A numerical study of variable coefficient elliptic Cauchy problem via projection method

In this paper, we investigate a numerical method for the solution of an inverse problem of recovering lacking data on some part of the boundary of a domain from the Cauchy data on other part for a variable coefficient elliptic Cauchy problem. In the process, the Cauchy problem is transformed into the problem of solving a compact linear operator equation. As a remedy to the ill-posedness of the problem, we use a projection method which allows regularization solely by discretization. The discretization level plays the role of regularization parameter in the case of projection method. The balancing principle is used for the choice of an appropriate discretization level. Several numerical examples show that the method produces a stable good approximate solution.     

Deadlock-free motion planning using the Laplace potential field

In this paper we introduce a motion planning method which uses an artificial potential field obtained by solving Laplace's differential equation. A potential field based on Laplace's equation has no minimal point; therefore, path planning is performed without falling into local minima. Furthermore, we propose an application of the motion planning method for recursive motion planning in an uncertain environment. We illustrate the robot motion generated by the proposed method with simulation examples.     

Indirect Boundary Integral Equation Method for the Cauchy Problem of the Laplace Equation

In this paper, we examine the Cauchy problem of the Laplace equation. Motivated by the incompleteness of the single-layer potential function method, we investigate the double-layer potential function method. Through the use of a layer approach to the solution, we devise a numerical method for approximating the solution of the Cauchy problem, which are well known to be highly ill-posed in nature. The ill-posedness is dealt with Tikhonov regularization, whilst the optimal regularization parameter is chosen by Morozov discrepancy principle. Convergence and stability estimates of the proposed method are then given. Finally, some examples are given for the efficiency of the proposed method. Especially, when the single-layer potential function method does not give accurate results for some problems, it is shown that the proposed method is effective and stable.     

Numerical solution of laplace's equation on non-simply connected regions on R 2

We consider the interior Dirichlet problem for Laplace's equation on a non-simply connected two-dimensional regions with smooth boundaries.The solution is sought as the real part of a holomorphic function on the region, given as Cauchy-type integral.The approximate double layer density function is found by solving a system of Fredholm integral equations of second kind.Because of the non-uniqueness of the solution of the system we solve it using a technique based on the solution of the “Modified Dirichlet problem”.The Nystrom's method coupled with the trapezoidal rule is used as numerical integration scheme.The linear system derived from the integral equation is solved using the conjugate gradient applied to the normal equation.Theoretical and computational details of the method are presented.     

Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation

In this paper, we investigate a Cauchy problem for the semi-linear elliptic equation. This problem is well known to be severely ill-posed and regularization methods are required. We use a modified quasi-boundary value method to deal with it, and a convergence estimate for the regularized solution is obtained under an a priori bound assumption for the exact solution. Finally, some numerical results show that our given method works well.     

Analysis of torsion of orthotropic bars using personal computers

A simple procedure for analysis of torsion of orthotropic bars is presented. It considers an equivalent isotropic region for which the warping function satisfies Laplace's equation. The method requires minimum computer time and memory and is suited to personal computers. To demonstrate the accuracy as well as the simplicity of the method, a straightforward algorithm and an example problem are included.     

On the numerical solution of the eigenvalue problem of the laplace operator by a capacitance matrix method

The problem of finding several eigenfunctions and eigenvalues of the interior Dirichlet problem for Laplace's equation on arbitrary bounded plane regions is considered. Two fast algorithms are combined: an iterative Block Lanczos method and a capacitance matrix method. The capacitance matrix is generated and factored only once for a given problem. In each iteration of the Block Lanczos method, a discrete Helmholtz equation is solved twice on a rectangle at a cost of the order ofn ² log₂ n operations wheren is the number of mesh points across the rectangle in which the region is imbedded.     

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

In this paper, a Cauchy problem for the Helmholtz equation is considered. It is known that such a problem is severely ill-posed, i.e. the solution does not depend continuously on the given Cauchy data. We propose a quasi-reversibility regularization method to solve it. Convergence estimates are established under two different a priori assumptions for an exact solution. Numerical results obtained by two different schemes show that our proposed methods work well.     

Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method

In this paper, we consider the identification of a corrosion boundary for the two-dimensional Laplace equation. A boundary collocation method is proposed for determining the unknown portion of the boundary from the Cauchy data on a part of the boundary. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing the Tikhonov regularization technique, while the regularization parameter is provided by the generalized cross-validation criterion. Numerical examples show that the proposed method is reasonable and feasible.     

An iterative method for the Cauchy problem in linear elasticity with fading regularization effect

In this paper, an iterative method for solving the Cauchy problem in linear elasticity is introduced. This problem consists in recovering missing data (displacements and forces) on some parts of a domain boundary from the knowledge of overspecified data (displacements and forces) on the remaining parts. The algorithm reads as a least square fitting of the given data, with a regularization term whose effect fades as the iterations go on. So the algorithm converges to the solution of the Cauchy problem. Numerical simulations using the finite element method highlight the algorithm’s efficiency, accuracy, robustness to noisy data as well as its ability to deblur noisy data.     

The density function reconstruction of surface sources from a single Cauchy measurement

The inverse problem of reconstructing sources is explored when a single boundary Cauchy data is postulated on the potential. We are particularly involved in sources supported by (hyper-)surfaces. Mild assumptions are required on the location of these supports and the calculation of the charge density function is then aimed. We consider a variational formulation, based on a duplication artifice of the potential and we check the symmetry and the positive definiteness of the weak problem. Because of the severe ill-posedness, the use of a regularization is mandatory for a safe approximation of the solution. Lavrentiev’s method is therefore recommended in the context owing to the symmetry and the positivity. We check why that regularization turns out to be a Tikhonov method for some underlying shadow equation that is not needed in computations and is therefore never explicitly constructed. Results stated in a wide literature for the Tikhonov regularization applies as well to our variational problem. An important consequence is that the Morozov Discrepancy Principle, we use for the selection of the regularization parameter yields a convergent strategy. Now, that the Discrepancy Principle requires the residual of that inaccessible ‘shadow equation’, we explain how the Kohn–Vogelius function allows for the computation of that residual.     

A method for the numerical solution of some elliptic boundary value problems for a strip

A boundary integral equation for the numerical solution of a class of elliptic boundary value problems for a strip is derived. The equation should be particularly useful for the solution of an important class of problems governed by Laplace's equation and also for the solution of relevant problems in anisotropic thermostatics and elastostatics     

The backward problem for a time-fractional diffusion-wave equation in a bounded domain

This paper is devoted to solve the backward problem for a time-fractional diffusion-wave equation in a bounded domain. Based on the series expression of the solution for the direct problem, the backward problem for searching the initial data is converted into solving the Fredholm integral equation of the first kind. The existence, uniqueness and conditional stability for the backward problem are investigated. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of the regularized solution for the backward problem. Furthermore, the convergence rate for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical results for five examples in one-dimensional case and two-dimensional case show that the proposed method is efficient and stable.     

New gauss–seidel like block iterative methods to solve discrete boundary value problems

The concept of new Gauss–Seidel like iterative methods, which was introduced in [3], will be extended so as to obtain a class of convergent Gauss–Seidel like block iterative methods to solve linear matrix equations Ax=b with an M-Matrix A. New block iterative methods will be applied to finite difference approximations of the Laplace's equation on a square (“model problem” [8]) which surpass even the block successive overrelaxation iterative method with optimum relaxation factor in this example.     

Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation

In this paper we consider the Monte Carlo solution of the Cauchy problem for a nonlinear parabolic equation. Using the fundamental solution of the heat equation, we obtain a nonlinear integral equation with solution the same as the original partial differential equation. On the basis of this integral representation, we construct a probabilistic representation of the solution to our original Cauchy problem. This representation is based on a branching stochastic process that allows one to directly sample the solution to the full nonlinear problem. Along a trajectory of these branching stochastic processes we build an unbiased estimator for the solution of original Cauchy problem. We then provide results of numerical experiments to validate the numerical method and the underlying stochastic representation.     

New wavelet preconditioner for solving boundary integral equations over nonsmooth boundaries

Matrices obtained by wavelet discretisation of partial differential equations or boundary integral equations (BIEs) are typically sparse with a finger-like sparsity pattern, in contrast to matrices obtained by traditional single scale discretisation, which are dense. In some cases diagonal preconditioning is sufficient, but there are effective preconditioners that can be used when this is not the case. Here, we consider the particular case of BIEs whose boundary has a geometrical singularity, and propose a new preconditioning strategy based on permutations of the unknowns. The strategy's performance is analysed and compared with related techniques for the double layer equation for Laplace's equation.     

Two regularization methods and the order optimal error estimates for a sideways parabolic equation

In this paper, we consider an inverse heat conduction problem which appears in some applied subjects. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. A Tikhonov type's regularization method and a Fourier regularization method are applied to formulate regularized solutions which are stably convergent to the exact ones with order optimal error estimates. A numerical example shows that the computational effect of these methods are all satisfactory.     

Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality

The purpose is to perform a sharp analysis of the Lavrentiev method applied to the regularization of the ill-posed Cauchy problem, set in the Steklov-Poincaré variational framework. Global approximation results have been stated earlier that demonstrate that the Lavrentiev procedure yields a convergent strategy. However, no convergence rates are available unless a source condition is assumed on the exact Cauchy solution. We pursue here bounds on the approximation (bias) and the noise propagation (variance) errors away from the incomplete boundary where instabilities are located. The investigation relies on a Carleman inequality that enables enhanced local convergence rates for both bias and variance errors without any particular smoothness assumption on the exact solution. These improved results allows a new insight on the behavior of the Lavrentiev solution, look similar to those established for the Quasi-Reversibility method in [Inverse Problems 25, 035005, 2009]. There is a case for saying that this sort of ??super-convergence?? is rather inherent to the nature of the Cauchy problem and any reasonable regularization procedure would enjoy the same locally super-convergent behavior.     

Some iterative algorithms for the obstacle problems

In this paper we propose some iterative algorithms for the obstacle problems discretized by the finite difference method. We rewrite the obstacle problem to an equivalent complementarity problem. We use the regularization trick to non-smooth absolute function. Then we apply the Newton's method to obtain an iterative algorithm. Some other two algorithms based on this algorithm are derived. Numerical experiments show the effective of the algorithm.