Method For Nonlinear Integral Equations Research Articles

Spectral approximation methods for nonlinear integral equations with non-smooth kernels

Spectral approximation methods for nonlinear integral equations with non-smooth kernels

Recovery of an infinite rough surface by a nonlinear integral equation method from phaseless near-field data

Abstract This paper is concerned with the two-dimensional inverse acoustic scattering by an unbounded, sound-soft rough surface. We propose a nonlinear integral equation method using multi-frequency phaseless near-field data associated with point sources to reconstruct the shape and location of the rough surface, yielding a fast imaging algorithm. Numerical examples are presented to show the effectiveness of the inverse algorithm.

A simplified iteratively regularized projection method for nonlinear ill-posed problems

In this paper, we propose a projection method for the numerical solution of the simplified iteratively regularized Gauss-Newton method of nonlinear integral equations for which an a posteriori stopping rule is proposed to terminate the iteration. Such methods require only the computation of the Fréchet derivative at the initial approximation. Thus the computational work is considerably reduced. Under certain mild conditions, we give the convergence analysis and derive the order optimality. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method.

Inverse obstacle scattering for elastic waves in the time domain

This paper concerns an inverse elastic scattering problem which is to determine a rigid obstacle from time domain scattered field data for a single incident plane wave. By using the Helmholtz decomposition, we reduce the initial-boundary value problem for the time domain Navier equation to a coupled initial-boundary value problem for wave equations, and prove the uniqueness of the solution for the coupled problem by employing the energy method. The retarded single layer potential is introduced to establish a set of coupled boundary integral equations, and uniqueness is discussed for the solution of these boundary integral equations. Based on the convolution quadrature method for time discretization, the coupled boundary integral equations are reformulated into a system of boundary integral equations in the s-domain, and then a convolution quadrature based nonlinear integral equation method is proposed for the inverse problem. Numerical experiments are presented to show the feasibility and effectiveness of the proposed method.

A Modified Minimal Error Method for Solving Nonlinear Integral Equations via Multiscale Galerkin Methods

A modified minimal error method for nonlinear integral equations is established. Such method, combined with the discrepancy principle as stopping rule, is a regularization method, which yields convergence to an exact solution when the nonlinear operator F satisfies certain conditions. Furthermore, the convergence of the modified minimal error method via multiscale Galerkin methods is also proved. Finally, numerical results show the accuracy and efficiency of the proposed method.

Inverse obstacle scattering for acoustic waves in the time domain

<p style='text-indent:20px;'>This paper concerns an inverse acoustic scattering problem which is to determine the location and shape of a rigid obstacle from time domain scattered field data. An efficient convolution quadrature method combined with nonlinear integral equation method is proposed to solve the inverse problem. In particular, replacing the classic Fourier transform with the convolution quadrature method for time discretization, the boundary integral equations for the Helmholtz equation with complex wave numbers can be obtained to guarantee the numerically approximate causality property of the scattered field under some condition. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed method.</p>

An inverse acoustic-elastic interaction problem with phased or phaseless far-field data

Consider the scattering of a time-harmonic acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous acoustic medium. This paper is concerned with an inverse acoustic-elastic interaction problem, which is to determine the location and shape of the elastic obstacle by using either the phased or phaseless far-field data. By introducing the Helmholtz decomposition, the model problem is reduced to a coupled boundary value problem of the Helmholtz equations. The jump relations are studied for the second derivatives of the single-layer potential in order to deduce the corresponding boundary integral equations. The well-posedness is discussed for the solution of the coupled boundary integral equations. An efficient and high order Nyström-type discretization method is proposed for the integral system. A numerical method of nonlinear integral equations is developed for the inverse problem. For the case of phaseless data, we show that the modulus of the far-field pattern is invariant under a translation of the obstacle. To break the translation invariance, an elastic reference ball technique is introduced. We prove that the inverse problem with phaseless far-field pattern has a unique solution under certain conditions. In addition, a numerical method of the reference ball technique based nonlinear integral equations is proposed for the phaseless inverse problem. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed methods.

Projection and multi projection methods for nonlinear integral equations on the half-line

In this article, Galerkin and collocation methods and their iterated versions are discussed to solve the nonlinear Hammerstein type integral equation on the half-line for both convolution and non-convolution kernels using the space of piecewise polynomial subspace. We prove that the approximate and iterated approximate solution in Galerkin and collocation methods converges with the order O ( n − r ) and O ( n − 2 r ) respectively in the infinity norm, where n − 1 is the maximum norm of the partition and r denotes the order of the piecewise polynomial employed in the approximation. We improve these results further by considering multi-projection methods. In fact we show that iterated approximate solution in multi-Galerkin, multi-collocation converges with the order O ( n − 4 r ) . Numerical results are presented to confirm the theoretical results.

Simultaneous recovery of an infinite rough surface and the impedance from near-field data

ABSTRACTIn this paper, we consider an inverse rough surface scattering problem in near-field optical imaging. This problem is actually to reconstruct the scattering surface as well as its impedance coefficient from multifrequency near-field data, and can be reduced into an integral scheme by employing an integral representation. We solve this integral scheme by a non-linear integral equation method, and further develop a fast inversion algorithm for reconstructing both the rough surface and the impedance coefficient. Numerical experiments are presented to illustrate the effectiveness of the algorithm.

Adapted Newton-Kantorovich Methods for Nonlinear Integral Equations

For this work, the main idea is to make an adapted modification to the Newton-Kantorovich method destined to solve a nonlinear integral equations, so that by this technical method we obtain a simple application to this solution. Moreover, we compare the numerical results obtained by this method against ones obtained by another authors. This comparison showed the efficiency of this method.

Asymptotic error analysis of projection and modified projection methods for nonlinear integral equations

Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a smooth kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree $\leq r$, previous authors have established an order $r + 1$ convergence for the Galerkin solution, $2 r + 2$ for the iterated Galerkin solution, $ 3 r + 3$ for the modified projection solution and $ 4 r + 4$ for the iterated modified projection solution. Equivalent results have also been established for the interpolatory projection at Gauss points. In this paper, the iterated Galerkin/iterated collocation solution and the iterated modified projection solution are shown to have asymptotic series expansions. The Richardson extrapolation can then be used to improve the order of convergence to $ 2 r + 4$ in the case of the iterated Galerkin/iterated collocation method and to $ 4 r + 6$ in the case of the iterated modified projection method. Numerical results are given to illustrate this improvement in the orders of convergence.

Integral Equations of the Stefan Problem and Their Application in Modeling of Thawing Soil

The paper offers a new numerical method for solving the Stefan problem, based on the reduction of the initial boundary value problem for nonlinear integral equation equivalent to the minimum dimension. A feature of the method is that the time-varying function of phase transition is used as a major unknown function. This parameterization is an efficient computational technique enabling the considerably simplified method of solution. Computational qualities of this method are investigated by the example of solving the practical problems of thawing frozen soils under thermal action produced by various ways. The method advantages are most evident in solving problems in an unbounded domain, as unlike traditional finite difference method there is no need to truncate the system of equations with a large possibly infinite dimension. Despite the complexity to construct Green's function, in the case of variable properties of the environment there is a broad class of problems concerning the simulation of phase transitions. Their solution can be successfully found by described method of non-linear integral equations. The method also provides additional convenience in Richardson extrapolation and in solution of Stephen inverse problems, in particular, problems of front control of phase transition.

Boundary effects on the supersymmetric sine-Gordon model through light-cone lattice approach

We discussed subspaces of the N=1 supersymmetric sine-Gordon model with Dirichlet boundaries through light-cone lattice regularization. In this paper, we showed, unlike the periodic boundary case, both of Neveu–Schwarz (NS) and Ramond (R) sectors of a superconformal field theory were obtained. Using a method of nonlinear integral equations for auxiliary functions defined by eigenvalues of transfer matrices, we found that an excitation state with an odd number of particles is allowed for a certain value of a boundary parameter even on a system consisting of an even number of sites. In a small-volume limit where conformal invariance shows up in the theory, we derived conformal dimensions of states constructed through the lattice-regularized theory. The result shows existence of the R sector, which cannot be obtained from the periodic system, while a winding number is restricted to an integer or a half-integer depending on boundary parameters.

A nonlinear integral equation method for the inverse scattering problem by sound-soft rough surfaces

In this paper, we consider the two-dimensional (2D) inverse acoustic scattering by a sound-soft, infinite rough surface. Our approach to this problem is based on the nonlinear integral equation method which was developed by Kress and Rundell for bounded obstacles. By using multiple frequency data, we derive a stable and accurate reconstruction method to image the profile of the rough surface. Numerical experiments are presented to illustrate the effectiveness of the inversion method.

Modified projection and the iterated modified projection methods for nonlinear integral equations

Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a smooth kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree $\leq r$, previous authors have established an order $r + 1$ convergence for the Galerkin solution and $2 r + 2$ for the iterated Galerkin solution. Equivalent results have also been established for the interpolatory projection at Gauss points. In this paper, a modified projection method is shown to have convergence of order $ 3 r + 3$ and one step of iteration is shown to improve the order of convergence to $ 4 r + 4$. The size of the system of equations that must be solved, in implementing this method, remains the same as for the Galerkin method.

Nonlinear integral equations for the black hole sigma model

It was previously established that the critical staggered XXZ spin chain provides a lattice regularization of the black hole conformal field theory (CFT). We reconsider the continuum limit of this spin chain with the exact method of nonlinear integral equations (NLIEs), paying particular attention to the effects of a singular integration kernel. With the help of the NLIEs, we rederive the continuous black hole spectrum, but also numerically match the density of states of the spin chain with that of the CFT, which is a new result. Finally, we briefly discuss the integrable structure of the black hole CFT and the identification of its massive integrable perturbation on the lattice.

Reconstruction of biologic tissue conductivity from noisy magnetic field by integral equation method

Magnetic resonance electrical impedance tomography (MREIT) is a new technique to recover the conductivity of biologic tissue from the induced magnetic flux density. This paper proposes an inversion scheme for recovering the conductivity from one component of the magnetic field based on the nonlinear integral equation method. To apply magnetic fields corresponding to two incoherent injected currents, an alternative iteration scheme is proposed to update the conductivity. For each magnetic field, the regularizing technique on the finite dimensional space is applied to solve an ill-posed linear system. Compared with the well-developed harmonic Bz method, the advantage of this inversion scheme is its stability, since no differential operation is required on the noisy magnetic field. Numerical implementations are given to show the convergence of the iteration and its validity for noisy input data.

Separating impact on an elliptic cylinder floating on the surface of an ideal incompressible finite-depth fluid

The method of nonlinear integral equations of the Hammerstein type is applied to numerically analyze the problem of the separating impact on an elliptic cylinder floating on the surface of an ideal incompressible finite-depth fluid. The bottom effect and the influence of the kinematic and geometric parameters on the zone of fluid particle separation from the cylinder surface are studied.

Multigrid method for nonlinear integral equations

In this paper, we introduce a multigrid method for solving the nonliear Urysohn integral equation. The algorithm is derived from a discrete resolvent equation which approximates the continuous resolvent equation of the nonlinear Urysohn integral equation. The algorithm is mathematically equivalent to Atkinson’s adaptive twogrid iteration. But the two are different computationally. We show the convergence of the algorithm and its equivalence to Atkinson’s adaptive twogrid iteration. In our numerical example, we compare our algorithm to other multigrid methods for solving the nonliear Urysohn integral equation including the nonlinear multigrid method introduced by Hackbush.

The discrete collocation method for nonlinear integral equations

The collocation method for solving linear and nonlinear integral equations results in many integrals which must be evaluated numerically. In this paper, we give a general framework for discrete collocation methods, in which all integrals are replaced by numerical integrals. In some cases, the collocation method leads to solutions which are superconvergent at the collocation node points. We consider generalizations of these results, to obtain similar results for discrete collocation solutions. Lastly, we consider a variant due to Kumar and Sloan for the collocation solution of Hammerstein integral equations