Invertible linear relations generated by an integral equation with Nevanlinna measure

We define families of maximal and minimal relations generated by integral equations with Nevanlinna operator measure and non-selfadjoint operator measure. We prove that if a restriction of a maximal relation is continuously invertible, then the inverse operator is integral. We study the case when the convergence of non-selfadjoint operator measures implies the convergence of the corresponding integral operators inverse to restrictions of maximal relations, and establish a sufficient condition for the validity of this implication. The obtained results are applicable to the study of differential equations with singular potentials.     

Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line

1.IntroductionInthispaPer,westudynumericalsolutionstointegralequationsofthesecondkinddefinedonthehalfline.Morepreciselyweconsidertheequationy(t)+Iooa(t,s)y(s)ds=g(t),OS相似文献   

Quadrature-iteration method and quadrature-splitting method for Volterra integral equations of the second kind

We construct and analyze for convergence a quadrature-iteration method for Volterra integral equations of the second kind and a quadrature-splitting method for linear equations. The iteration processes producing an approximate solution are accelerated, because the integral operator is approximated by a quadrature operator with an arbitrarily small residual operator.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 11–16, 1992;     

Compactness principles in nonlinear operator approximation theory

This paper is concerned with the approximate solution of nonlinear operator equations in abstract settings and with applications to integral and differential equations. A given operator with certain continuity and compactness or inverse compactness properties is a suitable limit of a sequence of operators with analogous properties which hold uniformly or asymptotically. Both fixed point equations and inhomogeneous equations are treated. Solutions of approximate problems converge to solutions of the given problem. This is an appropriate type of set convergence when solutions are not unique.     

求解第一类Fredholm积分方程的多层迭代算法

本文先把正则化后的第二类积分方程分解为等价的一对不含积分算子K^*K、仅含积分算子K以及K^*的方程组, 再用截断投影方法离散方程组, 采用多层迭代算法求解截断后的等价方程组, 并给出了后验参数的选择方法, 确保近似解达到最优.与传统全投影方法相比, 减少了积分计算的维数, 保持了最优收敛率. 最后, 算例说明了算法的有效性.     

求解Lavrentiev迭代方程的多尺度快速配置算法

考虑了第一类Fredholm积分方程的求解.采用有矩阵压缩策略的多尺度配置方法来离散Lavrentiev迭代方程,在积分算子是弱扇形紧算子时,给出近似解的先验误差估计,并给出了改进的后验参数的选择方法,得到了近似解的收敛率.最后,举例说明算法的有效性.     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

Uniform convergence of the rectangle method for singular integral equation with the Hölder density

We study an approximate method for solving singular integral equations. It implies an approximation of a singular operator by means of a compound quadrature formula similar to the rectangle one. The corresponding systems of linear algebraic equations are solvable if so is the integral equation, while its coefficients satisfy the strong ellipticity condition. Under these restrictions we obtain a bound for the rate of convergence of solutions of systems of linear equations to the solution of the considered integral equation in the uniform vector norm.     

Weak convergence of approximate solutions of stochastic equations with applications to random differential and integral equations ∗

In this paper, we considerably extend our earlier result about convergence in distribution of approximate solutions: of random operator equations, where the stochastic inputs and the underlying deterministic equation are simultaneously approximated. As a by-product, we obtain convergence results for approximate solutions of equations between spaces of probability measures. We apply our results to random Fredholm integral equations of the second kind and to a random [nbar]onlinear elliptic boundary value problem.     

Adaptive Wavelet BEM for Boundary Integral Equations: Theory and Numerical Experiments

We are concerned with the numerical treatment of boundary integral equations by the adaptive wavelet boundary element method. In particular, we consider the second kind Fredholm integral equation for the double layer potential operator on patchwise smooth manifolds contained in ?³. The corresponding operator equations are treated by adaptive implementations that are in complete accordance with the underlying theory. The numerical experiments demonstrate that adaptive methods really pay off in this setting. The observed convergence rates fit together very well with the theoretical predictions based on the Besov regularity of the exact solution.     

A remark on the single scattering preconditioner applied to boundary integral equations

This article deals with boundary integral equation preconditioning for the multiple scattering problem. The focus is put on the single scattering preconditioner, corresponding to the diagonal part of the integral operator, for which two results are proved. Indeed, after applying this geometric preconditioner, it appears that, firstly, every direct integral equations become identical to each other, and secondly, that the indirect integral equation of Brakhage–Werner becomes equal to the direct integral equations, up to a change of basis. These properties imply in particular that the convergence rate of a Krylov subspaces solver will be exactly the same for every preconditioned integral equations. To illustrate this, some numerical simulations are provided at the end of the paper.     

On nonlinear Fredholm integral equations with non-differentiable Nemystkii operator

From decomposition method for operators, we consider a Newton-Steffensen iterative scheme for approximating a solution of nonlinear Fredholm integral equations with non-differentiable Nemystkii operator. By means of a convergence study of the iterative scheme applied to this type of nonlinear Fredholm integral equations, we obtain domains of existence and uniqueness of solution for these equations. In addition, we illustrate this study with a numerical experiment.     

A residual method for solving nonlinear operator equations and their application to nonlinear integral equations using symbolic computation

A derivative-free residual method for solving nonlinear operator equations in real Hilbert spaces is discussed. This method uses in a systematic way the residual as search direction, but it does not use first order information. Furthermore a convergence analysis and numerical results of the new method applied to nonlinear integral equations using symbolic computation are presented.     

DISCRETE REGULARIZATION FOR OPERATOR EQUATION OF HAMMERSTEIN'S TYPE

The purpose of tthis note is to study a convergence for a method in form of combination of discrete approximations with regularization for solving operator equations of Hammerstein's type in Banach spaces. For illustration, an example in the theory of nonlinear integral equations is given.     

Discretization Methods for Three-Dimensional Singular Integral Equations of Electromagnetism

Theorems providing the convergence of approximate solutions of linear operator equations to the solution of the original equation are proved. The obtained theorems are used to rigorously mathematically justify the possibility of numerical solution of the 3D singular integral equations of electromagnetism by the Galerkin method and the collocation method.     

Semilocal convergence for Halley’s method under weak Lipschitz condition

The semilocal convergence properties of Halley’s method for nonlinear operator equations are studied under the hypothesis that the second derivative satisfies some weak Lipschitz condition. The method employed in the present paper is based on a family of recurrence relations which will be satisfied by the involved operator. An application to a nonlinear Hammerstein integral equation of the second kind is provided.     

On spline Galerkin methods for singular integral equations with piecewise continuous coefficients

Summary This article analizes the convergence of the Galerkin method with polynomial splines on arbitrary meshes for systems of singular integral equations with piecewise continuous coefficients inL ² on closed or open Ljapunov curves. It is proved that this method converges if and, for scalar equations and equidistant partitions, only if the integral operator is strongly elliptic (in some generalized sense). Using the complete asymptotics of the solution, we provide error estimates for equidistant and for special nonuni-form partitions.     

Approximation Spaces in the Numerical Analysis of Operator Equations

We show that generalized approximation spaces can be used to prove stability and convergence of projection methods for certain types of operator equations in which unbounded operators occur. Besides the convergence, we also get orders of convergence by this approach, even in case of non-uniformly bounded projections. We give an example in which weighted uniform convergence of the collocation method for an easy Cauchy singular integral equation is studied.     

Multilevel algorithms for ill-posed problems

Summary In this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations. Convergence estimates are established for multilevel algorithms applied to Tikhonov type regularization methods. Our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter. The algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations.Dedicated to Jim Bramble on the occasion of his sixtieth birthday     

DISCRETE REGULARIZATION FOR OPERATOR EQUATION OF HAMMERSTEIN＇S TYPE

51.IntroductionIn[5]wepresentedamethodofregularizationforsolvingtheoperatorequationofHammerstein'stypewhereboththeoperatorsFZ'X*-XandFI:X-X*arenonlinear,hemicontinuousandmonotone;Xisareflexive,strictlyconvexBanachspacehavingtheE-property,i.e.weakconvergenceandconvergenceofnormsforanysequenceinXfOllowitstrongconvergence,andX*denotestheadjointspaceofX.Obviously,anyuniformlyconvexBanachspacehastheE-property.FromnowonwesupposethatXandX*areuniformlyconvex.Further,forthesakeofsimplicitynorms…