The Spring-Damping Regularization Method and the Lie-Group Shooting Method for Inverse Cauchy Problems

The inverse Cauchy problems for elliptic equations, such as the Laplace equation, the Poisson equation, the Helmholtz equation and the modified Helmholtz equation, defined in annular domains are investigated. The outer boundary of the annulus is imposed by overspecified boundary data, and we seek unknown data on the inner boundary through the numerical solution by a spring-damping regularization method and its Lie-group shooting method (LGSM). Several numerical examples are examined to show that the LGSM can overcome the ill-posed behavior of inverse Cauchy problem against the disturbance from random noise, and the computational cost is very cheap.     

A coupled complex boundary method for the Cauchy problem

Considered in this paper is a Cauchy problem governed by an elliptic partial differential equation. In the Cauchy problem, one wants to recover the unknown Neumann and Dirichlet data on a part of the boundary from the measured Neumann and Dirichlet data, usually contaminated with noise, on the remaining part of the boundary. The Cauchy problem is an inverse problem with severe ill-posedness. In this paper, a coupled complex boundary method (CCBM), originally proposed in [Cheng XL, Gong RF, Han W, et al. A novel coupled complex boundary method for solving inverse source problems. Inverse Prob. 2014;30:055002], is applied to solve the Cauchy problem stably. With the CCBM, all the data, including the known and unknown ones on the boundary are used in a complex Robin boundary on the whole boundary. As a result, the Cauchy problem is transferred into a complex Robin boundary problem of finding the unknown data such that the imaginary part of the solution equals zero in the domain. Then the Tikhonov regularization is applied to the resulting new formulation. Some theoretical analysis is performed on the CCBM-based Tikhonov regularization framework. Moreover, through the adjoint technique, a simple solver is proposed to compute the regularized solution. The finite-element method is used for the discretization. Numerical results are given to show the feasibility and effectiveness of the proposed method.     

A multiple-scale Trefftz method for an incomplete Cauchy problem of biharmonic equation

In this paper we numerically solve both the direct and the inverse Cauchy problems of biharmonic equation by using a multiple-scale Trefftz method (TM). The approximate solution is expressed to be a linear combination of T-complete bases, and the unknown coefficients are determined to satisfy the boundary conditions, by solving a resultant linear equations system. We introduce a better multiple-scale in the T-complete bases by using the concept of equilibrated norm of the coefficient matrix, such that the explicit formulas of these multiple scales can be derived. The condition number of the coefficient matrix can be significantly reduced upon using these better scales; hence, the present multiple-scale Trefftz method (MSTM) can effectively solve the inverse Cauchy problem without needing of the overspecified data, which is an incomplete Cauchy problem. Numerical examples reveal the efficiency that the new method can provide a highly accurate numerical solution even the problem domain might have a corner singularity, and the given boundary data are subjected to a large random noise.     

Boundary knot method for some inverse problems associated with the Helmholtz equation

The boundary knot method is an inherently meshless, integration‐free, boundary‐type, radial basis function collocation technique for the solution of partial differential equations. In this paper, the method is applied to the solution of some inverse problems for the Helmholtz equation, including the highly ill‐posed Cauchy problem. Since the resulting matrix equation is badly ill‐conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the L‐curve method. Numerical results are presented for both smooth and piecewise smooth geometry. The stability of the method with respect to the noise in the data is investigated by using simulated noisy data. The results show that the method is highly accurate, computationally efficient and stable, and can be a competitive alternative to existing methods for the numerical solution of the problems. Copyright © 2005 John Wiley & Sons, Ltd.     

The MFS for numerical boundary identification in two-dimensional harmonic problems

In this study, we briefly review the applications of the method of fundamental solutions to inverse problems over the last decade. Subsequently, we consider the inverse geometric problem of identifying an unknown part of the boundary of a domain in which the Laplace equation is satisfied. Additional Cauchy data are provided on the known part of the boundary. The method of fundamental solutions is employed in conjunction with regularization in order to obtain a stable solution. Numerical results are presented and discussed.     

Direct method of solution for general boundary value problem of the Laplace equation

A general boundary value problem for two-dimensional Laplace equation in the domain enclosed by a piecewise smooth curve is considered. The Dirichlet and the Neumann data are prescribed on respective parts of the boundary, while there is the second part of the boundary on which no boundary data are given. There is the third part of the boundary on which the Robin condition is prescribed. This problem of finding unknown values along the whole boundary is ill posed. In this sense we call our problem an inverse boundary value problem. In order for a solution to be identified the inverse problem is reformulated in terms of a variational problem, which is then recast into primary and adjoint boundary value problems of the Laplace equation in its conventional form. A direct method for numerical solution of the inverse boundary value problem using the boundary element method is presented. This method proposes a non-iterative and unified treatment of conventional boundary value problem, the Cauchy problem, and under- or over-determined problems.     

Identification of a Corroded Boundary and Its Robin Coefficient

An inverse geometric problem for two-dimensional Helmholtz-type equations arising in corrosion detection is considered. This problem involves determining an unknown corroded portion of the boundary of a two-dimensional domain and possibly its surface heat transfer (impedance) Robin coefficient from one or two pairs of boundary Cauchy data (boundary temperature and heat flux), and is solved numerically using the meshless method of fundamental solutions. A nonlinear unconstrained minimisation of the objective function is regularised when noise is added into the input boundary data. The stability of the numerical results is investigated for several test examples, with respect to noise in the input data and various values of the regularisation parameters.     

Stable Boundary and Internal Data Reconstruction in Two-Dimensional Anisotropic Heat Conduction Cauchy Problems Using Relaxation Procedures for an Iterative MFS Algorithm

We investigate two algorithms involving the relaxation of either the given boundary temperatures (Dirichlet data) or the prescribed normal heat fluxes (Neumann data) on the over-specified boundary in the case of the iterative algorithm of Kozlov91 applied to Cauchy problems for two-dimensional steady-state anisotropic heat conduction (the Laplace-Beltrami 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 chosen according to the generalized cross-validation (GCV) criterion. The iterative MFS algorithms with relaxation are tested for over-, equally and under-determined Cauchy problems associated with the steady-state anisotropic heat conduction in various two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.     

Regularized MFS-Based Boundary Identification in Two-Dimensional Helmholtz-Type Equations

We study the stable numerical identification of an unknown portion of the boundary on which a given boundary condition is provided and additional Cauchy data are given on the remaining known portion of the boundary of a two-dimensional domain for problems governed by either the Helmholtz or the modified Helmholtz equation. This inverse geometric problem is solved using the method of fundamental solutions (MFS) in conjunction with the Tikhonov regularization method. The optimal value for the regularization parameter is chosen according to Hansen's L-curve criterion. The stability, convergence, accuracy and efficiency of the proposed method are investigated by considering several examples.     

Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation

In this paper, several boundary element regularization methods, such as iterative, conjugate gradient, Tikhonov regularization and singular value decomposition methods, for solving the Cauchy problem associated to the Helmholtz equation are developed and compared. Regularizing stopping criteria are developed and the convergence, as well as the stability, of the numerical methods proposed are analysed. The Cauchy problem for the Helmholtz equation can be regularized by various methods, such as the general regularization methods presented in this paper, but more accurate results are obtained by classical methods, such as the singular value decomposition and the Tikhonov regularization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical boundary identification for Helmholtz-type equations

We study the identification of an unknown portion of the boundary of a two-dimensional domain occupied by a material satisfying Helmholtz-type equations from additional Cauchy data on the remaining known portion of the boundary. This inverse geometric problem is approached using the boundary element method (BEM) in conjunction with the Tikhonov first-order regularization procedure, whilst the choice of the regularization parameter is based on the L-curve criterion. The numerical results obtained show that the proposed method produces a convergent and stable solution     

Numerical solutions of boundary detection problems using modified collocation Trefftz method and exponentially convergent scalar homotopy algorithm

In this paper, the boundary detection problem, which is governed by the Laplace equation, is analyzed by the modified collocation Trefftz method (MCTM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the boundary detection problem, the Cauchy data is given on part of the boundary and the Dirichlet boundary condition on the other part of the boundary, whose spatial position is unknown a priori. By adopting the MCTM, which is meshless and integral-free, the numerical solution is expressed by a linear combination of the T-complete functions of the Laplace equation. The use of a characteristic length in MCTM can stabilize the numerical procedure and ensure highly accurate solutions. Since the coefficients of MCTM and the position of part of the boundary are unknown, to collocate the boundary conditions will yield a system of nonlinear algebraic equations; the ECSHA, which is exponentially convergent, is adopted to solve the system of nonlinear algebraic equations. Several numerical examples are provided to demonstrate the ability and accuracy of the proposed meshless scheme. In addition, the consistency of the proposed scheme is validated by adding noise into the boundary conditions.     

Numerical identification for impedance coefficient by a MFS-based optimization method

In this paper, we propose a new numerical method to solve an inverse impedance problem for Laplace's equation. The Robin coefficient in the impedance boundary condition is recovered from Cauchy data on a part of boundary. A crucial step is to transform the problem into an optimization problem based on the MFS and Tikhonov regularization. Then the popular conjugate gradient method is used to solve the minimization problem. We compare several stopping rules in the iteration procedure and try to find an accurate and stable approximation. Numerical results for four examples in 2D and 3D cases will show the effectiveness of the proposed method.     

Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method

In this paper, the inverse Cauchy problem for Laplace equation defined in an arbitrary plane domain is investigated by using the collocation Trefftz method (CTM) with a better postconditioner. We first introduce a multiple-scale R_k in the T-complete functions as a set of bases to expand the trial solution. Then, the better values of R_k are sought by using the concept of an equilibrated matrix, such that the resulting coefficient matrix of a linear system to solve the expansion coefficients is best-conditioned from a view of postconditioner. As a result, the multiple-scale R_k can be determined exactly in a closed-form in terms of the collocated points used in the collocation to satisfy the boundary conditions. We test the present method for both the direct Dirichlet problem and the inverse Cauchy problem. A significant reduction of the condition number and the effective condition number can be achieved when the present CTM is used, which has a good efficiency and stability against the disturbance from large random noise, and the computational cost is much saving. Some serious cases of the inverse Cauchy problems further reveal that the unknown data can be recovered very well, although the overspecified data are provided only at a 20% of the overall boundary.     

A complex variable boundary element method: Development

A generalized boundary integral equation method for the solution of the Laplace equation is developed based on the Cauchy integral theorem for analytical complex variable functions. Although the approach is complicated by the utilization of complex variable theory, the resulting model involves direct integration along straight-line boundary segments (elements) rather than using quadrature formulae that are required in current real variable boundary element formulations. Previously published boundary integral equation methods based on the Cauchy integral theorem are shown to be a subset of the generalized model theory developed in this paper.     

A Lie-Group Adaptive Method for Imaging a Space-Dependent Rigidity Coefficient in an Inverse Scattering Problem of Wave Propagation

We are concerned with the reconstruction of an unknown space-dependent rigidity coefficient in a wave equation. This problem is known as one of the inverse scattering problems. Based on a two-point Lie-group equation we develop a Lie-group adaptive method (LGAM) to solve this inverse scattering problem through iterations, which possesses a special character that by using onlytwo boundary conditions and two initial conditions, as those used in the direct problem, we can effectively reconstruct the unknown rigidity function by aself-adaption between the local in time differential governing equation and the global in time algebraic Lie-group equation. The accuracy and efficiency of the present LGAM are assessed by comparing the imaged results with some postulated exact solutions. By means of LGAM, it is quite versatile to handle the wave inverse scattering problem for the image of the rigidity coefficient without needing any extra information from the wave motion.     

Modified boundary Tikhonov-type regularization method for the Cauchy problem of a semi-linear elliptic equation

In this paper, we consider a Cauchy problem for the semi-linear elliptic equation and use a modified boundary Tikhonov-type regularization method to overcome its ill-posedness. The existence, uniqueness and stability of the regularization solution are proven. Under an a-priori bound assumption for the exact solution, a convergence estimate of Hölder type for this method is obtained. Finally, an iterative scheme is proposed to calculate the regularization solution, numerical results show that this method works well.     

On Solving the Direct/Inverse Cauchy Problems of Laplace Equation in a Multiply Connected Domain,Using the Generalized Multiple-Source-Point Boundary-Collocation Trefftz Method &Characteristic Lengths

In this paper, a multiple-source-point boundary-collocation Trefftz method, with characteristic lengths being introduced in the basis functions, is proposed to solve the direct, as well as inverse Cauchy problems of the Laplace equation for a multiply connected domain. When a multiply connected domain with genus p (p>1) is considered, the conventional Trefftz method (T-Trefftz method) will fail since it allows only one source point, but the representation of solution using only one source point is impossible. We propose to relax this constraint by allowing many source points in the formulation. To set up a complete set of basis functions, we use the addition theorem of Bird and Steele (1992), to discuss how to correctly set up linearly-independent basis functions for each source point. In addition, we clearly explain the reason why using only one source point will fail, from a theoretical point of view, along with a numerical example. Several direct problems and inverse Cauchy problems are solved to check the validity of the proposed method. It is found that the present method can deal with both direct and inverse problems successfully. For inverse problems, the present method does not need to use any regularization technique, or the truncated singular value decomposition at all, since the use of a characteristic length can significantly reduce the ill-posed behavior. Here, the proposed method can be viewed as a general Trefftz method, since the conventional Trefftz method (T-Trefftz method) and the method of fundamental solutions (F-Trefftz method) can be considered as special cases of the presently proposed method.     

The method of fundamental solutions for a biharmonic inverse boundary determination problem

In this paper, a nonlinear inverse boundary value problem associated to the biharmonic equation is investigated. This problem consists of determining an unknown boundary portion of a solution domain by using additional data on the remaining known part of the boundary. The method of fundamental solutions (MFS), in combination with the Tikhonov zeroth order regularization technique, are employed. It is shown that the MFS regularization numerical technique produces a stable and accurate numerical solution for an optimal choice of the regularization parameter. A. Zeb on study leave visiting the University of Leeds.