Table of contents

We show that the Newton-Kantorovich and distorted Born methods for the computational solution of the nonlinear inverse scattering problem are equivalent. This was already shown for the discrete matrix case. Here we present an analysis based on the analytic representations of the integral operators. We first briefly review both methods and then show that they are equivalent.

The problem of velocity reconstruction in conducting fluids from measurements of induced magnetic fields and electric potentials is discussed in spherical geometry. Under the special case that the externally applied magnetic field is uniform and homogeneous throughout the fluid, the non-uniqueness problem is treated in detail. Assuming kinetic energy minimization for the moving fluid, it is shown that the velocity field can be reconstructed completely.

The reduction technique is used in conjunction with SL (3) algebra to generate a new nonlinear Klein-Gordon-type integrable system. The corresponding inverse problem is solved with the help of the dressing operator approach. The same method is also suitable to discuss the hidden symmetries of the system.

Taking as a guide the case of the set of monoparametric families y= h (x )+c , for which Szebehely's equation can be solved by quadratures for the potential V (x ,y ) generating the given set of orbits, we propose the following programmed motion problem : can we manage so as to have members of the given set inside a preassigned domain T²of the xyplane?

We come to understand that, among the various inequalities by means of which Tcan be ascribed, the simplest is b (x ,y ) 0 where, for each h (x ), the function b (x ,y ) is related to the kinetic energy of the moving point (equations (19)-(21)). We then proceed to show that, in general, if b (x ,y ) satisfies two conditions (equations (39) and (40)), the answer to our question is affirmative: on the grounds of the given appropriate b (x ,y ), a function h (x ) is found, associated with a certain potential V (x ,y ) creating members of the family y= h (x )+cinside the region b (x ,y ) 0.

Some special cases which stem from the method are studied separately. The limitations and also the promising features of the method developed to face the above inverse problem are discussed.

An inverse problem is considered to identify the geometry of discontinuities in a conductive material ²with conductivity 1+(k -1)_Dfrom the measurements taken on the boundary , where D and kis a known positive constant. We propose an efficient numerical algorithm for the problem to identify Din a certain class by a single measurement. Several numerical results are discussed.

In this paper a new procedure is established to obtain the element values of capacitors and resistances of a first Cauer network directly from the poles and zeros of the impedance function using the technique for constructing tridiagonal symmetric matrices from spectral data. This method is illustrated numerically.

A new (scalar) spectral decomposition is found for the Dirac system in two dimensions associated to the focusing Davey-Stewartson II (DSII) equation. The discrete spectrum in the spectral problem corresponds to eigenvalues embedded into a two-dimensional essential spectrum. We show that these embedded eigenvalues are structurally unstable under small variations of the initial data. This instability leads to the decay of localized initial data into continuous wavepackets prescribed by the nonlinear dynamics of the DSII equation.

An algorithm for reconstructing analytic functions from exponentially spaced samples is considered. Approximation errors and stability estimates are obtained.

In this paper, we develop a linear sampling method for the inverse scattering of time-harmonic plane waves by open arcs. We derive a characterization of the scatterer in terms of the spectral data of the scattering matrix analogously to the case of the scattering by bounded open domains. Numerical examples show that this theoretical result also leads to a very fast visualization technique for the unknown arc.

We consider the conductivity problem in two dimensions. We show that a complex-valued coefficient , whose imaginary part is small, can be recovered from the knowledge of the Dirichlet-to-Neumann map.

This paper describes a nonlinear inverse method which allows the determination of the second- and third-order elastic constants for a caesium dihydrogen phosphate lattice via ultrasonic velocity measurements. This analysis is based on a genetic algorithm. The efficiency and accuracy of the method and the influence of measurement errors are discussed.

Consider the identification of a planar crack located deep inside a heterogeneous conducting body. Low-frequency electromagnetic waves are used to penetrate the crack. An equivalent current distribution model is adopted for the crack identification by using the boundary measurements of the fields. In this paper, a new computational method is introduced for solving the inverse problem. Our method is based on a low-frequency asymptotic analysis of Maxwell's equations. A crucial ingredient of our approach is the use of the quasi-static limit for the construction of special test functions rather than for modelling the direct problem. Uniqueness and stability results for the inverse problem are also established. Our analysis indicates that the constructive method has good convergence properties and is computationally attractive.

The connection between the structure of the second-order system describing the Euclidean sigma-model equations associated with the generalized Weierstrass system and the possibility of the construction of some classes of solutions is discussed. It is shown using the conditional symmetry method that the auto-Bäcklund transformation (auto-BT) for the generalized Weierstrass system can be constructed. The permutability theorem for this auto-BT is formulated. New classes of nonsplitting multisoliton solutions are obtained through the use of the theorem of permutability.

We examine the scattering of time-harmonic electromagnetic waves in an inhomogeneous medium. Given two smooth refractive indices whose difference is small with respect to a C² -norm, we show that the maximum norm of the difference of the refractive indices can be estimated by the difference of certain boundary operators associated with the refractive indices. Moreover, using a very strong norm on the far-field patterns, we reconstruct the boundary operators from the far-field patterns and derive continuous dependence of the refractive index on the far-field pattern.

In this paper we apply the approximate inverse to a one-dimensional inverse heat conduction problem. We give results about the regularization effect of the approximate inverse and also some error estimate if the solution fulfils some smoothness condition. Then we transfer our theory to an algorithm in pseudo-code which is tested in a numerical example.

In this paper we consider some Newton-type methods in Hilbert scales to solve nonlinear inverse problems. Under certain conditions we obtain the error estimates when the iteration is terminated in an a posteriorimanner. Finally we present the numerical examples to verify the theoretical results.

The elliptic systems method (ESM) developed by the authors is applied to the numerical solution of an inverse problem for the parabolic equation with incomplete data collection. Several cases of data collection are considered, with focus on the scenario of back-reflected data. The principal idea consists of the introduction of a weight function into the elliptic differential operator resulting from the ESM. This function reduces the influence of parts of the boundary where data are not available. A theoretical estimate of the difference between solution with the incomplete data alone and one with `almost complete' data collection is obtained using Carleman estimates. Applications to laser imaging of small abnormalities in human organs, the atmosphere and the ocean are discussed.

For decades, the Radon transform has been used as an approximate model for two-dimensional (2D) positron emission tomography (PET). Since this model assumes that detector tubes are represented by lines (hence have no area), PET reconstruction algorithms need to be modified to account for the nonzero width of detectors. To date, these modifications have been obtained by computational methods, so fail to exhibit any inherent mathematical structure of the PET transform which takes emission intensity to detector tube means. This paper contains a precise mathematical representation of this PET transform and exploits this representation to propose a new method for reconstructing PET images. This representation is achieved by expressing the probability that an emission at a point is detected in a detector tube, in terms of the Green function and Poisson kernel for Laplace's equation on the unit disc. This new PET transform involves four weighted line integrals of the emission intensity function, instead of the single unweighted line integral defining the 2D Radon transform. Despite the complexity of this model, a reconstruction method is obtained by using classical orthogonal series representations of the emission intensity and detection means in terms of circular harmonics, Bessel functions and Chebyshev polynomials.

We consider the inverse problems for the regular Dirac operator in its classical formulations: from a given spectrum and norming constants, from a given two spectra and from one spectrum. In contrast to the known Gel'fand-Levitan approach we obtain explicit formulae for the solutions of the inverse problems corresponding to the variation of a finite number of the given spectral data.

In this paper we describe a new method for constructing integrable systems of differential equations. We are looking for systems in two variables in such forms that the reduction v= uleads us to a single equation in u . We give a complete classification of such systems that reduce to Korteweg-de Vries-type equations. Furthermore, we present an extensive (and complete for the systems of the Sawada-Kotera and Kaup-Kupershmidt types) classification of fifth-order equations in the same weighting. We show that the scalar integrable equations give rise to large classes of integrable systems. Moreover, we present a previously unknown example of a system that can be written in biHamiltonian form in infinitely many different ways, thereby solving the problem of the number of biHamiltonian forms that can have a differential equation. Finally, we present examples of nondegenerate systems possessing degenerate symmetries, which is impossible in the scalar case.