Table of contents

The Second International Conference on Inverse Problems: Recent Theoretical Developments and Numerical Approaches was held at Fudan University, Shanghai from 16–21 June 2004. The first conference in this series was held at the City University of Hong Kong in January 2002 and it was agreed to hold the conference once every two years in a Pan-Pacific Asian country. The next conference is scheduled to be held at Hokkaido University, Sapporo, Japan in July 2006. The purpose of this series of biennial conferences is to establish and develop constant international collaboration, especially among the Pan-Pacific Asian countries.

In recent decades, interest in inverse problems has been flourishing all over the globe because of both the theoretical interest and practical requirements. In particular, in Asian countries, one is witnessing remarkable new trends of research in inverse problems as well as the participation of many young talents. Considering these trends, the second conference was organized with the chairperson Professor Li Tat-tsien (Fudan University), in order to provide forums for developing research cooperation and to promote activities in the field of inverse problems.

Because solutions to inverse problems are needed in various applied fields, we entertained a total of 92 participants at the second conference and arranged various talks which ranged from mathematical analyses to solutions of concrete inverse problems in the real world.

This volume contains 18 selected papers, all of which have undergone peer review. The 18 papers are classified as follows:

1. Surveys: four papers give reviews of specific inverse problems.

2. Theoretical aspects: six papers investigate the uniqueness, stability, and reconstruction schemes.

3. Numerical methods: four papers devise new numerical methods and their applications to inverse problems.

4. Solutions to applied inverse problems: four papers discuss concrete inverse problems such as scattering problems and inverse problems in atmospheric sciences and oceanography.

Last but not least is our gratitude. As editors we would like to express our sincere thanks to all the plenary and invited speakers, the members of the International Scientific Committee and the Advisory Board for the success of the conference, which has given rise to this present volume of selected papers. We would also like to thank Mr Wang Yanbo, Miss Wan Xiqiong and the graduate students at Fudan University for their effective work to make this conference a success.

The conference was financially supported by the NFS of China, the Mathematical Center of Ministry of Education of China, E-Institutes of Shanghai Municipal Education Commission (No E03004) and Fudan University, Grant 15340027 from the Japan Society for the Promotion of Science, and Grant 15654015 from the Ministry of Education, Cultures, Sports and Technology.

We consider a hyperbolic equation p(x, t) ∂_t²u(x, t) = Δu(x, t) + ∑_k=1ⁿq_k(x, t) ∂_ku + q_{n + 1}(x, t) ∂_tu + r(x, t)u in ⁿ × with p ∊ C¹ and q₁, ... , q_n+1, r ∊ L^∞. Let Γ be a part of the boundary of a domain and let ν(x) be the inward unit normal vector to Γ at x. Then we prove the conditional stability in the unique continuation near a point x₀ across Γ if ∇p(x₀, t) · ν(x₀) < 0 and the radius of the osculating ball at x₀ is large for −∇p(x₀, t) · ν(x₀). Next we prove the conditional stability in the inverse problem of determining a coefficient r(x) from Cauchy data on Γ over a time interval. The key is a Carleman estimate in level sets of paraboloid shapes.

We provide a survey of important properties of the generalized polarization tensors which are the basic building blocks for the full asymptotic expansions of the boundary voltage perturbations due to the presence of a small conductivity inclusion inside a conductor. These properties may be used to design efficient algorithms for reconstructing conductivity inclusions of small volume.

We study the inverse boundary value problems for the Schrödinger equations with Yang-Mills potentials in a bounded domain Ω₀ ⊂ Rⁿ containing finite number of smooth obstacles Ω_j , 1 ⩽ j ⩽ r. We prove that the Dirichlet-to-Neumann operator on ∂Ω₀ determines the gauge equivalence class of the Yang-Mills potentials. We also prove that the metric tensor can be recovered up to a diffeomorphism that is identity on ∂Ω₀.

We propose an iterative regularization method for the solution of inverse medium scattering problems which takes advantage of factorizations of the far field operator. Numerical experiments with two and three dimensional acoustic and electromagnetic medium scattering problems show that our method is very competitve compared to other iterative regularization methods.

This paper reviews various kinds of inverse problems in atmospheric science and oceanography, and introduces powerful methods to treat these problems–variational data assimilation (VAR) and improved discrepancy principle, and discusses some essential difficulties in VAR. Due to the ill–posedness of these problems, the regularization method is also applied, i.e., additional terms are added to the cost functional as a stabilized functional with physical meaning. Inversions of four specific problems, such as the inversion of one-dimensional sea temperature model, the inversion of parameters in an ENSO cycle model, the inversion of wind field with single-Doppler data, and the inversion of satellite remote sensing, indicate that, adoption of the regularization method in VAR will overcome the ill-posedness, constrain calculational oscillations in iteration, and speed up convergence of solutions.

We deal with an inverse problem of determining a coefficient a(x, t) of principal part for second order parabolic equations with non-divergent form when the solution is known. Such a problem has important applications in a large fields of applied science. We propose a well-posed approximate algorithm to identify the coefficient. The existence, uniqueness and stability of such solutions a(x, t) are proved. A necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced. Our numerical simulations show that the coefficient is recovered very well.

In this paper, we discuss the inverse problem of determining the piecewise continuous coefficients in a wave equation from the interior measurements. It is proved that the coefficients can be uniquely determined in the domain where the shear wave reaches.

Reconstruction procedure for identifying an unknown polygonal cavity inside a homogeneous, isotropic conductive medium from two boundary measurements is given. The assumptions of the cavity is that there are at most one concave vertex between two convex vertices and the vertices are well separated. The Neumann data of the boundary measurement can change sign at most once and the sets on which one of them become positive and negative must be disjoint.

We consider the identification of a difiusion parameter in a second order elliptic equation in two dimensions by interior measurements. The difiusion parameter is assumed to have discontinuities. For its reconstruction we propose two regularization algorithms–one is based on a moving grid approach and the other on a Hamilton-Jacobi type equation. The numerical realization via an inexact iteratively regularized Gauss-Newton method shows that these algorithms can effectively identify the discontinuities.

In this paper, we will review some recent theoretical and algorithmic developments in parameter identifications in partial differential equations by our research group, focusing on such aspects as variational formulations, convergence analysis, choice strategies of regularization parameters and algorithmic implementations.

We establish a Carleman estimate for two-dimensional Maxwell's equations in isomagnetic anisotropic media and prove an energy estimate by boundary data.

We review three reconstruction methods for inverse obstacle scattering problems. We will analyse the relation between the Kirsch-Kress potential method 1986, the range test of Kusiak, Potthast and Sylvester (2003) and the singular sources method of Potthast (2000). In particular, we show that the range test is a logical extension of the Kirsch-Kress method into the category of sampling methods employing the tool of domain sampling. Then we will show how a multi-wave version of the range test can be set up and we will work out its relation to the singular sources method. Numerical examples and demonstrations will be provided.

Real numbers are usually represented in the computer as a finite number of digits hexa-decimal floating point numbers. Accordingly the numerical analysis is often suffered from rounding errors. The rounding errors particularly deteriorate the precision of numerical solution in inverse and ill-posed problems. We attempt to use a multi-precision arithmetic for reducing the rounding error evil. The use of the multi-precision arithmetic system is by the courtesy of Dr Fujiwara of Kyoto University. In this paper we try to show effectiveness of the multi-precision arithmetic by taking two typical examples; the Cauchy problem of the Laplace equation in two dimensions and the shape identification problem by inverse scattering in three dimensions. It is concluded from a few numerical examples that the multi-precision arithmetic works well on the resolution of those numerical solutions, as it is combined with the high order finite difference method for the Cauchy problem and with the eigenfunction expansion method for the inverse scattering problem.

This paper reviews the latest impedance imaging technique called Magnetic Resonance Electrical Impedance Tomography (MREIT) providing information on electrical conductivity and current density distributions inside an electrically conducting domain such as the human body. The motivation for this research is explained by discussing conductivity changes related with physiological and pathological events, electromagnetic source imaging and electromagnetic stimulations. We briefly summarize the related technique of Electrical Impedance Tomography (EIT) that deals with cross-sectional image reconstructions of conductivity distributions from boundary measurements of current-voltage data. Noting that EIT suffers from the ill-posed nature of the corresponding inverse problem, we introduce MREIT as a new conductivity imaging modality providing images with better spatial resolution and accuracy. MREIT utilizes internal information on the induced magnetic field in addition to the boundary current-voltage measurements to produce three-dimensional images of conductivity and current density distributions. Mathematical theory, algorithms, and experimental methods of current MREIT research are described. With numerous potential applications in mind, future research directions in MREIT are proposed.

In this paper we discuss inverse boundary value problems for quasilinear elliptic equations in connection with the nonlinear conductivity problem and inverse problems in nonlinear material sciences.

In this paper the inverse problem of magnetotelluric (MT) for 2-D are solved by using genetic algorithm (GA). Some quick and effective algorithm for solving forward problem is designed to reduce the computation time of GA. It is shown that as long as the genetic operators and some control parameters are chosen correctly, with some auxiliary techniques very good results can be obtained. Some difficulties encountered in other optimization methods can be overcome.

Based on the cubic spline theory, we propose in this paper a regularization method for reconstructing numerical differentiation from one-dimensional scattered noisy data. Under two different choice rules for a suitable regularization parameter, the regularized solutions can be derived. Convergence results for the approximate derivative are given. Numerical experiments verify that the proposed strategy with the Morozov's discrepancy principle for the choice of regularization parameter is effective and stable.

In this paper, we study the reconstruction of the chirality admittance of periodic chiral structures. The media is homogeneous and the structure is periodic in one direction and invariant in another direction. The electromagnetic fields inside the chiral medium are governed by Maxwell equations together with the Drude-Born-Fedorov equations. We simplify the problem to a two-dimensional inverse scattering problem and simply present the general recursive linearization procedure(continuous homotopy method) for solving the problem.