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.     

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.     

Modified regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. It is known that such problem is severely ill-posed. We propose a modified regularization method to solve it based on the solution given by the method of separation of variables. Convergence estimates are presented under two different a-priori bounded assumptions for the exact solution. Finally, numerical examples are given to show the effectiveness of the proposed numerical method.     

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.     

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.     

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.     

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.     

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.     

A regularization method for solving the radially symmetric backward heat conduction problem

This work is devoted to solving the radially symmetric backward heat conduction problem, starting from the final temperature distribution. The problem is ill-posed: the solution (if it exists) does not depend continuously on the given data. A modified Tikhonov regularization method is proposed for solving this inverse problem. A quite sharp estimate of the error between the approximate solution and the exact solution is obtained with a suitable choice of regularization parameter. A numerical example is presented to verify the efficiency and accuracy of the method.     

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 wavelet-Galerkin method for high order numerical differentiation

Numerical differentiation is a classical ill-posed problem. In this paper, we propose a wavelet-Galerkin method for high order numerical differentiation. By an appropriate choice of the regularization parameter an order optimal stability estimate of Hölder type is obtained. Some numerical examples show that the method is effective and stable.     

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.     

A Trefftz/MFS mixed-type method to solve the Cauchy problem of the Laplace equation

By using the multiple-scale Trefftz method (MSTM) to solve the Cauchy problem of the Laplace equation in an arbitrary bounded domain, we may lose the accuracy several orders when the noise being imposed on the specified Cauchy data is quite large. In addition to the linear equations obtained from the MSTM, the fundamental solutions play as the test functions being inserted into a derived boundary integral equation. Therefore, after merely supplementing a few linear equations in the mixed-type method (MTM), which is a well organized combination of the Trefftz method and the method of fundamental solutions (MFS), we can improve the ill-conditioned behavior of the linear equations system and hence increase the accuracy of the solution for the Cauchy problem significantly, as explored by two numerical examples.     

A note on a Cauchy problem for the Laplace equation: Regularization and error estimates

In this paper, a Cauchy problem for the Laplace equation is investigated. Based on the fundamental solution to the elliptic equation, we propose to solve this problem by the truncation method, which generates well-posed problem. Then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proved. Error estimates for this method are provided together with a selection rule for the regularization parameter. The numerical results show that our proposed numerical methods work effectively. This work extends to earlier results in Qian et al. (2008) [14] and Hao et al. (2009) [5].     

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over‐specified boundary in the case of the alternating iterative algorithm of Kozlov et al. (USSR Comput Math Math Phys 31 (1991), 45–52) applied to the Cauchy problem for the two‐dimensional modified Helmholtz equation. The two mixed, well‐posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is selected according to the generalized cross‐validation criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the modified Helmholtz equation in two‐dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A new method for numerical differentiation based on direct and inverse problems of partial differential equations

In this paper, we mainly study a numerical differentiation problem which aims to approximate the second order derivative of a single variable function from its noise data. By transforming the problem into a combination of direct and inverse problems of partial differential equations (heat conduction equations), a new method that we call the PDEs-based numerical differentiation method is proposed. By means of the finite element method and the Tikhonov regularization, implementations of the proposed PDEs-based method are presented with a posterior strategy for choosing regularization parameters. Numerical results show that the PDEs-based numerical differentiation method is highly feasible and stable with respect to data noise.     

A Tikhonov regularization method for Cauchy problem based on a new relaxation model

In this paper, we consider a Cauchy problem of recovering both missing value and flux on inaccessible boundary from Dirichlet and Neumann data measured on the remaining accessible boundary. Associated with two mixed boundary value problems, a regularized Kohn-Vogelius formulation is proposed. With an introduction of a relaxation parameter, the Dirichlet boundary conditions are approximated by two Robin ones. Compared to the existing work, weaker regularity is required on the Dirichlet data. This makes the proposed model simpler and more efficient in computation. A series of theoretical results are established for the new reconstruction model. Several numerical examples are provided to show feasibility and effectiveness of the proposed method. For simplicity of the statements, we take Poisson equation as the governed equation. However, the proposed method can be applied directly to Cauchy problems governed by more general equations, even other linear or nonlinear inverse problems.     

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.     

The quasi-reversibility method to solve the Cauchy problems for parabolic equations

In this paper, on the one hand, we take the conventional quasi-reversibility method to obtain the error estimates of approximate solutions of the Cauchy problems for parabolic equations in a sub-domain of QT with strong restrictions to the measured boundary data. On the other hand, weakening the conditions on the measured data, then combining the duality method in optimization with the quasi-reversibility method, we solve the Cauchy problems for parabolic equations in the presence of noisy data. Using this method, we can get the proper regularization parameter ε that we need in the quasi-reversibility method and obtain the convergence rate of approximate solutions as the noise of amplitude δ tends to zero.