Table of contents

We consider an electrical network in which the series of resistor, inductor and capacitor are connected in parallel. When the finite measurements of the current are given, we present a method to construct an RLC circuit which generates the current fitting the measurements. This method will be useful to synthesize the voltage (or current) source of an arbitrary waveform which is necessary in the measurement of damage of a semiconductor device caused by an electrostatic discharge.

For iterative methods for well-posed problems, invariance properties have been used to provide a unified framework for convergence analysis. We carry over this approach to iterative methods for nonlinear ill-posed problems and prove convergence with rates for the Landweber and the iteratively regularized Gauss-Newton methods. The conditions needed are weaker as far as the nonlinearity is concerned than those needed in earlier papers and apply also to severely ill-posed problems. With no additional effort, we can also treat multilevel versions of our methods.

Optical tomography is modelled by the inverse problem of the time-dependent linear transport equation in n spatial dimensions (n = 2,3). Based on the measurements which consist of some functionals of the outgoing density at the boundary for different sources , , two coefficients of the equation, the absorption coefficient and the scattering coefficient b(x), are reconstructed simultaneously inside . Starting out from some initial guess for these coefficients, the transport-backtransport (TBT) algorithm calculates the difference between the computed and the physically given measurements for a fixed source by solving a `direct' transport problem, and then transports these residuals back into the medium by solving a corresponding adjoint transport problem. The correction to the guess is calculated from the densities of the direct and the adjoint problem inside the medium. Doing this for all source positions , , one after the other yields one sweep of the algorithm. Numerical experiments are presented for the case when n = 2. They show that the TBT-method is able to reconstruct and to distinguish between scattering and absorbing objects in the case of large mean free path (which corresponds to x-ray tomography with scattering). In the case of very small mean free path (which corresponds to optical tomography), scattering and absorbing objects are located during the early sweeps, but phantoms are built up in the reconstructed scattering coefficient at positions where an absorber is situated and vice versa.

We study the inverse problem of synthesizing parameters of differential equations with singularities from incomplete spectral information. We establish properties of the spectral characteristics, obtain conditions for the solvability of such classes of inverse problems and provide algorithms for constructing the solution.

The class of inverse problems for a nonlinear elliptic variational inequality is considered. The nonlinear elliptic operator is assumed to be a monotone potential. The unknown coefficient of the operator depends on the gradient of the solution and belongs to a set of admissible coefficients which is compact in . It is shown that the nonlinear operator is pseudomonotone for the given class of coefficients. For the corresponding direct problem - coefficient convergence is proved. Based on this result the existence of a quasisolution of the inverse problem is obtained. As an important application an inverse diagnostic problem for an axially symmetric elasto-plastic body is considered. For this problem the numerical method and computational results are also presented.

The inverse problem in electrical impedance tomography (EIT) is severely ill-posed. Therefore, it is desirable to include any available a priori information in a numerical algorithm for its solution. If the conductivity inside the object is known to be piecewise constant this information can be directly incorporated in the algorithm by using boundary integral methods for the computation of the forward map. In this paper we will describe an implementation of this idea and compare the results with standard methods using both synthetic and measured data from the clinical applications of EIT.

The characterization of the local ill-posedness and the local degree of nonlinearity are of particular importance for the stable solution of nonlinear ill-posed problems. We present assertions concerning the interpendence between the ill-posedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity combined with source conditions can be used to characterize the local ill-posedness and to derive a posteriori estimates for nonlinear ill-posed problems. A posteriori estimates are widely used in finite element and multigrid methods for the solution of nonlinear partial differential equations, but these techniques are in general not applicable to inverse and ill-posed problems. Additionally we show for the well known Landweber method and the iteratively regularized Gauss-Newton method that they satisfy a posteriori estimates under source conditions; this can be used to prove convergence rate results. A numerical test is presented that confirms the theoretical assertions.

The iteratively regularized Gauss-Newton method is applied to solve an inverse transmission problem for the Helmholtz equation, which is known to be nonlinear and severely ill-posed. We first present a simplified proof of the characterization of the Fréchet derivative. Based on this result, we describe an efficient numerical implementation including an error analysis. Finally, we investigate the speed of convergence of the Newton iteration both for exact and for noisy data.

We consider a system

with a suitable boundary condition, where is a bounded domain, is a uniformly elliptic operator of the second order whose coefficients are suitably regular for , is fixed, and a function satisfies

Our inverse problems are determinations of g using overdetermining data or , where and . Our main result is the Lipschitz stability in these inverse problems. We also consider the determination of , in the case of with given R satisfying on . Finally, we discuss an upper estimation of our overdetermining data by means of f.

We investigate possibilities of choosing reasonable regularization parameters for the output least squares formulation of linear inverse problems. Based on the Morozov and damped Morozov discrepancy principles, we propose two iterative methods, a quasi-Newton method and a two-parameter model function method, for finding some reasonable regularization parameters in an efficient manner. These discrepancy principles require knowledge of the error level in the data of the considered inverse problems, which is often inaccessible or very expensive to achieve in real applications. We therefore propose an iterative algorithm to estimate the observation errors for linear inverse problems. Numerical experiments for one- and two-dimensional elliptic boundary value problems and an integral equation are presented to illustrate the efficiency of the proposed algorithms.

The retrieval of an unknown, possibly inhomogeneous, penetrable cylindrical obstacle buried entirely in a known homogeneous half-space - the constitutive material parameters of the obstacle and of its embedding obey a Maxwell model - is considered from single- or multiple-frequency aspect-limited data collected by ideal sensors located in air above the embedding half-space, when a small number of time-harmonic transverse electric (TE)-polarized line sources - the magnetic field H is directed along the axis of the cylinder - is also placed in air. The wavefield is modelled from a rigorous H-field domain integral-differential formulation which involves the dot product of the gradients of the single component of H and of the Green function of the stratified environment times a scalar-valued contrast function which contains the obstacle parameters (the frequency-independent, position-dependent relative permittivity and conductivity). A modified gradient method is developed in order to reconstruct the maps of such parameters within a prescribed search domain from the iterative minimization of a cost functional which incorporates both the error in reproducing the data and the error on the field built inside this domain. Non-physical values are excluded and convergence reached by incorporating in the solution algorithm, from a proper choice of unknowns, the condition that the relative permittivity be larger than or equal to 1, and the conductivity be non-negative. The efficiency of the constrained method is illustrated from noiseless and noisy synthetic data acquired independently. The importance of the choice of the initial values of the sought quantities, the need for a periodic refreshment of the constitutive parameters to avoid the algorithm providing inconsistent results, and the interest of a frequency-hopping strategy to obtain finer and finer features of the obstacle when the frequency is raised, are underlined. It is also shown that though either the permittivity map or the conductivity map can be obtained for a fair variety of cases, retrieving both of them may be difficult unless further information is made available.

We investigate the use of a functional analytical version of the Backus-Gilbert method as a reconstruction strategy to obtain specific information on the solution of linear and slightly nonlinear systems with Frechét derivable operators. Some a priori error estimates are shown and tested for two classes of problems: a nonlinear moment problem and a linear elliptic Cauchy problem. For this second class of problems a special version of the Green formula is developed in order to analyse the involved adjoint equations.

The sudden approximation is applied to invert structural data on randomly corrugated surfaces from inert atom scattering intensities. Several expressions relating experimental observables to surface statistical features are derived. The results suggest that atom (and in particular He) scattering can be used profitably to study hitherto unexplored forms of complex surface disorder.

Let us consider the Dirichlet problem for a semilinear elliptic equation . Sufficient regularity for enables us to derive the uniqueness for a from the Dirichlet to Neumann map under the condition .

A method of using a magnetopolariscope for reconstructing stress fields by means of the tomography technique is proposed. Stresses are determined within the framework of the Maxwell piezo-optic law (linear dependence of the permittivity tensor on stresses) and weak optical anisotropy. The path difference and isocline parameter values measured by the tomography technique in the presence and absence of a magnetic field are used as initial information. Up to now, the magnetopolariscope has been employed only to determine bending and membrane stresses in plane models. Their novel application allows us to determine the residual stress completely.

Let Q(x) be a continuous symmetric Jacobi matrix-valued even function on . It is shown that if each element in the Dirichlet spectrum of has multiplicity n, then there exists a scalar-valued function p(x) such that . This result is used to investigate vectorial Hill's operators with symmetric Jacobi matrix-valued potential functions, a theorem similar to the Borg theorem for scalar Hill's operators is proved.

We have devised an inverse algorithm for the reconstruction of underground velocity from seismic waveforms. The algorithm allows for the use of multiple transmitter-receiver pairs and a wide band of frequencies. The background medium velocity can be homogeneous or heterogeneous. The algorithm is an efficient one-step imaging process and requires no matrix inversion. It may provide a good estimate of the undergound velocity as long as the scattering is weak and the scatterer is small relative to the wavelength of the excitation field.

The analytical verification of the reconstruction algorithm makes use of a homogeneous Green function which is defined to be the difference between the causal Green function and its acausal counterpart. The homogeneous Green function stands for the backpropagation of the scattered field by a physical source. The inverse algorithm comprises four processing steps: data backpropagation, phase correction, low-pass filtering and imaging. The algorithm is derived for acoustic waves but has the potential to be extended to elastic waves.

For the n-dimensional integrable system with a twisted reduction, Darboux transformations given by Darboux matrices of degree two are constructed explicitly. These Darboux transformations are applied to the local isometric immersion of space forms with flat normal bundle and linearly-independent curvature normals to give the explicit expression of the position vector. Some examples are given from the trivial solutions and standard imbedding .

Using the nonlinear constraint and Darboux transformation methods, the localized solitons of the hyperbolic su(N) AKNS system are constructed. Here `hyperbolic su(N)' means that the first part of the Lax pair is where J is constant real diagonal and . When different solitons move in different velocities, each component of the solution U has at most peaks as . This corresponds to the solitons for the DSI equation. When all the solitons move in the same velocity, still has at most peaks if the phase differences are large enough.

The Newsletter is a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting forum by sending to our Bristol office material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome.

Book review

Rank-Deficient and Discrete Ill-Posed Problems, Numerical Aspects of Linear Inversion P C Hansen 1998 Philadelphia: SIAM 247 pp ISBN 0-89871-403-6 $44.00

For more than ten years, Per Christian Hansen has been one of the leading experts in the field of numerical linear algebra for discrete ill-posed problems. In this monograph he presents a detailed and comprehensive description of the state-of-the-art techniques for solving linear equations for this class of ill-conditioned problems.

In the first chapter, the author carefully distinguishes between rank-deficient and (discrete) ill-posed problems Ax=b in terms of the singular values of the matrix A. Rank-deficient problems are characterized by those matrices having a cluster of small singular values which are well separated from the other singular values. Ill-posed problems do not have this property, i.e. the singular values decay gradually to zero with no particular gap between them. They arise naturally from the discretization of integral equations of the first kind while rank-deficient matrices can be considered as perturbations of singular but well-conditioned matrices B in the sense that is not too large. (Here denotes the Moore - Penrose inverse of B.) In section 1.2 he summarizes basic facts on ill-posed problems described by integral equations of the first kind. Emphasis is placed on singular value decomposition as the basic tool for its analysis. The principle of regularization is introduced for both continuous and discrete problems. At the end of this chapter the author formulates four test examples: the first, from signal processing, leads to a rank-deficient problem, the second is the computation of the second derivative and is the discretization of a modestly ill-posed problem. The last two examples are discretizations of integral equations with analytic kernels and lead to highly ill-posed problems.

Chapter 2 presents the essential tools for the analysis of discrete ill-posed problems which are fundamental throughout the book. A central role is played, as indicated above, by singular value decomposition and its various generalizations including a brief survey on their numerical computations.

In chapter 3, rank-deficient problems are treated. For this class of problems the notion of the numerical -rank of a matrix makes sense and leads to a natural regularization concept. Perturbation bounds and the efficient computation by truncated SVD or QR decompositions are explained and illustrated by a number of numerical examples.

In contrast, the regularization of ill-posed problems derived from the discretization of integral equations of the first kind is more difficult. In chapter 4, the author introduces filter factors, the resolution matrix, and the L-curve approach as useful regularization tools.

The author distinguishes between direct and iterative regularization methods. In chapter 5 he studies the first class of methods among which the Tikhonov regularization is the most popular. For each of these methods he formulates perturbation bounds and discusses the numerical aspects. In the reviewer's opinion, a highlight of this chapter is the illustrative examples at the end where the different regularization schemes are compared with each other.

The iterative regularization methods are the subject of chapter 6. Emphasis is placed on conjugate gradient methods. The implementation of the standard CG algorithm by means of the Lanczos bidiagonalization algorithm is explained and various modifications are discussed. The author reviews results on regularization and convergence properties and devotes one section to the Lanczos bidiagonalization algorithm in finite precision.

Finally, chapter 7 surveys methods for choosing the regularization parameter, a problem of obvious importance. Classical discrepancy principles are discussed as well as more recent modifications, methods based on error estimation, generalized cross-validation, and the L-curve criterion.

An impressive bibliography of 378 references completes this monograph. As the author points out in the preface, this book is not intended to be an introduction into either the field of inverse problems or numerical linear algebra. It is not a textbook. The monograph contains almost no proofs but always references to the literature. It is the purpose of the book to give a survey of state-of-the-art numerical methods for solving rank-deficient or discrete ill-posed problems.

In the referee's opinion there is no other book around which serves this goal even nearly as well as this one. This work truly fills a gap in the literature on inverse problems and ill-posed problems and is strongly recommended for every applied mathematician or engineer who has to solve rank-deficient or ill-posed problems numerically.

A Kirsch University of Karlsruhe