Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green’s formulation, the problem can be transformed into a moment problem. Then we propose a modified Tikhonov regularization algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.     

On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation

In this paper, we consider the Cauchy problem for the Helmholtz equation in a rectangle, where the Cauchy data is given for y=0 and boundary data are for x=0 and x=π. The solution is sought in the interval 0<y≤1. A quasi-reversibility method is applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

A regularization method for a Cauchy problem of the Helmholtz equation

We investigate a Cauchy problem for the Helmholtz equation. A modified boundary method is used for solving this ill-posed problem. Some Hölder-type error estimates are obtained. The numerical experiment shows that the modified boundary method works well.     

Two regularization methods for the Cauchy problems of the Helmholtz equation

In this paper, the Cauchy problems for the Helmholtz equation are investigated. We propose two regularization methods to solve them. Convergence estimates are presented under an a-priori bounded assumption for the exact solution. Finally, the numerical examples show that the proposed numerical methods work effectively.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

A Herglotz wavefunction method for solving the inverse Cauchy problem connected with the Helmholtz equation

This paper is concerned with the Cauchy problem connected with the Helmholtz equation. On the basis of the denseness of Herglotz wavefunctions, we propose a numerical method for obtaining an approximate solution to the problem. We analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method.     

A fractional Tikhonov method for solving a Cauchy problem of Helmholtz equation

In this paper, we study a fractional Tikhonov regularization method (FTRM) for solving a Cauchy problem of Helmholtz equation in the frequency domain. On the one hand, the FTRM retains the advantage of classical Tikhonov method. On the other hand, our method can prevent the effect of oversmoothing of classical Tikhonov method and conveniently control the amount of damping. The convergence error estimates between the exact solution and its regularization approximation are constructed. Several interesting numerical examples are provided, which validate the effectiveness of the proposed method.     

Wavelets and regularization of the Cauchy problem for the Laplace equation

In this paper, a Cauchy problem for two-dimensional Laplace equation in the strip 0<x?1 is considered again. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data, a small perturbation in the data can cause a dramatically large error in the solution for 0<x?1. The stability of the solution is restored by using a wavelet regularization method. Moreover, some sharp stable estimates between the exact solution and its approximation in Hr(R)-norm is also provided.     

A Fourth-Order Modified Method for the Cauchy Problem of the Modified Helmholtz Equation

This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0 < x≤ 1, y ∈ R. The Cauchy data at x = 0 is given and the solution is then sought for the interval 0 < x ≤1. This problem is highly ill-posed and the solution (if it exists) does not depend continuously on the given data. In this paper, we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution. Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.     

Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation

In this paper, a Cauchy problem for the time fractional advection-dispersion equation (TFADE) is investigated. Such a problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order . We show that the Cauchy problem of TFADE is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. The convergence estimate is obtained under a priori bound assumptions for the exact solution. Numerical examples are given to show the effectiveness of the proposed numerical method.     

A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation

In this paper, we solve the Cauchy problem for an inhomogeneous Helmholtz-type equation with homogeneous Dirichlet and Neumann boundary condition. The proposed problem is ill-posed. Up to now, most investigations on this topic focus on very specific cases, and with Dirichlet boundary condition. Recently, we solve this problem in 2D for an inhomogeneous modified Helmholtz equation (2012). This work is a continuous expansion of our previous results. Herein we introduce a general filter regularization (GFR) method, and then from the GFR we deduce two concrete filters, which are a foundation to implement a numerical procedure. In addition, we develop a numerical model for solving this problem in three dimensional region. The proposed filter method has been verified by numerical experiments.     

求解椭圆方程柯西问题的修正吉洪诺夫正则化方法

研究了一类变系数椭圆方程的柯西问题,这类问题出现在很多实际问题领域.由于问题的不适定性,不可能通过经典的数值方法来求解上述问题,必须引入正则化手段.采用了一种修正吉洪诺夫正则化方法来求解上述问题.在一种先验和一种后验参数选取准则下,分别获得了问题的误差估计.数值例子进一步显示方法是稳定有效的.     

A regularization method for solving the Cauchy problem for the Helmholtz equation

In this paper, we investigate a Cauchy problem associated with Helmholtz-type equation in an infinite “strip”. This problem is well known to be severely ill-posed. The optimal error bound for the problem with only nonhomogeneous Neumann data is deduced, which is independent of the selected regularization methods. A framework of a modified Tikhonov regularization in conjunction with the Morozov’s discrepancy principle is proposed, it may be useful to the other linear ill-posed problems and helpful for the other regularization methods. Some sharp error estimates between the exact solutions and their regularization approximation are given. Numerical tests are also provided to show that the modified Tikhonov method works well.     

A regularization method for the cauchy problem of the modified Helmholtz equation

In the present paper, an iteration regularization method for solving the Cauchy problem of the modified Helmholtz equation is proposed. The a priori and a posteriori rule for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method. Copyright © 2014 John Wiley & Sons, Ltd.     

The a posteriori Fourier method for solving the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data

In the present paper, the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data in an infinite “strip” domain is considered. This problem is severely ill-posed, i.e., the solution does not depend continuously on the data. A conditional stability result is given. A new a posteriori Fourier method for solving this problem is proposed. The corresponding error estimate between the exact solution and its regularization approximate solution is also proved. Numerical examples show the effectiveness of the method and the comparison of numerical effect between the a posteriori and the a priori Fourier method are also taken into account.     

The Carleman formula for the Helmholtz equation on the plane

We consider the Cauchy problem for the Helmholtz equation in an arbitrary bounded planar domain with Cauchy data only on part of the boundary of the domain. We derive a Carleman-type formula for a solution to this problem and give a conditional stability estimate.     

Determining surface heat flux in the steady state for the Cauchy problem for the Laplace equation

In this paper, we consider the Cauchy problem for the Laplace equation, in a strip where the Cauchy data is given at x = 0 and the flux is sought in the interval 0<x?1. This problem is typical ill-posed: the solution (if it exists) does not depend continuously on the data. We study a modification of the equation, where a fourth-order mixed derivative term is added. Some error stability estimates for the flux are given, which show that the solution of the modified equation is approximate to the solution of the Cauchy problem for the Laplace equation. Furthermore, numerical examples show that the modified method works effectively.     

Fourier Moment Method with Regularization for the Cauchy Problem of Helmholtz Equation

In this paper, we consider the reconstruction of the wave field in abounded domain. By choosing a special family of functions, the Cauchy problemcan be transformed into a Fourier moment problem. This problem is ill-posed. Wepropose a regularization method for obtaining an approximate solution to the wavefield on the unspecified boundary. We also give the convergence analysis and errorestimate of the numerical algorithm. Finally, we present some numerical examples toshow the effectiveness of this method.     

An iterative method for a Cauchy problem for the heat equation

^** Email: tomjo{at}itn.liu.se An iterative method for reconstruction of the solution to aparabolic initial boundary value problem of second order fromCauchy data is presented. The data are given on a part of theboundary. At each iteration step, a series of well-posed mixedboundary value problems are solved for the parabolic operatorand its adjoint. The convergence proof of this method in a weightedL²-space is included.