in simulating topographic response without huge computation or truncation boundary error accumulation or special boundary condition requirements. Firstly, an induction coefficient is defined according to the topographic relief situation. Then the computational domain consisting of the target body, background medium and 3D topography is divided into reference model, background medium and the distribution of target body medium area. According to the characteristics of each sub-region, Anderson algorithm is an analytic solution based on Gaussian filtering, which is used to provide the primary field from the excited sources in surface. And then, the stable double conjugate gradient-fast Fourier transform is incorporated into integral equation algorithm to obtain the fast 3D terrain shaft frequency domain electromagnetic responses. By comparing the calculation results using the new algorithm presented in this paper with the analytical solutions of Anderson algorithm for half-space model with surface electromagnetic method, the precision and the efficiency of this new algorithm are demonstrated. And the ability to model the electromagnetic responses of 3D topography is shown by comparing with the published results of 3D boundary integral equation. Thus, the high accuracy and high efficiency of the new algorithm presented in this paper are validated. Finally, the influence of 3D valley topography on electromagnetic field response of surface to borehole electromagnetic (SBEM) observation system is presented and analyzed. It is observed that the response of SBEM is seriously disturbed by the field of 3D valley topography which is necessarily removed. The research results presented in this paper are of significance for guiding the identification and correction of electromagnetic topographic effect from 3D SBEM.

THE SIMULATION OF METAL-DIELECTRIC ANTENNA ON THE BASE OF COMBINED APPROACH

The structure of modern radio transmitting devices may include antennas formed of both metal and dielectric components-metal-dielectric antennas. They are compact enough and can be placed on various objects of equipment. The paper presents a simulation of a metal-dielectric antenna based on a combined approach. The scheme of antenna construction in different planes is given. The process of scattering of a plane electromagnetic wave on an antenna is considered. The combined algorithm including the method of the integral equation, parallel approach and genetic algorithm is developed. In this paper, the integral equation is used to determine the unknown surface electric currents on the antenna surface, it is solved on the basis of the method of moments. A parallel algorithm was used to speed up the calculations. The impedance matrix is represented as a block matrix. Each block has its own parallel stream. Taking into account the influence of a plane dielectric waveguide on the scattered field, a method associated with a generalized scattering matrix is used. To solve the problem of multi-alternative optimization associated with determining the linear dimensions of the antenna device at a given operating frequency of the antenna, a genetic algorithm is used. As a result, the dimensions of the designed antenna for the specified dimensions of its components are obtained В состав современных радиопередающих устройств могут входить антенны, сформированные как из металлических, так и диэлектрических компонентов - металлодиэлектрические антенны. Они являются достаточно компактными и могут размещаться на различных объектах техники. В работе проведено моделирование металлодиэлектрической антенны на основе комбинированного подхода. Приведена схема построения антенны в разных плоскостях. Рассмотрен процесс рассеяния плоской электромагнитной волны на антенне. Разработан комбинированный алгоритм, включающий метод интегрального уравнения, параллельный подход и генетический алгоритм. В работе интегральное уравнение применяется для того, чтобы определить неизвестные поверхностные электрические токи на поверхности антенны, оно решается на основе метода моментов. Для ускорения расчетов использовался параллельный алгоритм. Матрица импедансов представляется как блочная. Для каждого из блоков применяется свой параллельный поток. При учете влияния на рассеянное поле плоского диэлектрического волновода используется метод, связанный с обобщенной матрицей рассеяния. Для решения задачи многоальтернативной оптимизации, связанной с определением линейных размеров антенного устройства при заданной рабочей частотой антенны, используется генетический алгоритм. В результате получены размеры спроектированной антенны для заданных размеров входящих в ее состав компонентов

A hands-on approach to teaching system identification using first-order plus dead time modelling of step response data

This paper describes three step response-based system identification methods of increasing complexity, together with a range of exercises that will enhance student understanding of this area in an engaging and practical way. For illustration purposes and practicality, it is assumed that the model to be identified is of the first-order plus dead time type. The first method uses a popular graphical technique, which is easy to understand and apply, but inaccurate when the response data are not ideal. The second uses the Nelder–Mead simplex method, which is a more powerful technique and has the added benefit of introducing undergraduate students to the concepts of numerical optimisation. The third uses an integral equation algorithm. The latter two methods, which can be readily extended to other model structures and input types, are also demonstrated using experimental data obtained from a tank level control system.

A Fast and Spectrally Convergent Algorithm for Rational-Order Fractional Integral and Differential Equations

A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for ODEs [S. Olver and A. Townsend, SIAM Rev., 55 (2013), pp. 462--489] and involves constructing two different bases, one for the domain of the operator and one for the range of the operator. The bases are constructed from direct sums of suitably weighted ultraspherical or Jacobi polynomial expansions, for which explicit representations of fractional integrals and derivatives are known, and are carefully chosen so that the resulting operators are banded or almost banded. Geometric convergence is demonstrated for numerous model problems when the variable coefficients and right-hand side are sufficiently smooth.

A Design of Vertical Polarized Conformal Antenna and Its Array Based on UAV Structure

The designs of conformal antenna and its array usually need to combine with different carrier structures. As for cone or cylindrical carrier, through using of invasive weed optimization (IWO), the conformal array should be the optimized design as the antenna installation space requirements of cylindrical carrier; through adopting the antenna unit miniaturization method, it designs out a conformal antenna and makes it easier to be installed and the omnidirectional circular polarized beam is also realized. As for unmanned aircraft carrier, it designs out the low profile of conformal vertical polarization antenna unit; through using the electric field integral equation (EFIE) and electric/magnetic current combined-field integral equation (JMCFIE) and multilevel fast multipole algorithm (MLFMA), it analyzes the two different characteristics of unmanned aircraft carrier on the influence of antenna performance. The results obtained through experiments and JMCFIE analysis are in good agreement; according to unmanned aircraft carrier, it also designed out the antenna in X/K-band spectrum; through IWO, the antenna array is optimized; the test results show that the array scanning angle can reach the corresponding beam direction.

A boundary integral algorithm for the Laplace Dirichlet–Neumann mixed eigenvalue problem

We present a novel integral-equation algorithm for evaluation of Zaremba eigenvalues and eigenfunctions, that is, eigenvalues and eigenfunctions of the Laplace operator with mixed Dirichlet–Neumann boundary conditions; of course, (slight modifications of) our algorithms are also applicable to the pure Dirichlet and Neumann eigenproblems. Expressing the eigenfunctions by means of an ansatz based on the single layer boundary operator, the Zaremba eigenproblem is transformed into a nonlinear equation for the eigenvalue μ. For smooth domains the singular structure at Dirichlet–Neumann junctions is incorporated as part of our corresponding numerical algorithm—which otherwise relies on use of the cosine change of variables, trigonometric polynomials and, to avoid the Gibbs phenomenon that would arise from the solution singularities, the Fourier Continuation method (FC). The resulting numerical algorithm converges with high order accuracy without recourse to use of meshes finer than those resulting from the cosine transformation. For non-smooth (Lipschitz) domains, in turn, an alternative algorithm is presented which achieves high-order accuracy on the basis of graded meshes. In either case, smooth or Lipschitz boundary, eigenvalues are evaluated by searching for zero minimal singular values of a suitably stabilized discrete version of the single layer operator mentioned above. (The stabilization technique is used to enable robust non-local zero searches.) The resulting methods, which are fast and highly accurate for high- and low-frequencies alike, can solve extremely challenging two-dimensional Dirichlet, Neumann and Zaremba eigenproblems with high accuracies in short computing times—enabling, in particular, evaluation of thousands of eigenvalues and corresponding eigenfunctions for a given smooth or non-smooth geometry with nearly full double-precision accuracy.

Fully adaptive algorithms for multivariate integral equations using the non-standard form and multiwavelets with applications to the Poisson and bound-state Helmholtz kernels in three dimensions

We have developed and implemented a general formalism for fast numerical solution of time-independent linear partial differential equations as well as integral equations through the application of numerically separable integral operators in d ≥ 1 dimensions using the non-standard (NS) form. The proposed formalism is universal, compact and oriented towards the practical implementation into a working code using multiwavelets. The formalism is applied to the case of Poisson and bound-state Helmholtz operators in d = 3. Our algorithms are fully adaptive in the sense that the grid supporting each function is obtained on the fly while the function is being computed. In particular, when the function g = O f is obtained by applying an integral operator O, the corresponding grid is not obtained by transferring the grid from the input function f. This aspect has significant implications that will be discussed in the numerical section. The operator kernels are represented in a separated form with finite but arbitrary precision using Gaussian functions. Such a representation combined with the NS form allows us to build a sparse, banded representation of Green’s operator kernel. We have implemented a code for the application of such operators in a separated NS form to a multivariate function in a finite but, in principle, arbitrary number of dimensions. The error of the method is controlled, while the low complexity of the numerical algorithm is kept. The implemented code explicitly computes all the 22d components of the d-dimensional operator. Our algorithms are described in detail in the paper through pseudo-code examples. The final goal of our work is to be able to apply this method to build a fast and accurate Kohn–Sham solver for density functional theory.

Conical diffraction by multilayer gratings: a recursive integral equation approach

The paper is devoted to an integral equation algorithm for studying the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in ℝ2 coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with an arbitrary number of material layers we present the extension of a recursive procedure developed by Maystre for normal incidence, which transforms the problem to a sequence of equations with 2×2 operator matrices on each interface. Necessary and sufficient conditions for the applicability of the algorithm are derived.