A boundary‐element method with mesh refinement for a weakly singular integral equation

ABSTRACTThis paper gives two methods for the L1 analysis of sampled-data systems, by which we mean computing the L∞-induced norm of sampled-data systems. This is achieved by developing what we call the kernel approximation approach in the setting of sampled-data systems. We first consider the lifting treatment of sampled-data systems and give an operator theoretic representation of their input/output relation. We further apply the fast-lifting technique by which the sampling interval [0, h) is divided into M subintervals with an equal width, and provide methods for computing the L∞-induced norm. In contrast to a similar approach developed earlier called the input approximation approach, we use an idea of kernel approximation, in which the kernel function of an input operator and the hold function of an output operator are approximated by piecewise constant or piecewise linear functions. Furthermore, it is shown that the approximation errors in the piecewise constant approximation or piecewise linear approximation scheme converge to 0 at the rate of 1/M or 1/M2, respectively. In comparison with the existing input approximation approach, in which the input function (rather than the kernel function) of the input operator is approximated by piecewise constant or piecewise linear functions, we show that the kernel approximation approach gives improved computation results. More precisely, even though the convergence rates in the kernel approximation approach remain qualitatively the same as those in the input approximation approach, the newly developed former approach could lead to quantitatively improved approximation errors than the latter approach particularly when the piecewise linear approximation scheme is taken. Finally, a numerical example is given to demonstrate the effectiveness of the kernel approximation approach with this scheme.

In this paper, we consider the numerical solution of a class of optimal control problems involving variable time points in their cost functions. The control enhancing transform is first used to convert the optimal control problem with variable time points into an equivalent optimal control problem with fixed multiple characteristic time (MCT). Using the control parametrization technique, the time horizon is partitioned into several subintervals. Let the partition points also be taken as decision variables. The control functions are approximated by piecewise constant or piecewise linear functions in accordance with these variable partition points. We thus obtain a finite dimensional optimization problem. The control parametrization enhancing control transform (CPET) is again used to convert approximate optimal control problems with variable partition points into equivalent standard optimal control problems with MCT, where the control functions are piecewise constant or piecewise linear functions with pre-fixed partition points. The transformed problems are essentially optimal parameter selection problems with MCT. The gradient formulae for the objective function as well as the constraint functions with respect to relevant decision variables are obtained. Numerical examples are solved using the proposed method.

The Numerical Solution of Integral Equations on the Half-Line

The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schrodinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897–907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials, since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods.

A lookup table (LUT) is a computationally inexpensive piecewise function used to approximate computationally expensive mathematical functions. Evaluating a LUT can be as quick as using Horner’s method to evaluate a polynomial after looking up its coefficients. A common choice of LUT is a piecewise constant or piecewise linear function; however, high-degree interpolating polynomials can also be valuable. Here, we describe the functionality of FunC 2.0 , a C++ library designed to streamline the process of building, comparing, and implementing univariate LUTs in practical applications. In particular, FunC 2.0 can build relatively small LUTs satisfying user-provided absolute and relative tolerances for error. Furthermore, FunC 2.0 can build nonuniform LUTs, it provides utilities to quickly determine reasonable LUT bounds and tolerances for error, and it provides a way to quickly profile a set of LUTs. We demonstrate FunC ’s utility in application by reducing the total runtime of a simulation performed by the Canadian Hydrological Model (CHM). This simulation modelled the snow mass distribution across Western Canada over one month. Now, CHM can evaluate the mathematical function of interest about \(28\) times faster, allowing the necessary algorithm to finish \(2\) times faster, and the overall simulation is about \(9\%\) faster.

Consideration was given to the numerical solution of the problem of parametric identification of the processes obeying the parabolic equations using an example of the processes of underground oil filtration. The identified parameters belong to the given functional classes such as the piecewise constant and piecewise linear functions. In the problem, needed is not only to determine the values of the coefficients, but also to identify the constancy boundaries of the coefficients. For numerical solution of the problem, an approach was suggested based on reduction of the initial problem to that of finite-dimensional optimization with a special structure of constraints. Obtained were the formulas for the gradient of the objective functional in the discretized problem allowing one to apply the efficient methods of first-order optimization. The results of numerical experiments on the model problems were presented.

An efficient algorithm for the numerical evaluation of boundary integral equations

In this paper, the piecewise constant Block-Pulse functions and their operational matrices of integration have directly been used to solve a two-dimensional Fredholm-Volterra integral equation of second kind. This method presents a computational technique through converting this integral equation into a system of linear equations which can be easily solved by the known methods. Also the error analysis of this method will be considered. The efficiency and accuracy of the proposed method are illustrated by some examples.

This paper briefly describes the development of generalized collocation methods for solving coupled systems of nonlinear, one-dimensional, parabolic, partial differential equations. parabolic, partial differential equations. The methods are applied to several problems that are representative of fluid flow in a porous medium. The numerical results for the problems solved indicate that collocation methods using C piecewise polynomials with Gauss-Legendre collocation points polynomials with Gauss-Legendre collocation points are more efficient than conventional, second-order, finite-difference methods. In particular, for problems whose solutions are smooth, one obtains more accuracy per unit of computer time with increasingly higher-order collocation methods. For problems whose solutions are characterized by a steep, shock-like, transient wave front, one can conclude that increasing the order of the collocation method does not necessarily result in a more efficient method, but high-accuracy collocation methods still appear to be more efficient than conventional, second-order, finite-difference methods. Also, for a given accuracy, collocation methods generally require less computer storage than conventional, second-order, finite-difference methods. Finally, the collocation technique presented here is shown to be not directly applicable to the saturation equation for immiscible flow. Introduction The purpose of this paper is to investigate collocation methods for the numerical solution of nonlinear equations similar to those that describe the flow of a fully compressible fluid in a porous medium and those that describe the process by which one miscible liquid displaces another liquid in a porous medium. The objective is to determine if collocation methods are applicable and economically justified for solving these types of petroleum engineering problems. The conclusions petroleum engineering problems. The conclusions reached are based on the amount of computer time and computer storage that is required to obtain a specified accuracy. Since only two problems are solved, the results may not be extendable to more general problems; however, experience in solving a wide variety of problems indicates that this is the general behavior that might be expected when using collocation methods. Previously published results indicate how Galerkin, collocation, and finite-element methods can be used to solve various problems that arise in petroleum engineering. Most of these papers present a variety of arguments in favor of these present a variety of arguments in favor of these methods. The principal argument is that better answers can be obtained for the same computational effort than by finite-difference methods. Culham and Varga have presented the most convincing evidence in favor of Galerkin methods. However, they only considered piecewise linear and piecewise cubic-basis functions and they only considered one problem. Because of these limitations, it is not problem. Because of these limitations, it is not clear whether their conclusions can be extended to higher-order Galerkin methods or to other types of problems. Also, they did not consider collocation problems. Also, they did not consider collocation methods in any detail. This paper considers the use of increasingly higher-order collocation methods and compares the results with those obtained with the usual second-order, finite-difference method. The Galerkin method is not considered, even though that is the method with which collocation must ultimately compete. This was done for several reasons. A preliminary operation count indicates that Galerkin methods will not be competitive with collocation methods for most problems. Also, it is not clear how to implement the higher-order Galerkin methods to achieve maximum efficiency. These remarks indicate that the relative merits of collocation and Galerkin methods are still uncertain. PROBLEMS SOLVED PROBLEMS SOLVED Two problems are solved to serve as a basis for the comparisons. The first problem, called the "gas flow problem," is the same problem considered by Culham and Varga. This problem is nonlinear and includes a volumetric source term. SPEJ P. 345

Numerial Methods for Volterra Integral Equations with Singular Kernels

The key task of constructing stationary ground stations for information exchange via an ionosphere channel is to develop antenna systems which provide high spatial-frequency selectivity of transmitting and receiving signals with seasonal and daily fluctuations of electrophysical parameters of the propagation medium. An effective solution to this problem is based on the use of vertical asymmetric vibrators placed on the earth's surface on concentric circles as transmission and reception structures of arrays. The selected array configuration allows for a circular view of the space with a high directional coefficient under deployment area constraints within the transceiver station. The annular structure, which preserves the invariance of the structure when rotating in the azimuth plane around the central longitudinal axis at an angular distance between the vibrators, provides a circular view of the space with a high coefficient of directional action. The placement of antenna elements on several rings with a common center contributes to the expansion of the frequency range for scanning space compared to a single ring system with restrictions on the occupied area. Rational versions of arrays are established on the basis of electrodynamic analysis of relationships between parameters of structures and characteristics of signal transmission and reception. The purpose of the article is to establish patterns of change in directional properties of arrays with various electrical dimensions, the number of antenna elements and concentric circles. Using the apparatus of Hallen integral equations and the Krylov–Bogolyubov numerical method, electrodynamic analysis of ring arrays of vertical asymmetric vibrators for stationary terrestrial transmission and reception posts has been carried out to solve them. Setting of the boundary problem has been performed with application of principle of mirror images from flat platform, chosen for array rotation on earth surface, which has properties of ideally conducting material in the range of ionosphere propagation wavelengths. A system of integral equations with nuclei in the form of Green functions for point emitters in a homogeneous isotropic infinitely long medium relative to equivalent array currents has been obtained from the condition of turning to zero the tangential components of the superposition of intrinsic electric fields on the surfaces of vibrators and their equality to vertical projections of fields created by generators of electromotive forces in gaps for connecting outputs of a distribution line. According to results of numerical solution of integral equations by conversion to system of linear algebraic equations for calculation of complex amplitudes of equivalent currents of antenna elements at the approximation by sequences of piecewise constant basic functions, spatial-frequency selectivity of transmission-reception by arrays with certain parameters of structures has been investigated. Distribution of array currents is restored by summation of sequences of piecewise constant basis functions with specified weight coefficients found at partial inversion of matrix operator and control of achievable accuracy of calculations by residual error of boundary conditions of a boundary problem. Laws of change of array directivities at various electric dimensions, number of vibrators and density of their arrangement on concentric circles have been revealed. On the basis of electrodynamic analysis, relationships of characteristics and parameters of structures of ring arrays of vertical asymmetrical vibrators located on infinitely extended perfectly conducting flat surface have been established. Possibilities of increase of spatial-frequency selectivity of signals transmission-reception when propagating along ionosphere channel have been investigated. When the generators of external electromotive forces are included in the final axial breaks of the elements, the effective receiving-emitting surfaces of the vibrators decrease, the absolute values of currents decrease in comparison with the values achieved when excited by point sources of high-frequency oscillations. To increase the directivity of the array with a distribution line to a value achievable when excited by ideal sources in infinitesimal axial gaps, it is necessary to increase the lengths of the vibrators by the width of the gap. The obtained results make it possible to justify rational design alternatives of antenna arrays when simultaneously fulfilling tactical and technical requirements for functional properties and maximum permissible dimensions for deployment with stationary ground transceiver stations.

In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we devise and analyze a reliable and efficient a posteriori error estimator for a semilinear optimal control problem; control constraints are also considered. We consider a fully discrete scheme that discretizes the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. The devised error estimator can be decomposed as the sum of three contributions which are associated to the discretization of the state and adjoint equations and the control variable. We extend our results to a scheme that approximates the control variable with piecewise linear functions and also to a scheme that approximates the solution to a nondifferentiable optimal control problem. We illustrate the theory with two and three-dimensional numerical examples.

A general theory is given for discretized versions of the Galerkin method for solving Fredholm integral equations of the second kind.The discretized Galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the Galerkin method.The theoretical framework that is given parallels that of the regular Galerkin method, including the error analysis of the superconvergence of the iterated Galerkin and discrete Galerkin solutions.In some cases, the iterated discrete Galerkin solution is shown to coincide with the Nyström solution with the same numerical integration method.The paper concludes with applications to finite element Galerkin methods.1. Introduction.Consider the numerical solution of the Fredholm integral equation of the second kind, (1.1)x(s)-f K(s,t)x(t)da(t)=y(s), s e D. JDIn this paper we will define and analyze the discrete Galerkin method for the numerical solution of (1.1).The Galerkin method is a well-known procedure for the approximate solution of this equation (e.g., see [5, p. 62]); and the discrete Galerkin method results when the integrations of the Galerkin method are evaluated numerically.Before giving a more precise definition of the discrete Galerkin method, we review results for the Galerkin method.In Eq. (1.1), the region D is to be a closed subset of Rm, some m > 1; and the dimension of D can be less than m, for example, if D is a surface in R3.For the discrete Galerkin method, we will assume that K(s, t) is continuous for s, t e D, although that is not necessary for the discussion of Galerkin's method given below.The equation (1.1) is written symbolically as (1.2) (I-Jf)x=y, with the integral operator assumed to be a compact operator from L2(D) to L2(D) and from L°°(D) to C(D).Further, it is assumed that (1.1) is uniquely solvable in SC for all y e 3C, for both 3C= C(D) and SC= L2(D).Additional assumptions on D and K(s, t) will be given as they are needed in the applications presented later in the paper.Generally y e C(D), and this will imply x e C(D).

We consider a special case of the fourth Painlevé equation given by d 2 ƞ / dξ 2 = 3 ƞ 5 + 2ξ ƞ 3 + (1/4ξ 2 - v - 1/2 ) ƞ , (1) with v a parameter, and seek solutions ƞ (ξ; v ) satisfying the boundary condition ƞ (∞)=0. (2) Equation (1) arises as a symmetry reduction of the derivative nonlinear Schrödinger (DNLS) equation, which is a completely integrable soliton equation solvable by inverse scattering techniques. Solutions of equation (1), satisfying (2), are expressed in terms of the solutions of linear integral equations obtained from the inverse scattering formalism for the DNLS equation. We obtain exact ‘bound state’ solutions of equation (1) for v = n , a positive integer, using the integral equation representation, which decay exponentially as ξ→ ± ∞ and are the first example of such solutions for the Painlevé equations. Additionally, using Bäcklund transformations for the fourth Painlevé equation, we derive a nonlinear recurrence relation (commonly referred to as a Bäcklund transformation in the context of soliton equations) for equation (1) relating ƞ (ξ; v ) and ƞ (ξ; v + 1).

The MLFMM (Multilevel Fast Multipole Method) is an alternative formulation of the technology behind the MoM (Method of Moment) and is applicable to much larger structures than the MoM, making full-wave current-based solutions of electrically large structures a possibility. This fact implies that it can be applied to most large models that were previously treated with the MoM without having to change the mesh. In this paper we discuss the development and implementation of volumetric integral equations for both dielectric and magnetically permeable materials using curvilinear hexahedral elements. Both piecewise constant and higher order basis functions will be examined in the context of volumetric multilevel fast multipole method implementation. Comparisons with corresponding finite element-boundary integral solutions will also be presented and compared in terms of accuracy and efficiency.