Discretization Of Integral Equations Research Articles

Exact canonically conjugate momenta to quadrupole-type collective coordinates and derivation of nuclear quadrupole-type collective Hamiltonian

Exact canonically conjugate momenta Pi_{2mu} in quadrupole nuclear collective motions are proposed. The basic idea lies in the introduction of a discrete integral equation for the strict definition of canonically conjugate momenta to collective variables phi_{2mu}. A part of our collective Hamiltonian, the Pi_{2mu}-dependence of the kinetic part of the Hamiltonian, is given exactly. Further, phi_{2mu}-dependence of the kinetic part of the Hamiltonian is also given.

Research Spotlights

Accurately estimating solutions for models of physical systems with steep internal layers is an important but difficult numerical problem. For example, the formation of alloys---such as the creation of stainless steel---involves phase separation at the atomic level of its metallic components. This can be successfully modeled, but the delicate sharp interface structure makes computational methods for approximating solutions an ambitious problem. Sharp interface systems arise in the form of elliptic partial differential equations in an array of fluids problems, in semiconductor device simulations, and in microstructure evolution of materials. The Research Spotlights paper in this issue, “Norm-Preserving Discretization of Integral Equations for Elliptic PDEs with Internal Layers I: The One-Dimensional Case,” by Travis Askham and Leslie Greengard, concentrates on the numerical problem of accurate resolution of steep internal layers in one-dimensional problems. Common approaches include adaptive finite difference, finite element, and volume integral methods. This paper concentrates on the third approach, and in particular on adaptive volume integral methods that preserve $L^p$ norm. It considers the effect of the choice of $p$ on conditioning, showing in an example that the conditioning varies significantly with $p$. Well-conditioned discretization methods are shown to require both adaptive refinement within the steep internal layer and a careful choice of the value of $p$. This paper will appeal to SIAM readers with an interest in numerical methods for partial differential equations, materials science, fluid dynamics, or semiconductor physics.

Hierarchical quadrature for multidimensional singular integrals – Part II

In a previous part I, we introduced a method for the evaluation of singular integrals arising in the discretization of integral equations based on the repeated subdivision of domains. The integrals defined on these subdomains are classified such that a class of integrals can be expressed as a sum of regular integrals and representatives of other classes. A system of equations describes the relations between the classes. Furthermore the approximate value of the singular integrals only depends on the accuracy of the calculation of regular integrals. Part I left a gap on certain parameter configurations on which the mentioned system of equations is irregular. This paper shall close this gap. To this end, we introduce an alternative splitting strategy based on a modified Hadamard partie finie integral.

Norm-Preserving Discretization of Integral Equations for Elliptic PDEs with Internal Layers I: The One-Dimensional Case

We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various $L^p$ norms. We show that high-order accurate Nystrom discretization leads to well-conditioned finite-dimensional linear systems if and only if the discretization is both norm-preserving in a correctly chosen $L^p$ space and adaptively refined in the internal layer.

Automatic parameter setting for Arnoldi–Tikhonov methods

In the framework of iterative regularization techniques for large-scale linear ill-posed problems, this paper introduces a novel algorithm for the choice of the regularization parameter when performing the Arnoldi–Tikhonov method. Assuming that we can apply the discrepancy principle, this new strategy can work without restrictions on the choice of the regularization matrix. Moreover, this method is also employed as a procedure to detect the noise level whenever it is just overestimated. Numerical experiments arising from the discretization of integral equations and image restoration are presented.

Efficient Surface Integral Equation Using Travelling-Standing-Wave Vector Basis Function for Complex EM Scattering Problems

A new set of travelling-standing-wave (TSW) basis functions are proposed for curved triangular patch in this paper. This kind of bases is not only able to describe the standing component of induced surface current but also able to simulate the phase variation of induced current accurately. The authors used these basis functions to discrete surface integral equation (SIE) with a series of coarse meshes (even larger than 2λ) to calculate the electromagnetic (EM) scattering from the perfect electric conductor (PEC) targets with method of moment (MoM). Numerical experiments indicate that the MoM-TSW scheme even has more advantages than the MLFMA-CRWG scheme in the aspects of computational efficiency and memory storage.

High-order accurate methods for Nyström discretization of integral equations on smooth curves in the plane

Boundary integral equations and Nystrom discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weakly-singular kernel arises, in which case specialized quadratures that modify the matrix entries near the diagonal are needed to reach a high accuracy. We describe the construction of four different quadratures which handle logarithmically-singular kernels. Only smooth boundaries are considered, but some of the techniques extend straightforwardly to the case of corners. Three are modifications of the global periodic trapezoid rule, due to Kapur---Rokhlin, to Alpert, and to Kress. The fourth is a modification to a quadrature based on Gauss---Legendre panels due to Kolm---Rokhlin; this formulation allows adaptivity. We compare in numerical experiments the convergence of the four schemes in various settings, including low- and high-frequency planar Helmholtz problems, and 3D axisymmetric Laplace problems. We also find striking differences in performance in an iterative setting. We summarize the relative advantages of the schemes.

THE RESEARCH ON SHORT-RANGE TARGET HOLOGRAPHIC IMAGING ALGORITHM

Because the oversized, ultra short-range and arbitrary- shape goals cannot be imaged by Fourier transform algorithm, a Boundary Element Method (BEM) is presented for short-range millimeter wave holographic imaging. Through the discrete boundary integral equation, the discrete electromagnetic flelds on the source surface and holographic surface are obtained. They are linked by a transfer matrix. Finally, the discrete electromagnetic flelds obtain target holographic image. Due to the complexity of the transfer matrix, the Distributed Source Boundary Point Method (DSBPM) is introduced to calculate it, which greatly simplifles the calculation process. The simulation experiments of three-dimensional hemisphere imaging show the sensitivity of the imaging algorithm to test error, and regularization method has been proposed. The actual measurement of the four small metal balls verifles the validity of the imaging algorithm for large target imaging. The imaging results show that holographic imaging of the boundary element method can obtain high resolution and high amplitude accuracy.

Practical Application of the Equivalent Source Method as an Antenna Diagnostics Tool [AMTA Corner

Recently, significant efforts from different research groups have been aimed at the problem of reconstructing the extreme near field radiated by an antenna from measured field data. Among the different proposed solutions, the inverse-source or Equivalent-Current/Source Method (EQC), based on discretization of integral equations, has attracted considerable attention, due to a host of promising applications in antenna design and diagnostics. The integral-equation approach constitutes a complement to more-established tools, such as plane-wave or spherical-wave expansion. At the expense of a heavier computational burden, this method offers a greater generality and flexibility, since it allows reconstructing sources on arbitrary three-dimensional surfaces enclosing the antenna under test (AUT).

On Discrete Integrable Equations with Convex Variational Principles

We investigate the variational structure of discrete Laplace-type equations that are motivated by discrete integrable quad-equations. In particular, we explain why the reality conditions we consider should be all that are reasonable, and we derive sufficient conditions (that are often necessary) on the labeling of the edges under which the corresponding generalized discrete action functional is convex. Convexity is an essential tool to discuss existence and uniqueness of solutions to Dirichlet boundary value problems. Furthermore, we study which combinatorial data allow convex action functionals of discrete Laplace-type equations that are actually induced by discrete integrable quad-equations, and we present how the equations and functionals corresponding to (Q3) are related to circle patterns.

Integrability of Discrete Equations Modulo a Prime

We apply the 'almost good reduction' (AGR) criterion, which has been introduced in our previous (arXiv:1206.4456 and arXiv:1209.0223), to several classes of discrete integrable equations. We verify our conjecture that AGR plays the same role for maps of the plane define over simple finite fields as the notion of the singularity confinement does. We first prove that q-discrete analogues of the Painlev\'e III and IV equations have AGR. We next prove that the Hietarinta-Viallet equation, a non-integrable chaotic system also has AGR.

Guest editorial for the special issue on nontraditional differential and integral equation discretization schemes

Guest editorial for the special issue on nontraditional differential and integral equation discretization schemes

Discrete Integrable Equations over Finite Fields

Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a generalized discrete KdV equation related to a Yang-Baxter map. Explicit forms of soliton solutions and their periods over finite fields are obtained. Relation to the singularity confinement method is also discussed.

Optimal and Approximate Solutions of Singular Integral Equations by Means of Reproducing Kernels

A reproducing kernel method is proposed to obtain the optimal and approximate solutions of Carleman singular integral equations. Therefore, we will be mostly interested in singular integral equations with a Cauchy type kernel and whose coefficients are real or complex valued functions. The new method and corresponding concepts allow the analysis of associated discrete singular integral equations and corresponding inverse source problems in appropriate frameworks.

Lax Pairs for Discrete Integrable Equations via Darboux Transformations

A method is developed to construct discrete Lax pairs using Darboux transformations. More kinds of Lax pairs are found for some newly appeared discrete integrable equations, including the H1, the special H3 and the Q1 models in the Adler—Bobenko—Suris list and the closely related discrete and semi-discrete pKdV, pMKdV, SG and Liouville equations.

A direct solver with O(N) complexity for integral equations on one-dimensional domains

An algorithm for the direct inversion of the linear systems arising from Nyström discretization of integral equations on one-dimensional domains is described. The method typically has O(N) complexity when applied to boundary integral equations (BIEs) in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes’ equations. The scaling coefficient suppressed by the “big-O” notation depends logarithmically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and time-harmonic Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the ℋ- and ℋ 2-matrix arithmetic of Hackbusch and co-workers, and is closely related to previous work on Hierarchically Semi-Separable matrices.

Nonlinear Problems: Analytical and Computational Approach with Applications

The nonlinear sciences have been taking new dimensions with the passage of time. The thorough study of the literature witnesses the introduction of a number of new techniques which are being used to cope with the versatility of diversified physical problems, and in the same context, a number of numerical and analytical techniques have been developed to cope with the physical nature of such equations. The basic purpose of this special issue is to extend the applications of the relatively new approaches and theories for nonlinear problems of physical nature arising in mathematics, physics, and applied sciences and to see the latest developments in the same context. The authors were invited to submit original research articles as well as review articles that stimulated the continuing efforts in nonlinear equations and related theories. The special issue provided a forum for researches and scientists to communicate their recent developments and to present their novel results on nonlinear equations and related problems. The topics included in this special issue are solitary wave theory, solitons and compactons, geometric function theory and its applications, symmetries and separation of variables, Hirota’s bilinear method, Painleve test and analysis, HamiltonJacobi approach, variational inequalities and their applications, analytical methods such as variational iteration techniques and its various modifications, variation of parameters method, continuous and discrete integrable equations, numerical approaches to nolinear waves, and finally stability theory of solitary waves.

A Novel Parallelization Method for the Edge-Elements-Based Integral Equation

This letter introduces a new parallelization method for the computation of the discrete integral equation where edge elements are used as the discretization base. This new method does not need traditional mesh partitions and can be implemented with minimal programming efforts. The new method has been implemented using OpenMP for the robin boundary condition iteration (RBCI), but it applies to the method of moments as well. It is based on the subdivision of the discrete surface's elements into groups in such a way that elements belonging to the same group can be treated in parallel.

Acceleration of moment method using CUDA

PurposeThe purpose of this paper is to accelerate the time‐consuming task of assembling the impedance matrix resulting from the discretization of integral equations by the moment method, accelerated using massively parallel processing scheme.Design/methodology/approachThis paper provides several approaches for the implementation of moment method on compute unified device architecture (CUDA) capable general purpose video cards, as well as giving general implementation design patterns and a good overview on the topic.FindingsThe proposed method seems to be efficient in the light of the presented numerical results.Originality/valueThe subject of the paper is an evolving, considerably new aspect among computation techniques which could be of high interest for the scientific community.

Elliptic gamma-function and multi-spin solutions of the Yang–Baxter equation

We present a generalization of the master solution to the quantum Yang–Baxter equation (obtained recently in arXiv:1006.0651) to the case of multi-component continuous spin variables taking values on a circle. The Boltzmann weights are expressed in terms of the elliptic gamma-function. The associated solvable lattice model admits various equivalent descriptions, including an interaction-round-a-face formulation with positive Boltzmann weights. In the quasi-classical limit the model leads to a new series of classical discrete integrable equations on planar graphs.