Table of contents

The problem of determining the neural current inside the brain from measurements of the magnetic field outside the head is discussed. In particular the non-uniqueness question is completely resolved. Furthermore, it is shown that with the assumption of energy minimization, the current can be reconstructed uniquely.

We study an inverse parabolic problem with final overdetermination. First, we show that the solution to the direct problem depends analytically on the diffusion parameter. Then, using this fact and appropriate uniform bounds, we are able to prove generic local well-posedness of the inverse problem.

We consider the problem of determination of the regular Sturm - Liouville operator from two spectra in its classical statement according to Borg. In contrast to the known Gel'fand - Levitan approach we obtain explicit formulae for the solution of the inverse problem corresponding to the variation of a finite number of eigenvalues of the two spectra given.

In this paper, we analyse the discretization of the generalized radon transform/amplitude versus scattering angles (GRT/AVA) migration - inversion formula by means of quasi-Monte Carlo methods. These methods are efficient, in the sense that they require sparsely sampled measurements only, and accurate, which we have shown by theory and examples. Another feature of Monte Carlo methods is their ability to suppress effectively coherent noise associated with undesired wave phenomena in the inversion procedure. As examples, we carried out the associated integrations over and , and consistently found that quasi-random sequences achieve a prescribed accuracy with significantly fewer nodes.

We recall Krichever's construction of additional flows to Benney's hierarchy, attached to poles at finite distance of the Lax operator. Then we construct a `dispersionful' analogue of this hierarchy, in which the role of poles at finite distance is played by Miura fields. We connect this hierarchy with N-wave systems, and prove several facts about the latter (Lax representation, Chern - Simons-type Lagrangian, connection with Liouville equation, -functions).

This paper considers an inverse potential problem which seeks to recover the shape of an obstacle separating two different densities by measurements of the potential. A representation for the domain derivative of the corresponding operator is established and this allows the investigation of several iterative methods for the solution of this ill-posed problem.

We consider the inverse conductivity problem for the equation determining the unknown object D contained in a domain with one measurement on . The method in this paper is the layer potential technique. We find a representation formula for the solution to the equation using single layer potentials on D and . Using this representation formula, we prove that the location and size of a disk D contained in a simply connected bounded Lipschitz domain can be determined with one measurement corresponding to arbitrary non-zero Neumann data on . (Previously, it was known that a disk can be determined with one measurement if is assumed to be the half space.) We also prove a weaker version of the uniqueness for balls in with one measurement corresponding to a certain Neumann data.

The problem of quantitative non-destructive evaluation of corrosion in plates is considered. The inspection method uses boundary measurements of currents and voltages to determine the material loss caused by corrosion. The development of the method is based on linearization and the assumption that the plate is thin. The behaviour of the method is examined in numerical simulations.

The inverse scattering problem for the Schrödinger operator on the half-axis is studied. It is shown that this problem can be solved for the scattering matrices with arbitrary finite phase shift on the real axis if one admits potentials with long-range oscillating tails at infinity. The solution of the problem is constructed with the help of the Gelfand - Levitan - Marchenko procedure. The inverse problem has no unique solution for the standard set of scattering data which includes the scattering matrix, energies of the bound states and corresponding normalizing constants. This fact is related to zeros of the spectral density on the real axis. It is proven that the inverse problem has a unique solution in the defined class of potentials if the zeros of the spectral density are added to the set of scattering data.

We present a Newton method for solving the ill-posed and nonlinear inverse scattering problem. For its foundation we establish the Fréchet differentiability of the far-field operator with respect to the boundary and show that the Fréchet derivative can be obtained as the solution of a hypersingular integral equation. In order to derive the right-hand sides of this integral equation we use a quadrature method. We describe the numerical solution method and give some numerical results.

The inverse problem involving determination of the forcing function, depending not only on the unknown variable but also the space variable, is considered. However, the form of the function is assumed to be of the form f(u)+xg(u) assuming only an affine dependence on the space variable. The problem considered is defined on a finite space interval and non-negative time axis. The auxiliary conditions are prescribed at the boundaries of the space interval.

Given the data , i = 1,...,m, we consider the existence problem for the optimal parameters for the exponential function approximating these data in the sense of total least squares. We give sufficient conditions which guarantee the existence of such optimal parameters.

We consider the problem of recovering a reflecting surface such that for a given point source of light the directions of the reflected rays cover a prescribed region of a far field sphere and the density of the distribution of the reflected rays is a function prescribed in advance, where the aperture of the incident ray cone is also prescribed in advance. Mathematically this problem requires one to solve a nonlinear partial differential equation of Monge - Ampère type subject to a nonlinear boundary condition. Numerical computations for the problem have been carried out by several authors. In this paper we study the existence, uniqueness, and smoothness of the reflecting surfaces for the above problem.