Fuming Ma

Appropriate implementation of an invariant MFS for inverse boundary determination problem

In this paper, we are concerned with the inverse boundary determination problem from the Cauchy data connected with the Laplace equation in . We proposed a numerical method based on the method of fundamental solution. The major advantage of the invariant method of fundamental solution is keeping a very basic natural property under an invariant condition, i.e. the invariance under trivial coordinate changes in the problem description. This method combines the Tikhonov regularization method with Morozov discrepancy principle to solve an inverse problem. Some examples are given for numerical verification on the efficiency of the proposed method. It is shown that the proposed method is effective and stable even for the data with relatively high noise levels.

The reciprocity gap functional method for the inverse scattering problem for cavities

In this paper, we concern with determining the shape of a perfectly conducting cavity from the Cauchy data on a curve inside the cavity. The near-field linear sampling method (LSM), i.e. the reciprocity gap (RG) functional method, is employed to reconstruct the shape of the cavity. The equivalence of the RG method and the linear sampling method with mere the scattered field is established. But from the examples, we can see that the reconstructions are as satisfactory as the exterior scattering problems. I think this behavior is due to our reconstruction method since this method is due to the Cauchy data, but the LSM with mere the scattered field is used. Numerical tests show that the methods can provide qualitative information on the cavity. The numerical influence of the proposed method with respect to the wave numbers, the curve for the Cauchy data on which are measured, and the curve which is used to construct the single-layer potential function, respectively, are also analyzed with some examples. In particular, we give the examples of determining the cavity from the Cauchy data measured on a portion of the curve inside the cavity.

Time domain scattering and inverse scattering problems in a locally perturbed half-plane

This paper is concerned with efficient numerical methods for solving the time-dependent scattering and inverse scattering problems of acoustic waves in a locally perturbed half-plane. By symmetric continuation, the scattering problem is reformulated as an equivalent symmetric problem defined in the whole plane. The retarded potential boundary integral equation method is modified to solve the forward problem. Then we consider the inverse scattering problem of determinating the local perturbation from the measured scattered data. The time domain linear sampling method is employed to deal with the inverse problem. The computation schemes proposed in this paper are relatively simple and easy to implement. Several numerical examples are presented to show the effectiveness of the proposed methods.

Numerical method for inverse scattering in two-layered background in near-field optics

The inverse scattering problem in two-layered background that arises in near-field optics is discussed. The reconstruction of the scatterer from inhomogeneous medium deposited on a homogeneous substrate is presented, where the measured data only lies on a limited aperture. An error bound of the Born approximation is given. A moment method with truncated SVD is developed to handle the exponentially ill-posed problem with noisy measured data. Numerical experiments are presented to illustrate the resolution of the method, which are satisfying even under the perturbed data.

Finite element method with nonlocal boundary condition for solving the nondestructive testing problem of wood moisture content

Abstract In this paper, we use the finite element method (FEM) with nonlocal boundary condition for solving the nondestructive testing problem of wood moisture content based on a planar capacitance sensor model (i.e. the DtN-FEM for solving the mathematical model which is described by the exterior problems of a class of 3D Laplace equation with complicated boundary conditions). The original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. For numerical computation, a series of approximate problems with higher accuracy can be derived if one truncates the series term in the variational formulation, which is equivalent to the reduced problem. The error estimate is presented to show how the error depends on the finite element discretization and the accuracy of the approximate problem. Based on the numerical results, the relationship between the dielectric constant (DC) of tested wood and the capacitance value of the sensor is discussed. Finally, we applied a least squares fitting method (LSFM) to reconstruct the wood moisture content (WMC) from the data measured with a planar capacitance sensor. Compared with popular statistical methods, the hybrid experimental–computational method is more convenient and faster, and a large number of experiments are avoided, the costs of testing are reduced.

A meshless method for the Cauchy problem in linear elastodynamics

In this paper, we establish new density result for the Navier equation. Based on the denseness of the elastic single-layer potential functions, the Cauchy problem for the Navier equation is investigated. The ill-posedness of this problem is given via the compactness of the operator defined by the potential function. The method combines the Newton’s method and minimum norm solution with discrepancy principle to solve an inverse problem. Convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method.

An optimization method for acoustic inverse obstacle scattering problems with multiple incident waves

The inverse problem considered in this article is to determine the shape of a two-dimensional time-harmonic acoustic scatterer with Dirichlet boundary conditions from the knowledge of some far field patterns. Based on the optimization method due to Kirsch and Kress for the inverse scattering problem, we propose a new scheme by reformulating the cost functional via a technique of piecewise integration with respect to incident directions. Convergence analysis of this method is given. Numerical experiments show that our method accelerates the computations without losing the accuracy of the reconstructions for the full-aperture problems. The method is extended to the limited-aperture case by weighting the total fields with special factors. Numerical examples for limited-aperture problems are also presented which show that our method produces satisfactory results efficiently in the illuminated regions.