Numerical simulation for the shape reconstruction of a cavity

We consider the inverse problem of reconstructing the interior boundary curve of a cavity from the knowledge of the measurements on the exterior boundary. The domain derivative of the corresponding operator is presented, and this allows the investigation of the regularized Newton method for the solution of the ill‐posed and nonlinear problem. Numerical examples indicate the feasibility of our method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A moving pseudo‐boundary method of fundamental solutions for void detection

We propose a new moving pseudo‐boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as an inverse boundary value problem for harmonic functions. The algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The center of this radial polar parametrization is considered to be unknown. We also include the contraction and dilation factors to be part of the unknowns in the resulting nonlinear least‐squares problem. This approach addresses the major problem of locating the pseudo‐boundary in the MFS in a natural way, because the inverse problem in question is nonlinear anyway. The feasibility of this new method is illustrated by several numerical examples. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Carleman estimate for a strongly damped wave equation and applications to an inverse problem

In this paper, we establish a Carleman estimate for a strongly damped wave equation in order to solve a coefficient inverse problems of retrieving a stationary potential from a single time‐dependent Neumann boundary measurement on a suitable part of the boundary. This coefficient inverse problem is for a strongly damped wave equation. We prove the uniqueness and the local stability results for this inverse problem. The proof of the results relies on Carleman estimate and a certain energy estimates for hyperbolic equation with strongly damped term. Moreover, this method could be used for a similar inverse problem for an integro‐differential equation with hyperbolic memory kernel. Copyright © 2012 John Wiley & Sons, Ltd.     

Shape inverse problem for the two‐dimensional unsteady Stokes flow

In this article, the shape inverse problem for the two‐dimensional unsteady Stokes flow has been presented. We employ Piola transformation to bypass the divergence free condition for the flow and prove the differentiability of the solution to the initial boundary value problem. For the approximate solution of the ill‐posed and nonlinear problem, we propose a regularized Gauss‐Newton method. The numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Detecting inclusions with a generalized impedance condition from electrostatic data via sampling

In this paper, we derive a sampling method to solve the inverse shape problem of recovering an inclusion with a generalized impedance condition from electrostatic Cauchy data. The generalized impedance condition is a second order differential operator applied to the boundary of the inclusion. We assume that the Dirichlet‐to‐Neumann mapping is given from measuring the current on the outer boundary from an imposed voltage. A simple numerical example is given to show the effectiveness of the proposed inversion method for recovering the inclusion. We also consider the inverse impedance problem of determining the impedance parameters for a known material from the Dirichlet‐to‐Neumann mapping assuming the inclusion has been reconstructed where uniqueness for the reconstruction of the coefficients is proven.     

A coupled complex boundary method for an inverse conductivity problem with one measurement

We recently proposed in [Cheng, XL et al. A novel coupled complex boundary method for inverse source problems Inverse Problem 2014 30 055002] a coupled complex boundary method (CCBM) for inverse source problems. In this paper, we apply the CCBM to inverse conductivity problems (ICPs) with one measurement. In the ICP, the diffusion coefficient q is to be determined from both Dirichlet and Neumann boundary data. With the CCBM, q is sought such that the imaginary part of the solution of a forward Robin boundary value problem vanishes in the problem domain. This brings in advantages on robustness and computation in reconstruction. Based on the complex forward problem, the Tikhonov regularization is used for a stable reconstruction. Some theoretical analysis is given on the optimization models. Several numerical examples are provided to show the feasibility and usefulness of the CCBM for the ICP. It is illustrated that as long as all the subdomains share some portion of the boundary, our CCBM-based Tikhonov regularization method can reconstruct the diffusion parameters stably and effectively.     

A conformal mapping algorithm for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in where an unknown free boundary Γ₀ is determined by prescribed Cauchy data on Γ₀ in addition to a Dirichlet condition on the known boundary Γ₁. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary Γ₀ as the unknown boundary in the inverse problem to construct Γ₀ from the Dirichlet condition on Γ₀ and Cauchy data on the known boundary Γ₁. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ₁ by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Reconstruction of Elastic Inclusions of Small Volume via Dynamic Measurements

We consider the inverse problem of identifying locations and certain properties of the shapes of small elastic inclusions in a homogeneous background medium from dynamic boundary measurements for a finite interval in time. Using particular background solutions as weights, we present an asymptotic method based on appropriate averaging of the dynamic boundary measurements and propose non-iterative algorithms for solving our inverse problem.     

A wavelets method for plane elasticity problem

A Neumann boundary value problem of plane elasticity problem in the exterior circular domain is reduced into an equivalent natural boundary integral equation and a Poisson integral formula with the DtN method. Using the trigonometric wavelets and Galerkin method, we obtain a fast numerical method for the natural boundary integral equation which has an unique solution in the quotient space. We decompose the stiffness matrix in our numerical method into four circulant and symmetrical or antisymmetrical submatrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform (FFT) and the inverse fast Fourier transform (IFFT) instead of the inverse matrix. Examples are given for demonstrating our method has good accuracy of our method even though the exact solution is almost singular.     

A numerical method for solving an inverse thermoacoustic problem

In this paper, we consider an inverse problem of determining the initial condition of an initial boundary value problem for the wave equation with some additional information about solving a direct initial boundary value problem. The information is obtained from measurements at the boundary of the solution domain. The purpose of our paper is to construct a numerical algorithm for solving the inverse problem by an iterative method called a method of simple iteration (MSI) and to study the resolution quality of the inverse problem as a function of the number and location of measurement points. Three two-dimensional inverse problem formulations are considered. The results of our numerical calculations are presented. It is shown that the MSI decreases the objective functional at each iteration step. However, due to the ill-posedness of the inverse problem the difference between the exact and approximate solutions decreases up to some fixed number k _min, and then monotonically increases. This shows the regularizing properties of the MSI, and the iteration number can be considered a regularization parameter.     

Recovery of an interface from boundary measurement in an elliptic differential equation

We study the inverse problem of recovering an interior interface from a boundary measurement in an elliptic boundary value problem arising from a semiconductor transistor model. We set up a nonlinear least-squares formulation for solving the inverse problem, and establish the necessary derivatives with respect to the interface. We then propose both the Gauss–Newton iterative method and the conjugate gradient method for the least-squares problem, and present implementation of these methods using integral equations.     

The factorization method for cracks in inhomogeneous media

We consider the inverse scattering problem of determining the shape and location of a crack surrounded by a known inhomogeneous media. Both the Dirichlet boundary condition and a mixed type boundary conditions are considered. In order to avoid using the background Green function in the inversion process, a reciprocity relationship between the Green function and the solution of an auxiliary scattering problem is proved. Then we focus on extending the factorization method to our inverse shape reconstruction problems by using far field measurements at fixed wave number. We remark that this is done in a non intuitive space for the mixed type boundary condition as we indicate in the sequel.     

Approximation of inverse operators by a new family of high‐order iterative methods

The main goal of this paper is to approximate inverse operators by high‐order Newton‐type methods with the important feature of not using inverse operators. We analyse the semilocal convergence, the speed of convergence, and the efficiency of these methods. We determine that Chebyshev's method is the most efficient method and test it on two problems: one associated to the heat equation and the other one to a boundary value problem. We consider examples with matrices that are close to be singular and/or are badly conditioned. We check the robustness and the stability of the methods by considering situations with many steps and noised data. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability estimate for a multidimensional inverse spectral problem with partial spectral data

In Bellassoued, Choulli and Yamamoto (2009) [4] we proved a log-log type stability estimate for a multidimensional inverse spectral problem with partial spectral data for a Schrödinger operator, provided that the potential is known in a small neighbourhood of the boundary of the domain. In the present paper we discuss the same inverse problem. We show a log type stability estimate under an additional condition on potentials in terms of their X-ray transform. In proving our result, we follow the same method as in Alessandrini and Sylvester (1990) [1] and Bellassoued, Choulli and Yamamoto (2009) [4]. That is we relate the stability estimate for our inverse spectral problem to a stability estimate for an inverse problem consisting in the determination of the potential in a wave equation from a local Dirichlet to Neumann map (DN map in short).     

An iterative method based on equation decomposition for the fourth‐order singular perturbation problem

In this article, we propose an iterative method based on the equation decomposition technique ( 1 ) for the numerical solution of a singular perturbation problem of fourth‐order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second‐order elliptic equation and a second‐order singular perturbation problem. We prove that our approximate solution converges to the exact solution when the domain is a disc. Our numerical examples show the efficiency and accuracy of our method. Our iterative method works very well for singular perturbation problems, that is, the case of 0 < ε ? 1, and the convergence rate is very fast. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Direct and inverse problems for electromagnetic scattering by a doubly periodic structure with a partially coated dielectric

Consider the problem of scattering of electromagnetic waves by a doubly periodic Lipschitz structure. The medium above the structure is assumed to be homogenous and lossless with a positive dielectric coefficient. Below the structure there is a perfect conductor with a partially coated dielectric boundary. We first establish the well‐posedness of the direct problem in a proper function space and then obtain a uniqueness result for the inverse problem by extending Isakov's method. Copyright © 2009 John Wiley & Sons, Ltd.     

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

The paper deals with the Sturm–Liouville eigenvalue problem with the Dirichlet boundary condition at one end of the interval and with the boundary condition containing entire functions of the spectral parameter at the other end. We study the inverse problem, which consists in recovering the potential from a part of the spectrum. This inverse problem generalizes partial inverse problems on finite intervals and on graphs and also the inverse transmission eigenvalue problem. We obtain sufficient conditions for global solvability of the studied inverse problem, which prove its local solvability and stability. In addition, application of our main results to the partial inverse Sturm–Liouville problem on the star-shaped graph is provided.     

Inverse scattering for planar cracks via nonlinear integral equations

We present a Newton‐type method for reconstructing planar sound‐soft or perfectly conducting cracks from far‐field measurements for one time‐harmonic scattering with plane wave incidence. Our approach arises from a method suggested by Kress and Rundell (Inv. Probl. 2005; 21 (4):1207–1223) for an inverse boundary value problem for the Laplace equation. It was extended to inverse scattering problems for sound‐soft obstacles (Mathematical Methods in Scattering Theory and Biomedical Engineering. World Scientific: Singapore, 2006; 39–50) and for sound‐hard cracks (Inv. Probl. 2006; 22 (6)). In both cases it was shown that the method gives accurate reconstructions with reasonable stability against noisy data. The approach is based on a pair of nonlinear and ill‐posed integral equations for the unknown boundary. The integral equations are solved by linearization, i.e. by regularized Newton iterations. Numerical reconstructions illustrate the feasibility of the method. Copyright © 2007 John Wiley & Sons, Ltd.     

Estimating the Accuracy of a Method of Auxiliary Boundary Conditions in Solving an Inverse Boundary Value Problem for a Nonlinear Equation

An inverse boundary value problem for a nonlinear parabolic equation is considered. Two-sided estimates for the norms of values of a nonlinear operator in terms of those of a corresponding linear operator are obtained.On this basis, two-sided estimates for the modulus of continuity of a nonlinear inverse problem in terms of that of a corresponding linear problem are obtained. A method of auxiliary boundary conditions is used to construct stable approximate solutions to the nonlinear inverse problem. An accurate (to an order) error estimate for the method of auxiliary boundary conditions is obtained on a uniform regularization class.     

A smoothing Newton's method for the construction of a damped vibrating system from noisy test eigendata

In this paper we consider an inverse problem for a damped vibration system from the noisy measured eigendata, where the mass, damping, and stiffness matrices are all symmetric positive‐definite matrices with the mass matrix being diagonal and the damping and stiffness matrices being tridiagonal. To take into consideration the noise in the data, the problem is formulated as a convex optimization problem involving quadratic constraints on the unknown mass, damping, and stiffness parameters. Then we propose a smoothing Newton‐type algorithm for the optimization problem, which improves a pre‐existing estimate of a solution to the inverse problem. We show that the proposed method converges both globally and quadratically. Numerical examples are also given to demonstrate the efficiency of our method. Copyright © 2008 John Wiley & Sons, Ltd.